CINXE.COM

<!DOCTYPE html><html lang="en-US" class="html_boxed responsive av-preloader-disabled html_header_top html_logo_left html_main_nav_header html_menu_right html_custom html_header_sticky html_header_shrinking_disabled html_mobile_menu_phone html_header_searchicon html_content_align_center html_header_unstick_top_disabled html_header_stretch_disabled html_av-submenu-hidden html_av-submenu-display-click html_av-overlay-side html_av-overlay-side-classic html_av-submenu-noclone html_entry_id_25754 av-cookies-no-cookie-consent av-no-preview av-default-lightbox html_text_menu_active av-mobile-menu-switch-default"><head><meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"><meta charset="UTF-8" /> <script async src="https://www.googletagmanager.com/gtag/js?id=UA-2209856-4"></script> <script>window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'UA-2209856-4');</script> <script data-optimized="1" src="https://www.physicsforums.com/insights/wp-content/litespeed/js/2592eef92248026c6738059400d0d827.js?ver=0d827" defer type="text/javascript"></script> <meta name="viewport" content="width=device-width, initial-scale=1"><meta name='robots' content='index, follow, max-image-preview:large, max-snippet:-1, max-video-preview:-1' /><style>img:is([sizes="auto" i], [sizes^="auto," i]) { contain-intrinsic-size: 3000px 1500px }</style> <script type="text/javascript" id="wpp-js" src="https://www.physicsforums.com/insights/wp-content/plugins/wordpress-popular-posts/assets/js/wpp.js?ver=7.1.0" data-sampling="0" data-sampling-rate="100" data-api-url="https://www.physicsforums.com/insights/wp-json/wordpress-popular-posts" data-post-id="0" data-token="ac9d65413d" data-lang="0" data-debug="1"></script> <title>Physics Forums Insights: Expert Math and Physics Blog</title><meta name="description" content="Physics Forums expert math and physics blog. Read and learn from expert math and science articles, tutorials, and guides. Test your science knowledge with our quiz." /><link rel="canonical" href="https://www.physicsforums.com/insights/" /><meta property="og:locale" content="en_US" /><meta property="og:type" content="website" /><meta property="og:title" content="Physics Forums Insights: Expert Math and Physics Blog" /><meta property="og:description" content="Physics Forums expert math and physics blog. Read and learn from expert math and science articles, tutorials, and guides. Test your science knowledge with our quiz." /><meta property="og:url" content="https://www.physicsforums.com/insights/" /><meta property="og:site_name" content="Physics Forums Insights" /><meta property="article:publisher" content="https://www.facebook.com/physicsforums" /><meta property="article:modified_time" content="2023-05-26T11:45:28+00:00" /><meta property="og:image" content="https://www.physicsforums.com/insights/wp-content/uploads/2023/11/physics-forums-logo-og.webp" /><meta property="og:image:width" content="1200" /><meta property="og:image:height" content="630" /><meta property="og:image:type" content="image/webp" /><meta name="twitter:card" content="summary_large_image" /><meta name="twitter:site" content="@physicsforums" /> <script type="application/ld+json" class="yoast-schema-graph">{"@context":"https://schema.org","@graph":[{"@type":"WebPage","@id":"https://www.physicsforums.com/insights/","url":"https://www.physicsforums.com/insights/","name":"Physics Forums Insights: Expert Math and Physics Blog","isPartOf":{"@id":"https://www.physicsforums.com/insights/#website"},"about":{"@id":"https://www.physicsforums.com/insights/#organization"},"datePublished":"2019-09-27T16:29:49+00:00","dateModified":"2023-05-26T11:45:28+00:00","description":"Physics Forums expert math and physics blog. Read and learn from expert math and science articles, tutorials, and guides. Test your science knowledge with our quiz.","breadcrumb":{"@id":"https://www.physicsforums.com/insights/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https://www.physicsforums.com/insights/"]}]},{"@type":"BreadcrumbList","@id":"https://www.physicsforums.com/insights/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home"}]},{"@type":"WebSite","@id":"https://www.physicsforums.com/insights/#website","url":"https://www.physicsforums.com/insights/","name":"Physics Forums Insights","description":"Science and Math Articles, Tutorials and Guides","publisher":{"@id":"https://www.physicsforums.com/insights/#organization"},"alternateName":"PF Insights","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://www.physicsforums.com/insights/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https://www.physicsforums.com/insights/#organization","name":"Physics Forums","url":"https://www.physicsforums.com/insights/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https://www.physicsforums.com/insights/#/schema/logo/image/","url":"https://www.physicsforums.com/insights/wp-content/uploads/2018/05/facebook_image-1.png","contentUrl":"https://www.physicsforums.com/insights/wp-content/uploads/2018/05/facebook_image-1.png","width":205,"height":200,"caption":"Physics Forums"},"image":{"@id":"https://www.physicsforums.com/insights/#/schema/logo/image/"},"sameAs":["https://www.facebook.com/physicsforums","https://x.com/physicsforums","https://www.linkedin.com/company/physics-forums","https://www.pinterest.com/physicsforums"]}]}</script> <link rel="alternate" type="application/rss+xml" title="Physics Forums Insights » Feed" href="https://www.physicsforums.com/insights/feed/" /><link data-optimized="1" rel='stylesheet' id='xpress_wp_admin_style-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/9294efc37b077c18f8cbd596b1da1840.css?ver=a1840' type='text/css' media='all' /><link data-optimized="1" rel='stylesheet' id='wp-block-library-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/ad10b27e06583ae315af8368433fa1f7.css?ver=6d365' type='text/css' media='all' /><style id='global-styles-inline-css' type='text/css'>:root{--wp--preset--aspect-ratio--square: 1;--wp--preset--aspect-ratio--4-3: 4/3;--wp--preset--aspect-ratio--3-4: 3/4;--wp--preset--aspect-ratio--3-2: 3/2;--wp--preset--aspect-ratio--2-3: 2/3;--wp--preset--aspect-ratio--16-9: 16/9;--wp--preset--aspect-ratio--9-16: 9/16;--wp--preset--color--black: #000000;--wp--preset--color--cyan-bluish-gray: #abb8c3;--wp--preset--color--white: #ffffff;--wp--preset--color--pale-pink: #f78da7;--wp--preset--color--vivid-red: #cf2e2e;--wp--preset--color--luminous-vivid-orange: #ff6900;--wp--preset--color--luminous-vivid-amber: #fcb900;--wp--preset--color--light-green-cyan: #7bdcb5;--wp--preset--color--vivid-green-cyan: #00d084;--wp--preset--color--pale-cyan-blue: #8ed1fc;--wp--preset--color--vivid-cyan-blue: #0693e3;--wp--preset--color--vivid-purple: #9b51e0;--wp--preset--color--metallic-red: #b02b2c;--wp--preset--color--maximum-yellow-red: #edae44;--wp--preset--color--yellow-sun: #eeee22;--wp--preset--color--palm-leaf: #83a846;--wp--preset--color--aero: #7bb0e7;--wp--preset--color--old-lavender: #745f7e;--wp--preset--color--steel-teal: #5f8789;--wp--preset--color--raspberry-pink: #d65799;--wp--preset--color--medium-turquoise: #4ecac2;--wp--preset--gradient--vivid-cyan-blue-to-vivid-purple: linear-gradient(135deg,rgba(6,147,227,1) 0%,rgb(155,81,224) 100%);--wp--preset--gradient--light-green-cyan-to-vivid-green-cyan: linear-gradient(135deg,rgb(122,220,180) 0%,rgb(0,208,130) 100%);--wp--preset--gradient--luminous-vivid-amber-to-luminous-vivid-orange: linear-gradient(135deg,rgba(252,185,0,1) 0%,rgba(255,105,0,1) 100%);--wp--preset--gradient--luminous-vivid-orange-to-vivid-red: linear-gradient(135deg,rgba(255,105,0,1) 0%,rgb(207,46,46) 100%);--wp--preset--gradient--very-light-gray-to-cyan-bluish-gray: linear-gradient(135deg,rgb(238,238,238) 0%,rgb(169,184,195) 100%);--wp--preset--gradient--cool-to-warm-spectrum: linear-gradient(135deg,rgb(74,234,220) 0%,rgb(151,120,209) 20%,rgb(207,42,186) 40%,rgb(238,44,130) 60%,rgb(251,105,98) 80%,rgb(254,248,76) 100%);--wp--preset--gradient--blush-light-purple: linear-gradient(135deg,rgb(255,206,236) 0%,rgb(152,150,240) 100%);--wp--preset--gradient--blush-bordeaux: linear-gradient(135deg,rgb(254,205,165) 0%,rgb(254,45,45) 50%,rgb(107,0,62) 100%);--wp--preset--gradient--luminous-dusk: linear-gradient(135deg,rgb(255,203,112) 0%,rgb(199,81,192) 50%,rgb(65,88,208) 100%);--wp--preset--gradient--pale-ocean: linear-gradient(135deg,rgb(255,245,203) 0%,rgb(182,227,212) 50%,rgb(51,167,181) 100%);--wp--preset--gradient--electric-grass: linear-gradient(135deg,rgb(202,248,128) 0%,rgb(113,206,126) 100%);--wp--preset--gradient--midnight: linear-gradient(135deg,rgb(2,3,129) 0%,rgb(40,116,252) 100%);--wp--preset--font-size--small: 1rem;--wp--preset--font-size--medium: 1.125rem;--wp--preset--font-size--large: 1.75rem;--wp--preset--font-size--x-large: clamp(1.75rem, 3vw, 2.25rem);--wp--preset--spacing--20: 0.44rem;--wp--preset--spacing--30: 0.67rem;--wp--preset--spacing--40: 1rem;--wp--preset--spacing--50: 1.5rem;--wp--preset--spacing--60: 2.25rem;--wp--preset--spacing--70: 3.38rem;--wp--preset--spacing--80: 5.06rem;--wp--preset--shadow--natural: 6px 6px 9px rgba(0, 0, 0, 0.2);--wp--preset--shadow--deep: 12px 12px 50px rgba(0, 0, 0, 0.4);--wp--preset--shadow--sharp: 6px 6px 0px rgba(0, 0, 0, 0.2);--wp--preset--shadow--outlined: 6px 6px 0px -3px rgba(255, 255, 255, 1), 6px 6px rgba(0, 0, 0, 1);--wp--preset--shadow--crisp: 6px 6px 0px rgba(0, 0, 0, 1);}:root { --wp--style--global--content-size: 800px;--wp--style--global--wide-size: 1130px; }:where(body) { margin: 0; }.wp-site-blocks > .alignleft { float: left; margin-right: 2em; }.wp-site-blocks > .alignright { float: right; margin-left: 2em; }.wp-site-blocks > .aligncenter { justify-content: center; margin-left: auto; margin-right: auto; }:where(.is-layout-flex){gap: 0.5em;}:where(.is-layout-grid){gap: 0.5em;}.is-layout-flow > .alignleft{float: left;margin-inline-start: 0;margin-inline-end: 2em;}.is-layout-flow > .alignright{float: right;margin-inline-start: 2em;margin-inline-end: 0;}.is-layout-flow > .aligncenter{margin-left: auto !important;margin-right: auto !important;}.is-layout-constrained > .alignleft{float: left;margin-inline-start: 0;margin-inline-end: 2em;}.is-layout-constrained > .alignright{float: right;margin-inline-start: 2em;margin-inline-end: 0;}.is-layout-constrained > .aligncenter{margin-left: auto !important;margin-right: auto !important;}.is-layout-constrained > :where(:not(.alignleft):not(.alignright):not(.alignfull)){max-width: var(--wp--style--global--content-size);margin-left: auto !important;margin-right: auto !important;}.is-layout-constrained > .alignwide{max-width: var(--wp--style--global--wide-size);}body .is-layout-flex{display: flex;}.is-layout-flex{flex-wrap: wrap;align-items: center;}.is-layout-flex > :is(*, div){margin: 0;}body .is-layout-grid{display: grid;}.is-layout-grid > :is(*, div){margin: 0;}body{padding-top: 0px;padding-right: 0px;padding-bottom: 0px;padding-left: 0px;}a:where(:not(.wp-element-button)){text-decoration: underline;}:root :where(.wp-element-button, .wp-block-button__link){background-color: #32373c;border-width: 0;color: #fff;font-family: inherit;font-size: inherit;line-height: inherit;padding: calc(0.667em + 2px) calc(1.333em + 2px);text-decoration: none;}.has-black-color{color: var(--wp--preset--color--black) !important;}.has-cyan-bluish-gray-color{color: var(--wp--preset--color--cyan-bluish-gray) !important;}.has-white-color{color: var(--wp--preset--color--white) !important;}.has-pale-pink-color{color: var(--wp--preset--color--pale-pink) !important;}.has-vivid-red-color{color: var(--wp--preset--color--vivid-red) !important;}.has-luminous-vivid-orange-color{color: var(--wp--preset--color--luminous-vivid-orange) !important;}.has-luminous-vivid-amber-color{color: var(--wp--preset--color--luminous-vivid-amber) !important;}.has-light-green-cyan-color{color: var(--wp--preset--color--light-green-cyan) !important;}.has-vivid-green-cyan-color{color: var(--wp--preset--color--vivid-green-cyan) !important;}.has-pale-cyan-blue-color{color: var(--wp--preset--color--pale-cyan-blue) !important;}.has-vivid-cyan-blue-color{color: var(--wp--preset--color--vivid-cyan-blue) !important;}.has-vivid-purple-color{color: var(--wp--preset--color--vivid-purple) !important;}.has-metallic-red-color{color: var(--wp--preset--color--metallic-red) !important;}.has-maximum-yellow-red-color{color: var(--wp--preset--color--maximum-yellow-red) !important;}.has-yellow-sun-color{color: var(--wp--preset--color--yellow-sun) !important;}.has-palm-leaf-color{color: var(--wp--preset--color--palm-leaf) !important;}.has-aero-color{color: var(--wp--preset--color--aero) !important;}.has-old-lavender-color{color: var(--wp--preset--color--old-lavender) !important;}.has-steel-teal-color{color: var(--wp--preset--color--steel-teal) !important;}.has-raspberry-pink-color{color: var(--wp--preset--color--raspberry-pink) !important;}.has-medium-turquoise-color{color: var(--wp--preset--color--medium-turquoise) !important;}.has-black-background-color{background-color: var(--wp--preset--color--black) !important;}.has-cyan-bluish-gray-background-color{background-color: var(--wp--preset--color--cyan-bluish-gray) !important;}.has-white-background-color{background-color: var(--wp--preset--color--white) !important;}.has-pale-pink-background-color{background-color: var(--wp--preset--color--pale-pink) !important;}.has-vivid-red-background-color{background-color: var(--wp--preset--color--vivid-red) !important;}.has-luminous-vivid-orange-background-color{background-color: var(--wp--preset--color--luminous-vivid-orange) !important;}.has-luminous-vivid-amber-background-color{background-color: var(--wp--preset--color--luminous-vivid-amber) !important;}.has-light-green-cyan-background-color{background-color: var(--wp--preset--color--light-green-cyan) !important;}.has-vivid-green-cyan-background-color{background-color: var(--wp--preset--color--vivid-green-cyan) !important;}.has-pale-cyan-blue-background-color{background-color: var(--wp--preset--color--pale-cyan-blue) !important;}.has-vivid-cyan-blue-background-color{background-color: var(--wp--preset--color--vivid-cyan-blue) !important;}.has-vivid-purple-background-color{background-color: var(--wp--preset--color--vivid-purple) !important;}.has-metallic-red-background-color{background-color: var(--wp--preset--color--metallic-red) !important;}.has-maximum-yellow-red-background-color{background-color: var(--wp--preset--color--maximum-yellow-red) !important;}.has-yellow-sun-background-color{background-color: var(--wp--preset--color--yellow-sun) !important;}.has-palm-leaf-background-color{background-color: var(--wp--preset--color--palm-leaf) !important;}.has-aero-background-color{background-color: var(--wp--preset--color--aero) !important;}.has-old-lavender-background-color{background-color: var(--wp--preset--color--old-lavender) !important;}.has-steel-teal-background-color{background-color: var(--wp--preset--color--steel-teal) !important;}.has-raspberry-pink-background-color{background-color: var(--wp--preset--color--raspberry-pink) !important;}.has-medium-turquoise-background-color{background-color: var(--wp--preset--color--medium-turquoise) !important;}.has-black-border-color{border-color: var(--wp--preset--color--black) !important;}.has-cyan-bluish-gray-border-color{border-color: var(--wp--preset--color--cyan-bluish-gray) !important;}.has-white-border-color{border-color: var(--wp--preset--color--white) !important;}.has-pale-pink-border-color{border-color: var(--wp--preset--color--pale-pink) !important;}.has-vivid-red-border-color{border-color: var(--wp--preset--color--vivid-red) !important;}.has-luminous-vivid-orange-border-color{border-color: var(--wp--preset--color--luminous-vivid-orange) !important;}.has-luminous-vivid-amber-border-color{border-color: var(--wp--preset--color--luminous-vivid-amber) !important;}.has-light-green-cyan-border-color{border-color: var(--wp--preset--color--light-green-cyan) !important;}.has-vivid-green-cyan-border-color{border-color: var(--wp--preset--color--vivid-green-cyan) !important;}.has-pale-cyan-blue-border-color{border-color: var(--wp--preset--color--pale-cyan-blue) !important;}.has-vivid-cyan-blue-border-color{border-color: var(--wp--preset--color--vivid-cyan-blue) !important;}.has-vivid-purple-border-color{border-color: var(--wp--preset--color--vivid-purple) !important;}.has-metallic-red-border-color{border-color: var(--wp--preset--color--metallic-red) !important;}.has-maximum-yellow-red-border-color{border-color: var(--wp--preset--color--maximum-yellow-red) !important;}.has-yellow-sun-border-color{border-color: var(--wp--preset--color--yellow-sun) !important;}.has-palm-leaf-border-color{border-color: var(--wp--preset--color--palm-leaf) !important;}.has-aero-border-color{border-color: var(--wp--preset--color--aero) !important;}.has-old-lavender-border-color{border-color: var(--wp--preset--color--old-lavender) !important;}.has-steel-teal-border-color{border-color: var(--wp--preset--color--steel-teal) !important;}.has-raspberry-pink-border-color{border-color: var(--wp--preset--color--raspberry-pink) !important;}.has-medium-turquoise-border-color{border-color: var(--wp--preset--color--medium-turquoise) !important;}.has-vivid-cyan-blue-to-vivid-purple-gradient-background{background: var(--wp--preset--gradient--vivid-cyan-blue-to-vivid-purple) !important;}.has-light-green-cyan-to-vivid-green-cyan-gradient-background{background: var(--wp--preset--gradient--light-green-cyan-to-vivid-green-cyan) !important;}.has-luminous-vivid-amber-to-luminous-vivid-orange-gradient-background{background: var(--wp--preset--gradient--luminous-vivid-amber-to-luminous-vivid-orange) !important;}.has-luminous-vivid-orange-to-vivid-red-gradient-background{background: var(--wp--preset--gradient--luminous-vivid-orange-to-vivid-red) !important;}.has-very-light-gray-to-cyan-bluish-gray-gradient-background{background: var(--wp--preset--gradient--very-light-gray-to-cyan-bluish-gray) !important;}.has-cool-to-warm-spectrum-gradient-background{background: var(--wp--preset--gradient--cool-to-warm-spectrum) !important;}.has-blush-light-purple-gradient-background{background: var(--wp--preset--gradient--blush-light-purple) !important;}.has-blush-bordeaux-gradient-background{background: var(--wp--preset--gradient--blush-bordeaux) !important;}.has-luminous-dusk-gradient-background{background: var(--wp--preset--gradient--luminous-dusk) !important;}.has-pale-ocean-gradient-background{background: var(--wp--preset--gradient--pale-ocean) !important;}.has-electric-grass-gradient-background{background: var(--wp--preset--gradient--electric-grass) !important;}.has-midnight-gradient-background{background: var(--wp--preset--gradient--midnight) !important;}.has-small-font-size{font-size: var(--wp--preset--font-size--small) !important;}.has-medium-font-size{font-size: var(--wp--preset--font-size--medium) !important;}.has-large-font-size{font-size: var(--wp--preset--font-size--large) !important;}.has-x-large-font-size{font-size: var(--wp--preset--font-size--x-large) !important;} :where(.wp-block-post-template.is-layout-flex){gap: 1.25em;}:where(.wp-block-post-template.is-layout-grid){gap: 1.25em;} :where(.wp-block-columns.is-layout-flex){gap: 2em;}:where(.wp-block-columns.is-layout-grid){gap: 2em;} :root :where(.wp-block-pullquote){font-size: 1.5em;line-height: 1.6;}</style><link data-optimized="1" rel='stylesheet' id='wordpress-popular-posts-css-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/f37de7c74c7905b881abee3714c59450.css?ver=e678c' type='text/css' media='all' /><link data-optimized="1" rel='stylesheet' id='avia-merged-styles-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/cfb9c80d41094e4ad473a1e04dc00f70.css?ver=00f70' type='text/css' media='all' /><link data-optimized="1" rel='stylesheet' id='avia-single-post-25754-css' href='https://www.physicsforums.com/insights/wp-content/litespeed/css/8a0be2ada42d122c3695fe3713ef52f9.css?ver=f03e8' type='text/css' media='all' /> <script data-optimized="1" type="text/javascript" async='async' src="https://www.physicsforums.com/insights/wp-content/litespeed/js/22e6cada7a5696283f0dcecdd91a44c9.js?ver=a44c9" id="avia-head-scripts-js"></script> <style id="wpp-loading-animation-styles">@-webkit-keyframes bgslide{from{background-position-x:0}to{background-position-x:-200%}}@keyframes bgslide{from{background-position-x:0}to{background-position-x:-200%}}.wpp-widget-block-placeholder,.wpp-shortcode-placeholder{margin:0 auto;width:60px;height:3px;background:#dd3737;background:linear-gradient(90deg,#dd3737 0%,#571313 10%,#dd3737 100%);background-size:200% auto;border-radius:3px;-webkit-animation:bgslide 1s infinite linear;animation:bgslide 1s infinite linear}</style><link rel="profile" href="https://gmpg.org/xfn/11" /><link rel="alternate" type="application/rss+xml" title="Physics Forums Insights RSS2 Feed" href="https://www.physicsforums.com/insights/feed/" /><link rel="pingback" href="https://www.physicsforums.com/insights/xmlrpc.php" /><style type='text/css' media='screen'>#top #header_main > .container, #top #header_main > .container .main_menu .av-main-nav > li > a, #top #header_main #menu-item-shop .cart_dropdown_link{ height:70px; line-height: 70px; } .html_top_nav_header .av-logo-container{ height:70px; } .html_header_top.html_header_sticky #top #wrap_all #main{ padding-top:70px; }</style><link rel="icon" href="https://www.physicsforums.com/insights/wp-content/uploads/2016/08/favicon.ico" type="image/x-icon"><style type="text/css">.saboxplugin-wrap{-webkit-box-sizing:border-box;-moz-box-sizing:border-box;-ms-box-sizing:border-box;box-sizing:border-box;border:1px solid #eee;width:100%;clear:both;display:block;overflow:hidden;word-wrap:break-word;position:relative}.saboxplugin-wrap .saboxplugin-gravatar{float:left;padding:0 20px 20px 20px}.saboxplugin-wrap .saboxplugin-gravatar img{max-width:100px;height:auto;border-radius:0;}.saboxplugin-wrap .saboxplugin-authorname{font-size:18px;line-height:1;margin:20px 0 0 20px;display:block}.saboxplugin-wrap .saboxplugin-authorname a{text-decoration:none}.saboxplugin-wrap .saboxplugin-authorname a:focus{outline:0}.saboxplugin-wrap .saboxplugin-desc{display:block;margin:5px 20px}.saboxplugin-wrap .saboxplugin-desc a{text-decoration:underline}.saboxplugin-wrap .saboxplugin-desc p{margin:5px 0 12px}.saboxplugin-wrap .saboxplugin-web{margin:0 20px 15px;text-align:left}.saboxplugin-wrap .sab-web-position{text-align:right}.saboxplugin-wrap .saboxplugin-web a{color:#ccc;text-decoration:none}.saboxplugin-wrap .saboxplugin-socials{position:relative;display:block;background:#fcfcfc;padding:5px;border-top:1px solid #eee}.saboxplugin-wrap .saboxplugin-socials a svg{width:20px;height:20px}.saboxplugin-wrap .saboxplugin-socials a svg .st2{fill:#fff; transform-origin:center center;}.saboxplugin-wrap .saboxplugin-socials a svg .st1{fill:rgba(0,0,0,.3)}.saboxplugin-wrap .saboxplugin-socials a:hover{opacity:.8;-webkit-transition:opacity .4s;-moz-transition:opacity .4s;-o-transition:opacity .4s;transition:opacity .4s;box-shadow:none!important;-webkit-box-shadow:none!important}.saboxplugin-wrap .saboxplugin-socials .saboxplugin-icon-color{box-shadow:none;padding:0;border:0;-webkit-transition:opacity .4s;-moz-transition:opacity .4s;-o-transition:opacity .4s;transition:opacity .4s;display:inline-block;color:#fff;font-size:0;text-decoration:inherit;margin:5px;-webkit-border-radius:0;-moz-border-radius:0;-ms-border-radius:0;-o-border-radius:0;border-radius:0;overflow:hidden}.saboxplugin-wrap .saboxplugin-socials .saboxplugin-icon-grey{text-decoration:inherit;box-shadow:none;position:relative;display:-moz-inline-stack;display:inline-block;vertical-align:middle;zoom:1;margin:10px 5px;color:#444;fill:#444}.clearfix:after,.clearfix:before{content:' ';display:table;line-height:0;clear:both}.ie7 .clearfix{zoom:1}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-twitch{border-color:#38245c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-addthis{border-color:#e91c00}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-behance{border-color:#003eb0}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-delicious{border-color:#06c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-deviantart{border-color:#036824}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-digg{border-color:#00327c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-dribbble{border-color:#ba1655}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-facebook{border-color:#1e2e4f}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-flickr{border-color:#003576}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-github{border-color:#264874}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-google{border-color:#0b51c5}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-googleplus{border-color:#96271a}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-html5{border-color:#902e13}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-instagram{border-color:#1630aa}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-linkedin{border-color:#00344f}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-pinterest{border-color:#5b040e}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-reddit{border-color:#992900}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-rss{border-color:#a43b0a}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-sharethis{border-color:#5d8420}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-skype{border-color:#00658a}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-soundcloud{border-color:#995200}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-spotify{border-color:#0f612c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-stackoverflow{border-color:#a95009}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-steam{border-color:#006388}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-user_email{border-color:#b84e05}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-stumbleUpon{border-color:#9b280e}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-tumblr{border-color:#10151b}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-twitter{border-color:#0967a0}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-vimeo{border-color:#0d7091}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-windows{border-color:#003f71}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-whatsapp{border-color:#003f71}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-wordpress{border-color:#0f3647}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-yahoo{border-color:#14002d}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-youtube{border-color:#900}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-xing{border-color:#000202}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-mixcloud{border-color:#2475a0}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-vk{border-color:#243549}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-medium{border-color:#00452c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-quora{border-color:#420e00}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-meetup{border-color:#9b181c}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-goodreads{border-color:#000}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-snapchat{border-color:#999700}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-500px{border-color:#00557f}.saboxplugin-socials.sabox-colored .saboxplugin-icon-color .sab-mastodont{border-color:#185886}.sabox-plus-item{margin-bottom:20px}@media screen and (max-width:480px){.saboxplugin-wrap{text-align:center}.saboxplugin-wrap .saboxplugin-gravatar{float:none;padding:20px 0;text-align:center;margin:0 auto;display:block}.saboxplugin-wrap .saboxplugin-gravatar img{float:none;display:inline-block;display:-moz-inline-stack;vertical-align:middle;zoom:1}.saboxplugin-wrap .saboxplugin-desc{margin:0 10px 20px;text-align:center}.saboxplugin-wrap .saboxplugin-authorname{text-align:center;margin:10px 0 20px}}body .saboxplugin-authorname a,body .saboxplugin-authorname a:hover{box-shadow:none;-webkit-box-shadow:none}a.sab-profile-edit{font-size:16px!important;line-height:1!important}.sab-edit-settings a,a.sab-profile-edit{color:#0073aa!important;box-shadow:none!important;-webkit-box-shadow:none!important}.sab-edit-settings{margin-right:15px;position:absolute;right:0;z-index:2;bottom:10px;line-height:20px}.sab-edit-settings i{margin-left:5px}.saboxplugin-socials{line-height:1!important}.rtl .saboxplugin-wrap .saboxplugin-gravatar{float:right}.rtl .saboxplugin-wrap .saboxplugin-authorname{display:flex;align-items:center}.rtl .saboxplugin-wrap .saboxplugin-authorname .sab-profile-edit{margin-right:10px}.rtl .sab-edit-settings{right:auto;left:0}img.sab-custom-avatar{max-width:75px;}.saboxplugin-wrap .saboxplugin-gravatar img {-webkit-border-radius:50%;-moz-border-radius:50%;-ms-border-radius:50%;-o-border-radius:50%;border-radius:50%;}.saboxplugin-wrap .saboxplugin-gravatar img {-webkit-border-radius:50%;-moz-border-radius:50%;-ms-border-radius:50%;-o-border-radius:50%;border-radius:50%;}.saboxplugin-wrap .saboxplugin-gravatar img {-webkit-transition:all .5s ease;-moz-transition:all .5s ease;-o-transition:all .5s ease;transition:all .5s ease;}.saboxplugin-wrap .saboxplugin-gravatar img:hover {-webkit-transform:rotate(45deg);-moz-transform:rotate(45deg);-o-transform:rotate(45deg);-ms-transform:rotate(45deg);transform:rotate(45deg);}.saboxplugin-wrap {margin-top:0px; margin-bottom:0px; padding: 0px 0px }.saboxplugin-wrap .saboxplugin-authorname {font-size:18px; line-height:25px;}.saboxplugin-wrap .saboxplugin-desc p, .saboxplugin-wrap .saboxplugin-desc {font-size:14px !important; line-height:21px !important;}.saboxplugin-wrap .saboxplugin-web {font-size:14px;}.saboxplugin-wrap .saboxplugin-socials a svg {width:18px;height:18px;}</style> <script type="text/javascript">'use strict';var avia_is_mobile=!1;if(/Android|webOS|iPhone|iPad|iPod|BlackBerry|IEMobile|Opera Mini/i.test(navigator.userAgent)&&'ontouchstart' in document.documentElement){avia_is_mobile=!0;document.documentElement.className+=' avia_mobile '} else{document.documentElement.className+=' avia_desktop '};document.documentElement.className+=' js_active ';(function(){var e=['-webkit-','-moz-','-ms-',''],n='',o=!1,a=!1;for(var t in e){if(e[t]+'transform' in document.documentElement.style){o=!0;n=e[t]+'transform'};if(e[t]+'perspective' in document.documentElement.style){a=!0}};if(o){document.documentElement.className+=' avia_transform '};if(a){document.documentElement.className+=' avia_transform3d '};if(typeof document.getElementsByClassName=='function'&&typeof document.documentElement.getBoundingClientRect=='function'&&avia_is_mobile==!1){if(n&&window.innerHeight>0){setTimeout(function(){var e=0,o={},a=0,t=document.getElementsByClassName('av-parallax'),i=window.pageYOffset||document.documentElement.scrollTop;for(e=0;e<t.length;e++){t[e].style.top='0px';o=t[e].getBoundingClientRect();a=Math.ceil((window.innerHeight+i-o.top)*0.3);t[e].style[n]='translate(0px, '+a+'px)';t[e].style.top='auto';t[e].className+=' enabled-parallax '}},50)}}})();</script><style type="text/css">@font-face {font-family: 'entypo-fontello'; font-weight: normal; font-style: normal; font-display: swap; src: url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.woff2') format('woff2'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.woff') format('woff'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.ttf') format('truetype'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.svg#entypo-fontello') format('svg'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.eot'), url('https://www.physicsforums.com/insights/wp-content/themes/enfold/config-templatebuilder/avia-template-builder/assets/fonts/entypo-fontello.eot?#iefix') format('embedded-opentype'); } #top .avia-font-entypo-fontello, body .avia-font-entypo-fontello, html body [data-av_iconfont='entypo-fontello']:before{ font-family: 'entypo-fontello'; }</style></head><body id="top" class="home page-template-default page page-id-25754 boxed avia-mobile-no-animations rtl_columns av-curtain-numeric arial-websave arial avia-responsive-images-support" itemscope="itemscope" itemtype="https://schema.org/WebPage" ><div id='wrap_all'><header id='header' class='all_colors header_color light_bg_color av_header_top av_logo_left av_main_nav_header av_menu_right av_custom av_header_sticky av_header_shrinking_disabled av_header_stretch_disabled av_mobile_menu_phone av_header_searchicon av_header_unstick_top_disabled av_bottom_nav_disabled av_header_border_disabled' aria-label="Header" data-av_shrink_factor='50' role="banner" itemscope="itemscope" itemtype="https://schema.org/WPHeader" ><div id='header_main' class='container_wrap container_wrap_logo'><div class='container av-logo-container'><div class='inner-container'><span class='logo avia-standard-logo'><a href='https://www.physicsforums.com/insights/' class='' aria-label='Physics_Forums_Insights_logo' title='Physics_Forums_Insights_logo'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzMDAiIGhlaWdodD0iMTAwIiB2aWV3Qm94PSIwIDAgMzAwIDEwMCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png" height="100" width="300" alt='Physics Forums Insights' title='Physics_Forums_Insights_logo' /><noscript><img src="https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png" height="100" width="300" alt='Physics Forums Insights' title='Physics_Forums_Insights_logo' /></noscript></a></span><nav class='main_menu' data-selectname='Select a page' role="navigation" itemscope="itemscope" itemtype="https://schema.org/SiteNavigationElement" ><div class="avia-menu av-main-nav-wrap"><ul role="menu" class="menu av-main-nav" id="avia-menu"><li role="menuitem" id="menu-item-27276" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-top-level menu-item-top-level-1"><a href="https://www.physicsforums.com/insights/trending-physics-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a><ul class="sub-menu"><li role="menuitem" id="menu-item-26559" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-articles/physics-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics Articles</span></a></li><li role="menuitem" id="menu-item-26568" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-tutorials/physics-tutorials/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics Tutorials</span></a></li><li role="menuitem" id="menu-item-26564" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-guides/physics-guides/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics Guides</span></a></li><li role="menuitem" id="menu-item-26561" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-faqs/physics-faqs/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Physics FAQs</span></a></li></ul></li><li role="menuitem" id="menu-item-27288" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-top-level menu-item-top-level-2"><a href="https://www.physicsforums.com/insights/trending-math-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a><ul class="sub-menu"><li role="menuitem" id="menu-item-26558" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-articles/math-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math Articles</span></a></li><li role="menuitem" id="menu-item-26567" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-tutorials/math-tutorials/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math Tutorials</span></a></li><li role="menuitem" id="menu-item-26563" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-guides/math-guides/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math Guides</span></a></li><li role="menuitem" id="menu-item-26560" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-faqs/math-faqs/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Math FAQs</span></a></li></ul></li><li role="menuitem" id="menu-item-27287" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-top-level menu-item-top-level-3"><a href="https://www.physicsforums.com/insights/trending-bio-chem-tech-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Bio/Chem/Tech</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a><ul class="sub-menu"><li role="menuitem" id="menu-item-26556" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-articles/biology-chemistry-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Bio/Chem Articles</span></a></li><li role="menuitem" id="menu-item-26566" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-tutorials/computer-science-tutorials/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Computer Science Tutorials</span></a></li><li role="menuitem" id="menu-item-26565" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-guides/technology-guides/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Technology Guides</span></a></li></ul></li><li role="menuitem" id="menu-item-27286" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-has-children menu-item-top-level menu-item-top-level-4"><a href="https://www.physicsforums.com/insights/trending-education-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Education</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a><ul class="sub-menu"><li role="menuitem" id="menu-item-26557" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-articles/education-articles/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Education Articles</span></a></li><li role="menuitem" id="menu-item-26562" class="menu-item menu-item-type-taxonomy menu-item-object-category"><a href="https://www.physicsforums.com/insights/science-math-guides/education-guides/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Education Guides</span></a></li></ul></li><li role="menuitem" id="menu-item-26569" class="menu-item menu-item-type-taxonomy menu-item-object-category menu-item-top-level menu-item-top-level-5"><a href="https://www.physicsforums.com/insights/scientist-interviews/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Interviews</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a></li><li role="menuitem" id="menu-item-26570" class="menu-item menu-item-type-taxonomy menu-item-object-category menu-item-top-level menu-item-top-level-6"><a href="https://www.physicsforums.com/insights/science-math-quizzes/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Quizzes</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a></li><li role="menuitem" id="menu-item-26571" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-top-level menu-item-top-level-7"><a href="https://www.physicsforums.com/" itemprop="url" tabindex="0"><span class="avia-bullet"></span><span class="avia-menu-text">Forums</span><span class="avia-menu-fx"><span class="avia-arrow-wrap"><span class="avia-arrow"></span></span></span></a></li><li id="menu-item-search" class="noMobile menu-item menu-item-search-dropdown menu-item-avia-special" role="menuitem"><a aria-label="Search" href="?s=" rel="nofollow" data-avia-search-tooltip=" <search> <form role="search" action="https://www.physicsforums.com/insights/" id="searchform" method="get" class=""> <div> <input type="submit" value="" id="searchsubmit" class="button avia-font-entypo-fontello" title="Enter at least 3 characters to show search results in a dropdown or click to route to search result page to show all results" /> <input type="search" id="s" name="s" value="" aria-label='Search' placeholder='Search' required /> </div> </form> </search> " aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello'><span class="avia_hidden_link_text">Search</span></a></li><li class="av-burger-menu-main menu-item-avia-special " role="menuitem"> <a href="#" aria-label="Menu" aria-hidden="false"> <span class="av-hamburger av-hamburger--spin av-js-hamburger"> <span class="av-hamburger-box"> <span class="av-hamburger-inner"></span> <strong>Menu</strong> </span> </span> <span class="avia_hidden_link_text">Menu</span> </a></li></ul></div></nav></div></div></div><div class="header_bg"></div></header><div id='main' class='all_colors' data-scroll-offset='70'><div class='main_color container_wrap_first container_wrap sidebar_right' ><div class='container av-section-cont-open' ><main role="main" itemprop="mainContentOfPage" class='template-page content av-content-small alpha units'><div class='post-entry post-entry-type-page post-entry-25754'><div class='entry-content-wrapper clearfix'><div class='flex_column av-4vj50a-9128c539f73de6e396a5629f5b783829 av_one_full avia-builder-el-0 el_before_av_one_half avia-builder-el-first first flex_column_div av-zero-column-padding ' ><div class='av-special-heading av-k3qfpza8-bf2b316edd03ec0cb1fd766e4da2c96e av-special-heading-h1 blockquote modern-quote avia-builder-el-1 avia-builder-el-no-sibling av-linked-heading'><h1 class='av-special-heading-tag ' itemprop="headline" >The Expert Physics and Math Blog</h1><div class="special-heading-border"><div class="special-heading-inner-border"></div></div></div></div><div class='flex_column av-mwb6wq-f42f8b65270e9e758397899dfd7544fc av_one_half avia-builder-el-2 el_after_av_one_full el_before_av_one_half first flex_column_div av-zero-column-padding column-top-margin' ><section class='av_textblock_section av-k3qfn7qj-30245ff09c1fceecd439a6055e034b6d ' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><div class='avia_textblock' itemprop="text" ></div></section><div class='av-alb-blogposts template-blog av-blog-meta-html-info-disabled ' itemscope="itemscope" itemtype="https://schema.org/Blog" ><article class="post-entry post-entry-type-standard post-entry-47569 post-loop-1 post-parity-odd post-entry-last single-big with-slider post-47569 post type-post status-publish format-standard has-post-thumbnail hentry category-math-articles tag-projectile-motion" itemscope="itemscope" itemtype="https://schema.org/BlogPosting" ><div class="big-preview single-big" itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" ><a href="https://www.physicsforums.com/insights/brownian-motions-and-quantifying-randomness-in-physical-systems/" title="brownian-motion" ><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/08/brownian-motion.png" class="wp-image-47594 avia-img-lazy-loading-47594 attachment-entry_with_sidebar size-entry_with_sidebar wp-post-image" alt="brownian motion" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/08/brownian-motion.png" class="wp-image-47594 avia-img-lazy-loading-47594 attachment-entry_with_sidebar size-entry_with_sidebar wp-post-image" alt="brownian motion" /></noscript></a></div><div class="blog-meta"></div><div class='entry-content-wrapper clearfix standard-content'><header class="entry-content-header" aria-label="Post: Brownian Motions and Quantifying Randomness in Physical Systems"><h2 class='post-title entry-title ' itemprop="headline" ><a href="https://www.physicsforums.com/insights/brownian-motions-and-quantifying-randomness-in-physical-systems/" rel="bookmark" title="Permanent Link: Brownian Motions and Quantifying Randomness in Physical Systems">Brownian Motions and Quantifying Randomness in Physical Systems<span class="post-format-icon minor-meta"></span></a></h2><span class="post-meta-infos"><time class="date-container minor-meta updated" itemprop="datePublished" datetime="2024-08-26T07:14:45-05:00" >August 26, 2024</time><span class="text-sep">/</span><span class="comment-container minor-meta"><a href="https://www.physicsforums.com/insights/brownian-motions-and-quantifying-randomness-in-physical-systems/#comments" class="comments-link" >1 Comment</a></span><span class="text-sep">/</span><span class="blog-categories minor-meta">in <a href="https://www.physicsforums.com/insights/science-math-articles/math-articles/" rel="tag">Mathematics Articles</a></span><span class="text-sep">/</span><span class="blog-author minor-meta">by <span class="entry-author-link" itemprop="author" ><span class="author"><span class="fn"><a href="https://www.physicsforums.com/insights/author/ergospherical/" title="Posts by ergospherical" rel="author">ergospherical</a></span></span></span></span></span></header><div class="entry-content" itemprop="text" ><p>Stochastic calculus has come a long way since Robert Brown described the motion of pollen through a microscope in 1827. It’s now a key player in data science, quant finance, and mathematical biology. This article is drawn from notes I wrote for an undergraduate statistical physics course a few months ago. There won’t be any…</p></div><footer class="entry-footer"></footer><div class='post_delimiter'></div></div><div class="post_author_timeline"></div><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2024/08/brownian-motion.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>ergospherical</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>ergospherical</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2024-08-26T07:14:45-05:00" >2024-08-26 07:14:45</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-11-16 10:39:17</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Brownian Motions and Quantifying Randomness in Physical Systems</span></span></span></article></div></div></p><div class='flex_column av-1aemvbu-96c51015b9b5146610da837d471b9fcb av_one_half avia-builder-el-5 el_after_av_one_half el_before_av_one_full flex_column_div av-zero-column-padding column-top-margin' ><section class='av_textblock_section av-k3xly0fg-28caedcdbec7cc4607b15fc251ccf83b ' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><div class='avia_textblock' itemprop="text" ><h2>Trending Articles</h2><ul class="wpp-list"><li class=""><a href="https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/" target="_self"><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1MCIgaGVpZ2h0PSIzNSIgdmlld0JveD0iMCAwIDUwIDM1Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjY2ZkNGRiIi8+PC9zdmc+" data-src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35.png" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35@3x.png 3x" width="50" height="35" alt="planck" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"><noscript><img src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35.png" srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/4930-featured-50x35@3x.png 3x" width="50" height="35" alt="planck" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"></noscript></a> <a href="https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/" class="wpp-post-title" target="_self">What Planck Length Is and It’s Common Misconceptions</a> <span class="wpp-meta post-stats"><span class="wpp-views">316 views</span> | <span class="wpp-author">by <a href="https://www.physicsforums.com/insights/author/klotza/">Alex Klotz</a></span> | <span class="wpp-date">posted on September 9, 2015</span></span></li><li class=""><a href="https://www.physicsforums.com/insights/can-see-atom/" target="_self"><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1MCIgaGVpZ2h0PSIzNSIgdmlld0JveD0iMCAwIDUwIDM1Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjY2ZkNGRiIi8+PC9zdmc+" data-src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35.png" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35@3x.png 3x" width="50" height="35" alt="atoms" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"><noscript><img src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35.png" srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/5387-featured-50x35@3x.png 3x" width="50" height="35" alt="atoms" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"></noscript></a> <a href="https://www.physicsforums.com/insights/can-see-atom/" class="wpp-post-title" target="_self">Can We See an Atom?</a> <span class="wpp-meta post-stats"><span class="wpp-views">281 views</span> | <span class="wpp-author">by <a href="https://www.physicsforums.com/insights/author/klotza/">Alex Klotz</a></span> | <span class="wpp-date">posted on September 28, 2015</span></span></li><li class=""><a href="https://www.physicsforums.com/insights/what-is-a-tensor/" target="_self"><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1MCIgaGVpZ2h0PSIzNSIgdmlld0JveD0iMCAwIDUwIDM1Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjY2ZkNGRiIi8+PC9zdmc+" data-src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35.png" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35@3x.png 3x" width="50" height="35" alt="whatisatensor" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"><noscript><img src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35.png" srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/14577-featured-50x35@3x.png 3x" width="50" height="35" alt="whatisatensor" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"></noscript></a> <a href="https://www.physicsforums.com/insights/what-is-a-tensor/" class="wpp-post-title" target="_self">What Is a Tensor? The mathematical point of view.</a> <span class="wpp-meta post-stats"><span class="wpp-views">172 views</span> | <span class="wpp-author">by <a href="https://www.physicsforums.com/insights/author/fresh_42/">fresh_42</a></span> | <span class="wpp-date">posted on June 18, 2017</span></span></li><li class=""><a href="https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/" target="_self"><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1MCIgaGVpZ2h0PSIzNSIgdmlld0JveD0iMCAwIDUwIDM1Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjY2ZkNGRiIi8+PC9zdmc+" data-src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35.png" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35@3x.png 3x" width="50" height="35" alt="rules of quantum mechanics" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"><noscript><img src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35.png" srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/24410-featured-50x35@3x.png 3x" width="50" height="35" alt="rules of quantum mechanics" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"></noscript></a> <a href="https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/" class="wpp-post-title" target="_self">The 7 Basic Rules of Quantum Mechanics</a> <span class="wpp-meta post-stats"><span class="wpp-views">142 views</span> | <span class="wpp-author">by <a href="https://www.physicsforums.com/insights/author/a-neumaier/">Arnold Neumaier</a></span> | <span class="wpp-date">posted on May 11, 2019</span></span></li><li class=""><a href="https://www.physicsforums.com/insights/parallel-programming-on-an-nvidia-gpu/" target="_self"><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1MCIgaGVpZ2h0PSIzNSIgdmlld0JveD0iMCAwIDUwIDM1Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBmaWxsPSIjY2ZkNGRiIi8+PC9zdmc+" data-src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35.png" data-srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35@3x.png 3x" width="50" height="35" alt="programming gpu" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"><noscript><img src="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35.png" srcset="https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35.png, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35@1.5x.png 1.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35@2x.png 2x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35@2.5x.png 2.5x, https://www.physicsforums.com/insights/wp-content/uploads/wordpress-popular-posts/40425-featured-50x35@3x.png 3x" width="50" height="35" alt="programming gpu" class="wpp-thumbnail wpp_featured wpp_cached_thumb" decoding="async"></noscript></a> <a href="https://www.physicsforums.com/insights/parallel-programming-on-an-nvidia-gpu/" class="wpp-post-title" target="_self">Parallel Programming on an NVIDIA GPU</a> <span class="wpp-meta post-stats"><span class="wpp-views">133 views</span> | <span class="wpp-author">by <a href="https://www.physicsforums.com/insights/author/mark44/">Mark44</a></span> | <span class="wpp-date">posted on April 16, 2022</span></span></li></ul></div></section></div><div class='flex_column av-jqnjiy-4776d88a505f64f3733fd37335eb912d av_one_full avia-builder-el-7 el_after_av_one_half avia-builder-el-last first flex_column_div av-zero-column-padding column-top-margin' ><section class='av_textblock_section av-k3xmgddq-873c715fa01a03c22f6687c10e53cae4 ' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><div class='avia_textblock' itemprop="text" ><h2>Recent Entries</h2></div></section><div data-slideshow-options="{"animation":"fade","autoplay":false,"loop_autoplay":"once","interval":5,"loop_manual":"manual-endless","autoplay_stopper":false,"noNavigation":false,"show_slide_delay":90}" class='avia-content-slider avia-content-grid-active avia-content-slider1 avia-content-slider-odd avia-builder-el-9 el_after_av_textblock avia-builder-el-last av-slideshow-ui av-control-default av-no-slider-navigation av-slideshow-manual av-loop-once av-loop-manual-endless ' itemscope="itemscope" itemtype="https://schema.org/Blog" ><div class="avia-content-slider-inner"><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-46583 slide-entry-overview slide-loop-1 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/pbs-video-comment-what-if-physics-is-not-describing-reality/' data-rel='slide-1' class='slide-image' title='PBS Video Comment: “What If Physics IS NOT Describing Reality”'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/07/physics-of-reality.png" class="wp-image-46641 avia-img-lazy-loading-46641 attachment-portfolio size-portfolio wp-post-image" alt="physics of reality" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/07/physics-of-reality.png" class="wp-image-46641 avia-img-lazy-loading-46641 attachment-portfolio size-portfolio wp-post-image" alt="physics of reality" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: PBS Video Comment: "What If Physics IS NOT Describing Reality""><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/pbs-video-comment-what-if-physics-is-not-describing-reality/' title='PBS Video Comment: “What If Physics IS NOT Describing Reality”'>PBS Video Comment: “What If Physics IS NOT Describing Reality”</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-07-22T11:24:16-05:00" >July 22, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/pbs-video-comment-what-if-physics-is-not-describing-reality/#comments'>2 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >PBS Space Time produces some very good videos on the foundations of quantum mechanics (QM), so let me comment on their video What If Physics IS NOT Describing…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2024/07/physics-of-reality.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Mark Stuckey</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Mark Stuckey</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-07-22 11:24:16</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 14:10:09</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>PBS Video Comment: “What If Physics IS NOT Describing Reality”</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-46242 slide-entry-overview slide-loop-2 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/aspects-behind-the-concept-of-dimension-in-various-fields/' data-rel='slide-1' class='slide-image' title='Aspects Behind the Concept of Dimension in Various Fields'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/07/pvsnp.png" class="wp-image-41832 avia-img-lazy-loading-41832 attachment-portfolio size-portfolio wp-post-image" alt="pvsnp" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/07/pvsnp.png" class="wp-image-41832 avia-img-lazy-loading-41832 attachment-portfolio size-portfolio wp-post-image" alt="pvsnp" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Aspects Behind the Concept of Dimension in Various Fields"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/aspects-behind-the-concept-of-dimension-in-various-fields/' title='Aspects Behind the Concept of Dimension in Various Fields'>Aspects Behind the Concept of Dimension in Various Fields</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-06-21T12:42:18-05:00" >June 21, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/aspects-behind-the-concept-of-dimension-in-various-fields/#comments'>2 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Abstract It took until the last century for physicists and mathematicians in the Netherlands to question the Euclidean concept of dimension as length,…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/07/pvsnp.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-06-21 12:42:18</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-26 07:17:43</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Aspects Behind the Concept of Dimension in Various Fields</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-46007 slide-entry-overview slide-loop-3 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/views-on-complex-numbers/' data-rel='slide-1' class='slide-image' title='Views On Complex Numbers'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/06/complex-numbers-views.png" class="wp-image-46207 avia-img-lazy-loading-46207 attachment-portfolio size-portfolio wp-post-image" alt="complex numbers views" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/06/complex-numbers-views.png" class="wp-image-46207 avia-img-lazy-loading-46207 attachment-portfolio size-portfolio wp-post-image" alt="complex numbers views" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Views On Complex Numbers"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/views-on-complex-numbers/' title='Views On Complex Numbers'>Views On Complex Numbers</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-06-07T06:10:00-05:00" >June 7, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/views-on-complex-numbers/#comments'>70 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Abstract Why do we need yet another article about complex numbers? This is a valid question and I have asked it myself. I could mention that I wanted…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2024/06/complex-numbers-views.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-06-07 06:10:00</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-09-30 09:01:04</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Views On Complex Numbers</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-45831 slide-entry-overview slide-loop-4 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/addition-of-velocities-velocity-composition-in-special-relativity/' data-rel='slide-1' class='slide-image' title='Addition of Velocities (Velocity Composition) in Special Relativity'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/05/addition_of_velocities.png" class="wp-image-45978 avia-img-lazy-loading-45978 attachment-portfolio size-portfolio wp-post-image" alt="" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/05/addition_of_velocities.png" class="wp-image-45978 avia-img-lazy-loading-45978 attachment-portfolio size-portfolio wp-post-image" alt="" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Addition of Velocities (Velocity Composition) in Special Relativity"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/addition-of-velocities-velocity-composition-in-special-relativity/' title='Addition of Velocities (Velocity Composition) in Special Relativity'>Addition of Velocities (Velocity Composition) in Special Relativity</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-05-20T06:37:49-05:00" >May 20, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/addition-of-velocities-velocity-composition-in-special-relativity/#comments'>1 Comment</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >The "Addition of Velocities" formula (more correctly, the "Composition of Velocities" formula) in Special Relativity[tex]\frac{v_{AC}}{c}=\frac{ \frac{v_{AB}}{c}+\frac{v_{BC}}{c}…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2024/05/addition_of_velocities.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>robphy</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>robphy</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-05-20 06:37:49</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-09-30 09:15:04</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Addition of Velocities (Velocity Composition) in Special Relativity</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-45596 slide-entry-overview slide-loop-5 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/schrodingers-cat-and-the-qbit/' data-rel='slide-1' class='slide-image' title='Schrödinger’s Cat and the Qbit'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/03/shrodingers-cat.png" class="wp-image-45663 avia-img-lazy-loading-45663 attachment-portfolio size-portfolio wp-post-image" alt="Schrodinger's cat" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/03/shrodingers-cat.png" class="wp-image-45663 avia-img-lazy-loading-45663 attachment-portfolio size-portfolio wp-post-image" alt="Schrodinger's cat" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Schrödinger's Cat and the Qbit"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/schrodingers-cat-and-the-qbit/' title='Schrödinger’s Cat and the Qbit'>Schrödinger’s Cat and the Qbit</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-04-02T08:51:19-05:00" >April 2, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/schrodingers-cat-and-the-qbit/#comments'>27 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >The concept of quantum superposition (or superposition for short) is very counterintuitive, as Schr##\ddot{\text{o}}##dinger noted in 1935 writing [1],…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2024/03/shrodingers-cat.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Mark Stuckey</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Mark Stuckey</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-04-02 08:51:19</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-26 07:18:34</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Schrödinger’s Cat and the Qbit</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-45556 slide-entry-overview slide-loop-6 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/the-slinky-drop-experiment-analysed/' data-rel='slide-1' class='slide-image' title='The Slinky Drop Experiment Analysed'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/03/slinky-drop-experiment.png" class="wp-image-45639 avia-img-lazy-loading-45639 attachment-portfolio size-portfolio wp-post-image" alt="slinky drop experiment" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/03/slinky-drop-experiment.png" class="wp-image-45639 avia-img-lazy-loading-45639 attachment-portfolio size-portfolio wp-post-image" alt="slinky drop experiment" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: The Slinky Drop Experiment Analysed"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/the-slinky-drop-experiment-analysed/' title='The Slinky Drop Experiment Analysed'>The Slinky Drop Experiment Analysed</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-03-28T16:19:54-05:00" >March 28, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/the-slinky-drop-experiment-analysed/#comments'>1 Comment</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >The slinky drop is a rather simple experiment. In its most basic form, it requires only a popular toy for children, a stable hand, and a keen eye.…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2024/03/slinky-drop-experiment.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Orodruin</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Orodruin</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-03-28 16:19:54</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-09-30 09:01:52</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>The Slinky Drop Experiment Analysed</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-48162 slide-entry-overview slide-loop-7 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/infinite-universe/' data-rel='slide-1' class='slide-image' title='Is the Universe Finite, or Is It Infinite?'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/11/infinite-universe.png" class="wp-image-48165 avia-img-lazy-loading-48165 attachment-portfolio size-portfolio wp-post-image" alt="infinite universe" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/11/infinite-universe.png" class="wp-image-48165 avia-img-lazy-loading-48165 attachment-portfolio size-portfolio wp-post-image" alt="infinite universe" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Is the Universe Finite, or Is It Infinite?"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/infinite-universe/' title='Is the Universe Finite, or Is It Infinite?'>Is the Universe Finite, or Is It Infinite?</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-03-16T10:35:04-05:00" >March 16, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/infinite-universe/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Standard cosmological models come in three flavors, open, flat, and closed,[Carroll] whose spatial curvatures are negative, zero, and positive. The open…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2024/11/infinite-universe.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Greg Bernhardt</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Greg Bernhardt</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-03-16 10:35:04</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-11-16 10:45:50</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Is the Universe Finite, or Is It Infinite?</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-44868 slide-entry-overview slide-loop-8 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/how-to-solve-a-multi-atwood-machine-assembly/' data-rel='slide-1' class='slide-image' title='How to Solve a Multi-Atwood Machine Assembly'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/02/Multi-Atwood-Machine-Assembly.png" class="wp-image-45414 avia-img-lazy-loading-45414 attachment-portfolio size-portfolio wp-post-image" alt="Multi-Atwood Machine Assembly" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/02/Multi-Atwood-Machine-Assembly.png" class="wp-image-45414 avia-img-lazy-loading-45414 attachment-portfolio size-portfolio wp-post-image" alt="Multi-Atwood Machine Assembly" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: How to Solve a Multi-Atwood Machine Assembly"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/how-to-solve-a-multi-atwood-machine-assembly/' title='How to Solve a Multi-Atwood Machine Assembly'>How to Solve a Multi-Atwood Machine Assembly</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-02-16T10:18:12-06:00" >February 16, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/how-to-solve-a-multi-atwood-machine-assembly/#comments'>2 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >IntroductionThe figure on the right shows a "double-double" Atwood machine with three ideal pulleys and four masses. All pulleys are released from…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2024/02/Multi-Atwood-Machine-Assembly.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>kuruman</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>kuruman</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-02-16 10:18:12</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 10:52:43</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>How to Solve a Multi-Atwood Machine Assembly</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-44731 slide-entry-overview slide-loop-9 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/the-lambert-w-function-in-finance/' data-rel='slide-1' class='slide-image' title='The Lambert W Function in Finance'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2024/02/Lambert-W-Function-in-Finance.png" class="wp-image-45299 avia-img-lazy-loading-45299 attachment-portfolio size-portfolio wp-post-image" alt="Lambert W Function in Finance" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2024/02/Lambert-W-Function-in-Finance.png" class="wp-image-45299 avia-img-lazy-loading-45299 attachment-portfolio size-portfolio wp-post-image" alt="Lambert W Function in Finance" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: The Lambert W Function in Finance"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/the-lambert-w-function-in-finance/' title='The Lambert W Function in Finance'>The Lambert W Function in Finance</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-02-09T08:25:09-06:00" >February 9, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/the-lambert-w-function-in-finance/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Preamble The classical mathematician practically by instinct views the continuous process as the "real" process, and the discrete process as an approximation…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2024/02/Lambert-W-Function-in-Finance.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Neil Parker</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Neil Parker</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-02-09 08:25:09</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-02-09 08:25:09</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>The Lambert W Function in Finance</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-44769 slide-entry-overview slide-loop-10 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/why-division-by-zero-is-a-bad-idea/' data-rel='slide-1' class='slide-image' title='Why Division by Zero is a Bad Idea'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2015/09/infinity.png" class="wp-image-4849 avia-img-lazy-loading-4849 attachment-portfolio size-portfolio wp-post-image" alt="infinity" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2015/09/infinity.png" class="wp-image-4849 avia-img-lazy-loading-4849 attachment-portfolio size-portfolio wp-post-image" alt="infinity" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Why Division by Zero is a Bad Idea"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/why-division-by-zero-is-a-bad-idea/' title='Why Division by Zero is a Bad Idea'>Why Division by Zero is a Bad Idea</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2024-01-11T10:41:33-06:00" >January 11, 2024</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/why-division-by-zero-is-a-bad-idea/#comments'>33 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >A division by zero is primarily an algebraic question. The reasoning therefore follows the indirect pattern of most algebraic proofs: What if it was allowed? Then…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2015/09/infinity.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2024-01-11 10:41:33</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-11-04 08:08:37</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Why Division by Zero is a Bad Idea</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-44649 slide-entry-overview slide-loop-11 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/digital-filtering-and-exact-reconstruction-of-digital-audio/' data-rel='slide-1' class='slide-image' title='Digital Filtering and Exact Reconstruction of Digital Audio'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/12/digital-audio-1.png" class="wp-image-44679 avia-img-lazy-loading-44679 attachment-portfolio size-portfolio wp-post-image" alt="digital audio guide" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/12/digital-audio-1.png" class="wp-image-44679 avia-img-lazy-loading-44679 attachment-portfolio size-portfolio wp-post-image" alt="digital audio guide" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Digital Filtering and Exact Reconstruction of Digital Audio"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/digital-filtering-and-exact-reconstruction-of-digital-audio/' title='Digital Filtering and Exact Reconstruction of Digital Audio'>Digital Filtering and Exact Reconstruction of Digital Audio</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-12-14T14:50:05-06:00" >December 14, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/digital-filtering-and-exact-reconstruction-of-digital-audio/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction This elaborates some of the claims in my insights article on digital audio. The Sinc Function The first link in my insights article has…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/12/digital-audio-1.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Bill Hobba</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Bill Hobba</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-12-14 14:50:05</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 10:51:35</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Digital Filtering and Exact Reconstruction of Digital Audio</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-44584 slide-entry-overview slide-loop-12 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/digital-audio/' data-rel='slide-1' class='slide-image' title='Introduction to Modern Digital Audio Concepts'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/12/digital-audio-introduction.png" class="wp-image-44674 avia-img-lazy-loading-44674 attachment-portfolio size-portfolio wp-post-image" alt="digital audio intro" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/12/digital-audio-introduction.png" class="wp-image-44674 avia-img-lazy-loading-44674 attachment-portfolio size-portfolio wp-post-image" alt="digital audio intro" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Introduction to Modern Digital Audio Concepts"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/digital-audio/' title='Introduction to Modern Digital Audio Concepts'>Introduction to Modern Digital Audio Concepts</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-12-11T19:08:57-06:00" >December 11, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/digital-audio/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >IntroductionFirst, we need some background in Digital Signals. This can be mathematically quite advanced, but since I would like this…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/12/digital-audio-introduction.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Bill Hobba</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Bill Hobba</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-12-11 19:08:57</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 10:51:06</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Introduction to Modern Digital Audio Concepts</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-44496 slide-entry-overview slide-loop-13 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/series-in-mathematics-from-zeno-to-quantum-theory/' data-rel='slide-1' class='slide-image' title='Series in Mathematics: From Zeno to Quantum Theory'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/11/geometric-series.png" class="wp-image-44636 avia-img-lazy-loading-44636 attachment-portfolio size-portfolio wp-post-image" alt="geometric series" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/11/geometric-series.png" class="wp-image-44636 avia-img-lazy-loading-44636 attachment-portfolio size-portfolio wp-post-image" alt="geometric series" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Series in Mathematics: From Zeno to Quantum Theory"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/series-in-mathematics-from-zeno-to-quantum-theory/' title='Series in Mathematics: From Zeno to Quantum Theory'>Series in Mathematics: From Zeno to Quantum Theory</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-11-07T20:13:49-06:00" >November 7, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/series-in-mathematics-from-zeno-to-quantum-theory/#comments'>2 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/11/geometric-series.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-11-07 20:13:49</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-12-19 13:21:53</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Series in Mathematics: From Zeno to Quantum Theory</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-44438 slide-entry-overview slide-loop-14 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/epsilontic-limits-and-continuity/' data-rel='slide-1' class='slide-image' title='Epsilontic – Limits and Continuity'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/10/epsilontic.png" class="wp-image-44474 avia-img-lazy-loading-44474 attachment-portfolio size-portfolio wp-post-image" alt="Epsilontic limits and continuity" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/10/epsilontic.png" class="wp-image-44474 avia-img-lazy-loading-44474 attachment-portfolio size-portfolio wp-post-image" alt="Epsilontic limits and continuity" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Epsilontic - Limits and Continuity"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/epsilontic-limits-and-continuity/' title='Epsilontic – Limits and Continuity'>Epsilontic – Limits and Continuity</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-10-06T08:41:07-05:00" >October 6, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/epsilontic-limits-and-continuity/#comments'>1 Comment</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Abstract I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/10/epsilontic.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-10-06 08:41:07</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-10-06 08:41:07</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Epsilontic – Limits and Continuity</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-43029 slide-entry-overview slide-loop-15 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/the-poor-mans-milli-ohm-meter/' data-rel='slide-1' class='slide-image' title='The Poor Man’s Milli-Ohm Meter'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/Milli-Ohm.png" class="wp-image-44426 avia-img-lazy-loading-44426 attachment-portfolio size-portfolio wp-post-image" alt="Milli-Ohm" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/Milli-Ohm.png" class="wp-image-44426 avia-img-lazy-loading-44426 attachment-portfolio size-portfolio wp-post-image" alt="Milli-Ohm" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: The Poor Man's Milli-Ohm Meter"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/the-poor-mans-milli-ohm-meter/' title='The Poor Man’s Milli-Ohm Meter'>The Poor Man’s Milli-Ohm Meter</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-09-01T16:17:55-05:00" >September 1, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/the-poor-mans-milli-ohm-meter/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction In a previous article on measuring battery internal resistance, a simple technique for low-resistance measurement was outlined. In this article,…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/08/Milli-Ohm.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Neil Parker</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Neil Parker</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-09-01 16:17:55</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 10:49:47</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>The Poor Man’s Milli-Ohm Meter</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-44253 slide-entry-overview slide-loop-16 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/differential-equation-systems-and-nature/' data-rel='slide-1' class='slide-image' title='Differential Equation Systems and Nature'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/Differential-Equation-Systems-and-Nature.png" class="wp-image-44413 avia-img-lazy-loading-44413 attachment-portfolio size-portfolio wp-post-image" alt="Differential Equation Systems and Nature" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/Differential-Equation-Systems-and-Nature.png" class="wp-image-44413 avia-img-lazy-loading-44413 attachment-portfolio size-portfolio wp-post-image" alt="Differential Equation Systems and Nature" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Differential Equation Systems and Nature"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/differential-equation-systems-and-nature/' title='Differential Equation Systems and Nature'>Differential Equation Systems and Nature</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-08-27T07:20:35-05:00" >August 27, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/differential-equation-systems-and-nature/#comments'>12 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Abstract "Mathematics is the native language of nature." is a phrase that is often used when it comes to explaining why mathematics is all around in natural…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/08/Differential-Equation-Systems-and-Nature.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-08-27 07:20:35</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-01-09 07:51:58</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Differential Equation Systems and Nature</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-43710 slide-entry-overview slide-loop-17 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/precalculus-calculus-and-infinitesimals/' data-rel='slide-1' class='slide-image' title='Beginners Guide to Precalculus, Calculus and Infinitesimals'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/calc-precalc.png" class="wp-image-44245 avia-img-lazy-loading-44245 attachment-portfolio size-portfolio wp-post-image" alt="calc precalc" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/calc-precalc.png" class="wp-image-44245 avia-img-lazy-loading-44245 attachment-portfolio size-portfolio wp-post-image" alt="calc precalc" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Beginners Guide to Precalculus, Calculus and Infinitesimals"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/precalculus-calculus-and-infinitesimals/' title='Beginners Guide to Precalculus, Calculus and Infinitesimals'>Beginners Guide to Precalculus, Calculus and Infinitesimals</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-08-18T01:04:41-05:00" >August 18, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/precalculus-calculus-and-infinitesimals/#comments'>3 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction I am convinced students learn Calculus far too late. In my view, there has never been a good reason for this.In the US, they go through…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/08/calc-precalc.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Bill Hobba</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Bill Hobba</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-08-18 01:04:41</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-08-19 06:58:25</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Beginners Guide to Precalculus, Calculus and Infinitesimals</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-43790 slide-entry-overview slide-loop-18 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/how-to-apply-newtons-second-law-to-variable-mass-systems/' data-rel='slide-1' class='slide-image' title='How to Apply Newton’s Second Law to Variable Mass Systems'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/Variable-Mass-Systems.png" class="wp-image-44219 avia-img-lazy-loading-44219 attachment-portfolio size-portfolio wp-post-image" alt="Variable Mass Systems" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/Variable-Mass-Systems.png" class="wp-image-44219 avia-img-lazy-loading-44219 attachment-portfolio size-portfolio wp-post-image" alt="Variable Mass Systems" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: How to Apply Newton's Second Law to Variable Mass Systems"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/how-to-apply-newtons-second-law-to-variable-mass-systems/' title='How to Apply Newton’s Second Law to Variable Mass Systems'>How to Apply Newton’s Second Law to Variable Mass Systems</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-08-15T08:45:48-05:00" >August 15, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/how-to-apply-newtons-second-law-to-variable-mass-systems/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction The applicability of Newton's second law in the oft-quoted "general form" $$\begin{align}\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{ext}}\end{align}$$…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/08/Variable-Mass-Systems.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>kuruman</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>kuruman</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-08-15 08:45:48</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 09:54:33</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>How to Apply Newton’s Second Law to Variable Mass Systems</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-43536 slide-entry-overview slide-loop-19 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/what-are-numbers/' data-rel='slide-1' class='slide-image' title='What Are Numbers?'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/what-are-numbers.png" class="wp-image-44224 avia-img-lazy-loading-44224 attachment-portfolio size-portfolio wp-post-image" alt="what are numbers" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/08/what-are-numbers.png" class="wp-image-44224 avia-img-lazy-loading-44224 attachment-portfolio size-portfolio wp-post-image" alt="what are numbers" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: What Are Numbers?"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/what-are-numbers/' title='What Are Numbers?'>What Are Numbers?</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-08-14T22:43:45-05:00" >August 14, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/what-are-numbers/#comments'>7 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction When doing mathematics, we usually take for granted what natural numbers, integers, and rationals are. They are pretty intuitive. Going…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/08/what-are-numbers.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Bill Hobba</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Bill Hobba</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-08-14 22:43:45</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-11-01 18:53:56</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>What Are Numbers?</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-43587 slide-entry-overview slide-loop-20 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/introduction-to-the-world-of-algebras/' data-rel='slide-1' class='slide-image' title='Introduction to the World of Algebras'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/07/world-of-algebras.png" class="wp-image-43690 avia-img-lazy-loading-43690 attachment-portfolio size-portfolio wp-post-image" alt="world of algebras" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/07/world-of-algebras.png" class="wp-image-43690 avia-img-lazy-loading-43690 attachment-portfolio size-portfolio wp-post-image" alt="world of algebras" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Introduction to the World of Algebras"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/introduction-to-the-world-of-algebras/' title='Introduction to the World of Algebras'>Introduction to the World of Algebras</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-07-08T07:35:07-05:00" >July 8, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/introduction-to-the-world-of-algebras/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/07/world-of-algebras.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-07-08 07:35:07</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-07-22 09:22:21</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Introduction to the World of Algebras</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-43317 slide-entry-overview slide-loop-21 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/why-chatgpt-is-not-reliable/' data-rel='slide-1' class='slide-image' title='Why ChatGPT AI Is Not Reliable'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/07/chatgpt-reliable.png" class="wp-image-43679 avia-img-lazy-loading-43679 attachment-portfolio size-portfolio wp-post-image" alt="chatgpt-reliable" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/07/chatgpt-reliable.png" class="wp-image-43679 avia-img-lazy-loading-43679 attachment-portfolio size-portfolio wp-post-image" alt="chatgpt-reliable" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Why ChatGPT AI Is Not Reliable"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/why-chatgpt-is-not-reliable/' title='Why ChatGPT AI Is Not Reliable'>Why ChatGPT AI Is Not Reliable</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-07-05T06:08:15-05:00" >July 5, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/why-chatgpt-is-not-reliable/#comments'>138 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >I'll start with the simple fact: ChatGPT is not a reliable answerer to questions.To try to explain why from scratch would be a heavy lift, but fortunately,…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/07/chatgpt-reliable.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>PeterDonis</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>PeterDonis</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-07-05 06:08:15</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-11-10 13:43:06</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Why ChatGPT AI Is Not Reliable</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-43447 slide-entry-overview slide-loop-22 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/what-are-infinitesimals-simple-version/' data-rel='slide-1' class='slide-image' title='What Are Infinitesimals – Simple Version'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/05/Infinitesimals.png" class="wp-image-43343 avia-img-lazy-loading-43343 attachment-portfolio size-portfolio wp-post-image" alt="Infinitesimals" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/05/Infinitesimals.png" class="wp-image-43343 avia-img-lazy-loading-43343 attachment-portfolio size-portfolio wp-post-image" alt="Infinitesimals" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: What Are Infinitesimals - Simple Version"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/what-are-infinitesimals-simple-version/' title='What Are Infinitesimals – Simple Version'>What Are Infinitesimals – Simple Version</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-06-04T12:49:41-05:00" >June 4, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/what-are-infinitesimals-simple-version/#comments'>5 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/05/Infinitesimals.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Bill Hobba</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Bill Hobba</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-06-04 12:49:41</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 09:52:41</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>What Are Infinitesimals – Simple Version</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-43331 slide-entry-overview slide-loop-23 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/how-quantum-information-theory-solves-the-only-mystery-of-quantum-mechanics/' data-rel='slide-1' class='slide-image' title='How Quantum Information Theory Solves “the only mystery” of Quantum Mechanics'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/06/quantum-information2.png" class="wp-image-43433 avia-img-lazy-loading-43433 attachment-portfolio size-portfolio wp-post-image" alt="quantum information" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/06/quantum-information2.png" class="wp-image-43433 avia-img-lazy-loading-43433 attachment-portfolio size-portfolio wp-post-image" alt="quantum information" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: How Quantum Information Theory Solves "the only mystery" of Quantum Mechanics"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/how-quantum-information-theory-solves-the-only-mystery-of-quantum-mechanics/' title='How Quantum Information Theory Solves “the only mystery” of Quantum Mechanics'>How Quantum Information Theory Solves “the only mystery” of Quantum Mechanics</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-06-03T08:49:50-05:00" >June 3, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/how-quantum-information-theory-solves-the-only-mystery-of-quantum-mechanics/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >In Chapter 37 of "The Feynman Lectures on Physics Volume 1," Richard Feynman famously wrote that the mystery of wave-particle duality in the double-slit…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/06/quantum-information2.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Mark Stuckey</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Mark Stuckey</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-06-03 08:49:50</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-06-06 09:16:27</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>How Quantum Information Theory Solves “the only mystery” of Quantum Mechanics</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-43241 slide-entry-overview slide-loop-24 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/what-are-infinitesimals/' data-rel='slide-1' class='slide-image' title='What Are Infinitesimals – Advanced Version'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/05/Infinitesimals.png" class="wp-image-43343 avia-img-lazy-loading-43343 attachment-portfolio size-portfolio wp-post-image" alt="Infinitesimals" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/05/Infinitesimals.png" class="wp-image-43343 avia-img-lazy-loading-43343 attachment-portfolio size-portfolio wp-post-image" alt="Infinitesimals" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: What Are Infinitesimals - Advanced Version"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/what-are-infinitesimals/' title='What Are Infinitesimals – Advanced Version'>What Are Infinitesimals – Advanced Version</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-05-24T03:01:58-05:00" >May 24, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/what-are-infinitesimals/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction When I learned calculus, the intuitive idea of infinitesimal was used. These are real numbers so small that, for all practical purposes (say…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/05/Infinitesimals.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Bill Hobba</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Bill Hobba</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-05-24 03:01:58</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 09:51:21</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>What Are Infinitesimals – Advanced Version</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-43063 slide-entry-overview slide-loop-25 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/opinion-when-pro-scientists-explain-using-pop-science/' data-rel='slide-1' class='slide-image' title='When Pro Scientists Explain Using Pop Science'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/04/pro-scientist-pop-science.png" class="wp-image-43125 avia-img-lazy-loading-43125 attachment-portfolio size-portfolio wp-post-image" alt="Carl Sagan" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/04/pro-scientist-pop-science.png" class="wp-image-43125 avia-img-lazy-loading-43125 attachment-portfolio size-portfolio wp-post-image" alt="Carl Sagan" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: When Pro Scientists Explain Using Pop Science"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/opinion-when-pro-scientists-explain-using-pop-science/' title='When Pro Scientists Explain Using Pop Science'>When Pro Scientists Explain Using Pop Science</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-04-30T12:05:43-05:00" >April 30, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/opinion-when-pro-scientists-explain-using-pop-science/#comments'>19 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Abstract There is so much to say about the many endeavors by professional scientists to explain to us the world. The list is long: Carl Sagan, Harald…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/04/pro-scientist-pop-science.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-04-30 12:05:43</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-10-07 08:04:41</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>When Pro Scientists Explain Using Pop Science</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-42985 slide-entry-overview slide-loop-26 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/the-art-of-integration/' data-rel='slide-1' class='slide-image' title='The Art of Integration'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/04/art-of-integration.png" class="wp-image-43017 avia-img-lazy-loading-43017 attachment-portfolio size-portfolio wp-post-image" alt="art of integration" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/04/art-of-integration.png" class="wp-image-43017 avia-img-lazy-loading-43017 attachment-portfolio size-portfolio wp-post-image" alt="art of integration" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: The Art of Integration"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/the-art-of-integration/' title='The Art of Integration'>The Art of Integration</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-04-10T07:36:52-05:00" >April 10, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/the-art-of-integration/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Abstract My school teacher used to say "Everybody can differentiate, but it takes an artist to integrate." The mathematical reason behind this phrase…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/04/art-of-integration.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-04-10 07:36:52</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-01-07 06:01:07</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>The Art of Integration</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-42356 slide-entry-overview slide-loop-27 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/a-lesson-in-teaching-physics-you-cant-give-it-away/' data-rel='slide-1' class='slide-image' title='A Lesson In Teaching Physics: You Can’t Give It Away'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/03/teaching-physics.png" class="wp-image-42957 avia-img-lazy-loading-42957 attachment-portfolio size-portfolio wp-post-image" alt="teaching physics" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/03/teaching-physics.png" class="wp-image-42957 avia-img-lazy-loading-42957 attachment-portfolio size-portfolio wp-post-image" alt="teaching physics" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: A Lesson In Teaching Physics: You Can't Give It Away"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/a-lesson-in-teaching-physics-you-cant-give-it-away/' title='A Lesson In Teaching Physics: You Can’t Give It Away'>A Lesson In Teaching Physics: You Can’t Give It Away</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-03-16T08:06:33-05:00" >March 16, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/a-lesson-in-teaching-physics-you-cant-give-it-away/#comments'>1 Comment</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >A central principle of Physics Forums regarding homework help is not to provide solutions on demand but to guide students along a path to the answer. The…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/03/teaching-physics.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>kuruman</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>kuruman</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-03-16 08:06:33</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 09:47:41</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>A Lesson In Teaching Physics: You Can’t Give It Away</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-42892 slide-entry-overview slide-loop-28 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/an-overview-of-complex-differentiation-and-integration/' data-rel='slide-1' class='slide-image' title='An Overview of Complex Differentiation and Integration'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/03/integration_differentiation.png" class="wp-image-42943 avia-img-lazy-loading-42943 attachment-portfolio size-portfolio wp-post-image" alt="integration and complex differentiation" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/03/integration_differentiation.png" class="wp-image-42943 avia-img-lazy-loading-42943 attachment-portfolio size-portfolio wp-post-image" alt="integration and complex differentiation" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: An Overview of Complex Differentiation and Integration"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/an-overview-of-complex-differentiation-and-integration/' title='An Overview of Complex Differentiation and Integration'>An Overview of Complex Differentiation and Integration</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-03-14T08:24:25-05:00" >March 14, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/an-overview-of-complex-differentiation-and-integration/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Abstract I want to shed some light on complex analysis without getting all the technical details in the way which are necessary for the precise treatments…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/03/integration_differentiation.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-03-14 08:24:25</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-06-02 15:46:27</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>An Overview of Complex Differentiation and Integration</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-41870 slide-entry-overview slide-loop-29 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/how-to-measure-internal-resistance-of-a-battery/' data-rel='slide-1' class='slide-image' title='How to Measure Internal Resistance of a Battery'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/02/Measure-Internal-Resistance-of-Battery.png" class="wp-image-42846 avia-img-lazy-loading-42846 attachment-portfolio size-portfolio wp-post-image" alt="Measure Internal Resistance of Battery" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/02/Measure-Internal-Resistance-of-Battery.png" class="wp-image-42846 avia-img-lazy-loading-42846 attachment-portfolio size-portfolio wp-post-image" alt="Measure Internal Resistance of Battery" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: How to Measure Internal Resistance of a Battery"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/how-to-measure-internal-resistance-of-a-battery/' title='How to Measure Internal Resistance of a Battery'>How to Measure Internal Resistance of a Battery</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-02-14T08:49:27-06:00" >February 14, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/how-to-measure-internal-resistance-of-a-battery/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction A commonly encountered school-level Physics practical is the determination of the internal resistance of a battery - typically an AA or D…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/02/Measure-Internal-Resistance-of-Battery.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Neil Parker</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Neil Parker</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-02-14 08:49:27</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 09:46:45</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>How to Measure Internal Resistance of a Battery</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-42711 slide-entry-overview slide-loop-30 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/when-lie-groups-became-physics/' data-rel='slide-1' class='slide-image' title='When Lie Groups Became Physics'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/02/lie-group-physics.png" class="wp-image-42798 avia-img-lazy-loading-42798 attachment-portfolio size-portfolio wp-post-image" alt="lie group physics" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/02/lie-group-physics.png" class="wp-image-42798 avia-img-lazy-loading-42798 attachment-portfolio size-portfolio wp-post-image" alt="lie group physics" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: When Lie Groups Became Physics"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/when-lie-groups-became-physics/' title='When Lie Groups Became Physics'>When Lie Groups Became Physics</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-02-08T07:44:28-06:00" >February 8, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/when-lie-groups-became-physics/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/02/lie-group-physics.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-02-08 07:44:28</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-04-05 20:15:42</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>When Lie Groups Became Physics</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-29861 slide-entry-overview slide-loop-31 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/why-there-are-maximum-mass-limits-for-compact-objects/' data-rel='slide-1' class='slide-image' title='Why There Are Maximum Mass Limits for Compact Objects'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/01/white-dwarfs.png" class="wp-image-42721 avia-img-lazy-loading-42721 attachment-portfolio size-portfolio wp-post-image" alt="white dwarfs" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/01/white-dwarfs.png" class="wp-image-42721 avia-img-lazy-loading-42721 attachment-portfolio size-portfolio wp-post-image" alt="white dwarfs" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Why There Are Maximum Mass Limits for Compact Objects"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/why-there-are-maximum-mass-limits-for-compact-objects/' title='Why There Are Maximum Mass Limits for Compact Objects'>Why There Are Maximum Mass Limits for Compact Objects</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-01-27T08:01:59-06:00" >January 27, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/why-there-are-maximum-mass-limits-for-compact-objects/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >In this article, we will look at why there are maximum mass limits for objects that are supported against gravity by degeneracy pressure instead of kinetic…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/01/white-dwarfs.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>PeterDonis</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>PeterDonis</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-01-27 08:01:59</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-03-08 19:25:41</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Why There Are Maximum Mass Limits for Compact Objects</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-42435 slide-entry-overview slide-loop-32 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/oppenheimer-snyder-model-of-gravitational-collapse-implications/' data-rel='slide-1' class='slide-image' title='Oppenheimer-Snyder Model of Gravitational Collapse: Implications'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/01/gravity-collapse3.png" class="wp-image-42698 avia-img-lazy-loading-42698 attachment-portfolio size-portfolio wp-post-image" alt="gravity collapse" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/01/gravity-collapse3.png" class="wp-image-42698 avia-img-lazy-loading-42698 attachment-portfolio size-portfolio wp-post-image" alt="gravity collapse" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Oppenheimer-Snyder Model of Gravitational Collapse: Implications"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/oppenheimer-snyder-model-of-gravitational-collapse-implications/' title='Oppenheimer-Snyder Model of Gravitational Collapse: Implications'>Oppenheimer-Snyder Model of Gravitational Collapse: Implications</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-01-15T21:27:43-06:00" >January 15, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/oppenheimer-snyder-model-of-gravitational-collapse-implications/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn the last article in this series, we finished up with a metric for the Oppenheimer-Snyder…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/01/gravity-collapse3.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>PeterDonis</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>PeterDonis</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-01-15 21:27:43</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-11-24 11:58:00</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Oppenheimer-Snyder Model of Gravitational Collapse: Implications</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-14113 slide-entry-overview slide-loop-33 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/what-are-tensors-and-why-are-they-used-in-relativity/' data-rel='slide-1' class='slide-image' title='What Are Tensors and Why Are They Used in Relativity?'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2020/08/tensors-relativity.png" class="wp-image-44146 avia-img-lazy-loading-44146 attachment-portfolio size-portfolio wp-post-image" alt="tensors relativity" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2020/08/tensors-relativity.png" class="wp-image-44146 avia-img-lazy-loading-44146 attachment-portfolio size-portfolio wp-post-image" alt="tensors relativity" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: What Are Tensors and Why Are They Used in Relativity?"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/what-are-tensors-and-why-are-they-used-in-relativity/' title='What Are Tensors and Why Are They Used in Relativity?'>What Are Tensors and Why Are They Used in Relativity?</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-01-11T07:57:09-06:00" >January 11, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/what-are-tensors-and-why-are-they-used-in-relativity/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >If you try learning general relativity, and sometimes special relativity, on your own, you will undoubtedly run into tensors. This article will outline…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2020/08/tensors-relativity.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Isaac0427</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Isaac0427</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-01-11 07:57:09</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 09:45:39</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>What Are Tensors and Why Are They Used in Relativity?</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-42415 slide-entry-overview slide-loop-34 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/oppenheimer-snyder-model-of-gravitational-collapse-mathematical-details/' data-rel='slide-1' class='slide-image' title='Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/12/gravity-collapse2.png" class="wp-image-42659 avia-img-lazy-loading-42659 attachment-portfolio size-portfolio wp-post-image" alt="gravity collapse" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/12/gravity-collapse2.png" class="wp-image-42659 avia-img-lazy-loading-42659 attachment-portfolio size-portfolio wp-post-image" alt="gravity collapse" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/oppenheimer-snyder-model-of-gravitational-collapse-mathematical-details/' title='Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details'>Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-01-09T08:01:01-06:00" >January 9, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/oppenheimer-snyder-model-of-gravitational-collapse-mathematical-details/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsIn a previous article, I described in general terms the model of gravitational…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/12/gravity-collapse2.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>PeterDonis</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>PeterDonis</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-01-09 08:01:01</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-04-02 10:15:26</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Oppenheimer-Snyder Model of Gravitational Collapse: Mathematical Details</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-42488 slide-entry-overview slide-loop-35 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/when-discussing-the-twin-paradox-read-this-first/' data-rel='slide-1' class='slide-image' title='When Discussing the Twin Paradox: Read This First'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2023/01/twin-paradox.png" class="wp-image-42603 avia-img-lazy-loading-42603 attachment-portfolio size-portfolio wp-post-image" alt="twin paradox" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2023/01/twin-paradox.png" class="wp-image-42603 avia-img-lazy-loading-42603 attachment-portfolio size-portfolio wp-post-image" alt="twin paradox" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: When Discussing the Twin Paradox: Read This First"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/when-discussing-the-twin-paradox-read-this-first/' title='When Discussing the Twin Paradox: Read This First'>When Discussing the Twin Paradox: Read This First</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2023-01-02T08:55:15-06:00" >January 2, 2023</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/when-discussing-the-twin-paradox-read-this-first/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >This article is intended for anyone who wants to start a thread here at Physics Forums on the twin paradox. There are already many, many threads here on…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2023/01/twin-paradox.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>PeterDonis</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>PeterDonis</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2023-01-02 08:55:15</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-09-30 08:56:57</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>When Discussing the Twin Paradox: Read This First</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-23731 slide-entry-overview slide-loop-36 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/the-oppenheimer-snyder-model-of-gravitational-collapse-an-overview/' data-rel='slide-1' class='slide-image' title='The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/12/gravity-collapse.png" class="wp-image-42464 avia-img-lazy-loading-42464 attachment-portfolio size-portfolio wp-post-image" alt="gravity collapse" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/12/gravity-collapse.png" class="wp-image-42464 avia-img-lazy-loading-42464 attachment-portfolio size-portfolio wp-post-image" alt="gravity collapse" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/the-oppenheimer-snyder-model-of-gravitational-collapse-an-overview/' title='The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview'>The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-12-28T11:07:47-06:00" >December 28, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/the-oppenheimer-snyder-model-of-gravitational-collapse-an-overview/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Part 1: OverviewPart 2: Mathematical DetailsPart 3: ImplicationsMost people who have spent any time at all studying GR are familiar with the…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/12/gravity-collapse.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>PeterDonis</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>PeterDonis</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-12-28 11:07:47</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-09-30 08:59:43</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>The Oppenheimer-Snyder Model of Gravitational Collapse: An Overview</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-42298 slide-entry-overview slide-loop-37 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/subtleties-overlooked-in-friction-questions-object-slides-down-ramp/' data-rel='slide-1' class='slide-image' title='Subtleties Overlooked in Friction Questions: Object Slides Down Ramp'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/12/object-slide-down-ramp-physics-1.png" class="wp-image-42325 avia-img-lazy-loading-42325 attachment-portfolio size-portfolio wp-post-image" alt="object slide down ramp physics" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/12/object-slide-down-ramp-physics-1.png" class="wp-image-42325 avia-img-lazy-loading-42325 attachment-portfolio size-portfolio wp-post-image" alt="object slide down ramp physics" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Subtleties Overlooked in Friction Questions: Object Slides Down Ramp"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/subtleties-overlooked-in-friction-questions-object-slides-down-ramp/' title='Subtleties Overlooked in Friction Questions: Object Slides Down Ramp'>Subtleties Overlooked in Friction Questions: Object Slides Down Ramp</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-12-06T09:22:09-06:00" >December 6, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/subtleties-overlooked-in-friction-questions-object-slides-down-ramp/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Problem statement (simplified) An object slides down a ramp at angle θ to encounter level ground. Both surfaces have kinetic friction: μ' on the ramp,…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/12/object-slide-down-ramp-physics-1.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Derek Bolton</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Derek Bolton</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-12-06 09:22:09</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2022-12-21 00:14:58</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Subtleties Overlooked in Friction Questions: Object Slides Down Ramp</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-42189 slide-entry-overview slide-loop-38 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/classification-of-mathematics-by-42-branches/' data-rel='slide-1' class='slide-image' title='Classification of Mathematics by 42 Branches'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/10/math-classifications.png" class="wp-image-42262 avia-img-lazy-loading-42262 attachment-portfolio size-portfolio wp-post-image" alt="math classifications" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/10/math-classifications.png" class="wp-image-42262 avia-img-lazy-loading-42262 attachment-portfolio size-portfolio wp-post-image" alt="math classifications" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Classification of Mathematics by 42 Branches"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/classification-of-mathematics-by-42-branches/' title='Classification of Mathematics by 42 Branches'>Classification of Mathematics by 42 Branches</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-10-25T09:06:07-05:00" >October 25, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/classification-of-mathematics-by-42-branches/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" > I often read questions about our classification scheme that we use on physicsforums.com to sort posts by science fields and subjects, what has…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/10/math-classifications.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-10-25 09:06:07</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-09-30 08:54:18</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Classification of Mathematics by 42 Branches</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-42135 slide-entry-overview slide-loop-39 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/reduction-of-order-for-recursions/' data-rel='slide-1' class='slide-image' title='Reduction of Order For Recursions'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/10/recursion-math.png" class="wp-image-42187 avia-img-lazy-loading-42187 attachment-portfolio size-portfolio wp-post-image" alt="recursion" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/10/recursion-math.png" class="wp-image-42187 avia-img-lazy-loading-42187 attachment-portfolio size-portfolio wp-post-image" alt="recursion" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Reduction of Order For Recursions"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/reduction-of-order-for-recursions/' title='Reduction of Order For Recursions'>Reduction of Order For Recursions</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-10-04T09:30:24-05:00" >October 4, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/reduction-of-order-for-recursions/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >This is not meant as a full introduction to recursion relations but it should suffice for just about any level of the student.Most of us remember recursion…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/10/recursion-math.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>topsquark</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>topsquark</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-10-04 09:30:24</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-04-02 10:17:17</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Reduction of Order For Recursions</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-42071 slide-entry-overview slide-loop-40 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/counting-to-p-adic-calculus-all-number-systems-that-we-have/' data-rel='slide-1' class='slide-image' title='Counting to p-adic Calculus: All Number Systems That We Have'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/09/history-of-numbers.png" class="wp-image-42106 avia-img-lazy-loading-42106 attachment-portfolio size-portfolio wp-post-image" alt="history of numbers" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/09/history-of-numbers.png" class="wp-image-42106 avia-img-lazy-loading-42106 attachment-portfolio size-portfolio wp-post-image" alt="history of numbers" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Counting to p-adic Calculus: All Number Systems That We Have"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/counting-to-p-adic-calculus-all-number-systems-that-we-have/' title='Counting to p-adic Calculus: All Number Systems That We Have'>Counting to p-adic Calculus: All Number Systems That We Have</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-09-18T15:11:13-05:00" >September 18, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/counting-to-p-adic-calculus-all-number-systems-that-we-have/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >An entire book could easily be written about the history of numbers from ancient Babylon and India, over Abu Dscha'far Muhammad ibn Musa al-Chwarizmi (##\sim…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/09/history-of-numbers.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-09-18 15:11:13</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-08-15 09:04:11</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Counting to p-adic Calculus: All Number Systems That We Have</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-42024 slide-entry-overview slide-loop-41 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/intro-to-evariste-galois-and-galois-theory/' data-rel='slide-1' class='slide-image' title='Évariste Galois and His Theory'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/09/evariste-galois.png" class="wp-image-42059 avia-img-lazy-loading-42059 attachment-portfolio size-portfolio wp-post-image" alt="evariste galois" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/09/evariste-galois.png" class="wp-image-42059 avia-img-lazy-loading-42059 attachment-portfolio size-portfolio wp-post-image" alt="evariste galois" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Évariste Galois and His Theory"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/intro-to-evariste-galois-and-galois-theory/' title='Évariste Galois and His Theory'>Évariste Galois and His Theory</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-09-05T19:40:52-05:00" >September 5, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/intro-to-evariste-galois-and-galois-theory/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" > * Oct. 25th, 1811 † May 31st, 1832 ... or why squaring the circle is doomed. Galois died in a duel at the age of twenty. Yet, he gave…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/09/evariste-galois.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-09-05 19:40:52</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2023-08-15 09:04:12</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Évariste Galois and His Theory</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-41907 slide-entry-overview slide-loop-42 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/yardsticks-to-metric-tensor-fields/' data-rel='slide-1' class='slide-image' title='Yardsticks to Metric Tensor Fields'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/08/definition-differences.png" class="wp-image-41991 avia-img-lazy-loading-41991 attachment-portfolio size-portfolio wp-post-image" alt="definition differences" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/08/definition-differences.png" class="wp-image-41991 avia-img-lazy-loading-41991 attachment-portfolio size-portfolio wp-post-image" alt="definition differences" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Yardsticks to Metric Tensor Fields"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/yardsticks-to-metric-tensor-fields/' title='Yardsticks to Metric Tensor Fields'>Yardsticks to Metric Tensor Fields</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-08-27T10:57:47-05:00" >August 27, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/yardsticks-to-metric-tensor-fields/#comments'>1 Comment</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist,…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/08/definition-differences.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-08-27 10:57:47</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-09-30 09:20:21</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Yardsticks to Metric Tensor Fields</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-41842 slide-entry-overview slide-loop-43 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/programming-an-atmega8a-using-arduino/' data-rel='slide-1' class='slide-image' title='Programming an ATmega8A using Arduino'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/08/ATmega8A.png" class="wp-image-41898 avia-img-lazy-loading-41898 attachment-portfolio size-portfolio wp-post-image" alt="ATmega8A" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/08/ATmega8A.png" class="wp-image-41898 avia-img-lazy-loading-41898 attachment-portfolio size-portfolio wp-post-image" alt="ATmega8A" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Programming an ATmega8A using Arduino"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/programming-an-atmega8a-using-arduino/' title='Programming an ATmega8A using Arduino'>Programming an ATmega8A using Arduino</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-08-13T07:50:26-05:00" >August 13, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/programming-an-atmega8a-using-arduino/#comments'>2 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >If you are interested in programming and electronics, you probably do not need an introduction to Arduino. If you want to make your Arduino projects permanent,…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/08/ATmega8A.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Wrichik Basu</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Wrichik Basu</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-08-13 07:50:26</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2022-08-16 03:03:39</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Programming an ATmega8A using Arduino</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-41797 slide-entry-overview slide-loop-44 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/p-vs-np-conjecture-calculations-and-meaning/' data-rel='slide-1' class='slide-image' title='P vs. NP and what is a Turing Machine (TM)?'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/07/pvsnp.png" class="wp-image-41832 avia-img-lazy-loading-41832 attachment-portfolio size-portfolio wp-post-image" alt="pvsnp" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/07/pvsnp.png" class="wp-image-41832 avia-img-lazy-loading-41832 attachment-portfolio size-portfolio wp-post-image" alt="pvsnp" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: P vs. NP and what is a Turing Machine (TM)?"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/p-vs-np-conjecture-calculations-and-meaning/' title='P vs. NP and what is a Turing Machine (TM)?'>P vs. NP and what is a Turing Machine (TM)?</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-07-25T08:13:16-05:00" >July 25, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/p-vs-np-conjecture-calculations-and-meaning/#comments'>1 Comment</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >P or NP This article deals with the complexity of calculations and in particular the meaning of ##P\stackrel{?}{\neq}NP## Before we explain what P and…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/07/pvsnp.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-07-25 08:13:16</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2022-07-25 09:22:26</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>P vs. NP and what is a Turing Machine (TM)?</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-41743 slide-entry-overview slide-loop-45 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/quantum-computing-for-beginners/' data-rel='slide-1' class='slide-image' title='Quantum Computing for Beginners'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/07/quantum-computers-101.png" class="wp-image-41754 avia-img-lazy-loading-41754 attachment-portfolio size-portfolio wp-post-image" alt="quantum computers 101" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/07/quantum-computers-101.png" class="wp-image-41754 avia-img-lazy-loading-41754 attachment-portfolio size-portfolio wp-post-image" alt="quantum computers 101" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Quantum Computing for Beginners"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/quantum-computing-for-beginners/' title='Quantum Computing for Beginners'>Quantum Computing for Beginners</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-07-14T21:35:57-05:00" >July 14, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/quantum-computing-for-beginners/#comments'>5 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >This article provides an accessible introduction to quantum computing, a cutting-edge technology that processes certain types of information much faster…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/07/quantum-computers-101.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Bob Walance</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Bob Walance</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-07-14 21:35:57</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-11-16 08:15:06</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Quantum Computing for Beginners</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-41423 slide-entry-overview slide-loop-46 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/a-physics-misconception-with-gauss-law/' data-rel='slide-1' class='slide-image' title='A Physics Misconception with Gauss’ Law'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/06/gauss-law-misconceptions.png" class="wp-image-41690 avia-img-lazy-loading-41690 attachment-portfolio size-portfolio wp-post-image" alt="gauss law misconceptions" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/06/gauss-law-misconceptions.png" class="wp-image-41690 avia-img-lazy-loading-41690 attachment-portfolio size-portfolio wp-post-image" alt="gauss law misconceptions" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: A Physics Misconception with Gauss' Law"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/a-physics-misconception-with-gauss-law/' title='A Physics Misconception with Gauss’ Law'>A Physics Misconception with Gauss’ Law</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-06-12T09:34:50-05:00" >June 12, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/a-physics-misconception-with-gauss-law/#comments'>3 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction It is relatively common to see the following type of argument: The surface area is ##A## and the enclosed charge is ##Q##. The electric…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/06/gauss-law-misconceptions.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Orodruin</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Orodruin</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-06-12 09:34:50</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2022-06-12 09:34:50</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>A Physics Misconception with Gauss’ Law</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-41241 slide-entry-overview slide-loop-47 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/how-to-model-a-magnet-falling-through-a-conducting-pipe/' data-rel='slide-1' class='slide-image' title='How to Model a Magnet Falling Through a Conducting Pipe'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/06/model-magent2.png" class="wp-image-41681 avia-img-lazy-loading-41681 attachment-portfolio size-portfolio wp-post-image" alt="model magnet" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/06/model-magent2.png" class="wp-image-41681 avia-img-lazy-loading-41681 attachment-portfolio size-portfolio wp-post-image" alt="model magnet" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: How to Model a Magnet Falling Through a Conducting Pipe"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/how-to-model-a-magnet-falling-through-a-conducting-pipe/' title='How to Model a Magnet Falling Through a Conducting Pipe'>How to Model a Magnet Falling Through a Conducting Pipe</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-06-11T06:58:05-05:00" >June 11, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/how-to-model-a-magnet-falling-through-a-conducting-pipe/#comments'>7 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction In an earlier article, we examined a magnet falling through a solenoid. We argued that the point dipole model can account for the basic features…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/06/model-magent2.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>kuruman</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>kuruman</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-06-11 06:58:05</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2022-06-11 06:58:05</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>How to Model a Magnet Falling Through a Conducting Pipe</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-40829 slide-entry-overview slide-loop-48 slide-parity-odd av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/how-to-model-a-magnet-falling-through-a-solenoid/' data-rel='slide-1' class='slide-image' title='How to Model a Magnet Falling Through a Solenoid'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/05/model-magent.png" class="wp-image-41595 avia-img-lazy-loading-41595 attachment-portfolio size-portfolio wp-post-image" alt="model magnet" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/05/model-magent.png" class="wp-image-41595 avia-img-lazy-loading-41595 attachment-portfolio size-portfolio wp-post-image" alt="model magnet" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: How to Model a Magnet Falling Through a Solenoid"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/how-to-model-a-magnet-falling-through-a-solenoid/' title='How to Model a Magnet Falling Through a Solenoid'>How to Model a Magnet Falling Through a Solenoid</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-06-05T12:48:47-05:00" >June 5, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/how-to-model-a-magnet-falling-through-a-solenoid/#comments'>11 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Introduction Modeling a magnet realistically is a task best done numerically. Even the simplified model of two separated disks with uniform surface…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/05/model-magent.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>kuruman</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>kuruman</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-06-05 12:48:47</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2022-06-11 07:07:53</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>How to Model a Magnet Falling Through a Solenoid</span></span></span></article></div><div class="slide-entry-wrap"><article class='slide-entry flex_column post-entry post-entry-41012 slide-entry-overview slide-loop-49 slide-parity-odd av_one_third first real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/how-can-we-jump-when-the-ground-does-no-work/' data-rel='slide-1' class='slide-image' title='How Can We Jump When the Ground Does No Work?'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/05/why-can-we-jump.png" class="wp-image-41451 avia-img-lazy-loading-41451 attachment-portfolio size-portfolio wp-post-image" alt="why can we jump" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/05/why-can-we-jump.png" class="wp-image-41451 avia-img-lazy-loading-41451 attachment-portfolio size-portfolio wp-post-image" alt="why can we jump" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: How Can We Jump When the Ground Does No Work?"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/how-can-we-jump-when-the-ground-does-no-work/' title='How Can We Jump When the Ground Does No Work?'>How Can We Jump When the Ground Does No Work?</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-05-30T16:09:36-05:00" >May 30, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/how-can-we-jump-when-the-ground-does-no-work/#comments'>20 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >It is relatively common on Physics Forums to see arguments that are effectively similar to the following: When we jump off the ground, the ground does…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/05/why-can-we-jump.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Orodruin</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Orodruin</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-05-30 16:09:36</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-08-18 09:41:49</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>How Can We Jump When the Ground Does No Work?</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-40500 slide-entry-overview slide-loop-50 slide-parity-even av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/the-history-and-importance-of-the-riemann-hypothesis/' data-rel='slide-1' class='slide-image' title='The History and Importance of the Riemann Hypothesis'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/05/Riemann-Hypothesis-History.png" class="wp-image-41099 avia-img-lazy-loading-41099 attachment-portfolio size-portfolio wp-post-image" alt="Riemann Hypothesis History" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/05/Riemann-Hypothesis-History.png" class="wp-image-41099 avia-img-lazy-loading-41099 attachment-portfolio size-portfolio wp-post-image" alt="Riemann Hypothesis History" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: The History and Importance of the Riemann Hypothesis"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/the-history-and-importance-of-the-riemann-hypothesis/' title='The History and Importance of the Riemann Hypothesis'>The History and Importance of the Riemann Hypothesis</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-05-21T11:29:54-05:00" >May 21, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/the-history-and-importance-of-the-riemann-hypothesis/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Riemann Hypothesis History The Riemann Hypothesis is one of the most famous and long-standing unsolved problems in mathematics, specifically in the field…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/05/Riemann-Hypothesis-History.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>fresh_42</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>fresh_42</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-05-21 11:29:54</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2024-09-30 09:04:58</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>The History and Importance of the Riemann Hypothesis</span></span></span></article><article class='slide-entry flex_column post-entry post-entry-40779 slide-entry-overview slide-loop-51 slide-parity-odd post-entry-last av_one_third real-thumbnail posttype-post post-format-standard' itemscope="itemscope" itemtype="https://schema.org/CreativeWork" ><a href='https://www.physicsforums.com/insights/symmetry-arguments-and-the-infinite-wire-with-a-current/' data-rel='slide-1' class='slide-image' title='Symmetry Arguments and the Infinite Wire with a Current'><img data-lazyloaded="1" src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMTM1IiB2aWV3Qm94PSIwIDAgMjQwIDEzNSI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgZmlsbD0iI2NmZDRkYiIvPjwvc3ZnPg==" decoding="async" loading="lazy" width="240" height="135" data-src="https://www.physicsforums.com/insights/wp-content/uploads/2022/05/magnetism-current.png" class="wp-image-40916 avia-img-lazy-loading-40916 attachment-portfolio size-portfolio wp-post-image" alt="magnetism current" /><noscript><img decoding="async" loading="lazy" width="240" height="135" src="https://www.physicsforums.com/insights/wp-content/uploads/2022/05/magnetism-current.png" class="wp-image-40916 avia-img-lazy-loading-40916 attachment-portfolio size-portfolio wp-post-image" alt="magnetism current" /></noscript></a><div class="slide-content"><header class="entry-content-header" aria-label="Slide: Symmetry Arguments and the Infinite Wire with a Current"><h3 class='slide-entry-title entry-title ' itemprop="headline" ><a href='https://www.physicsforums.com/insights/symmetry-arguments-and-the-infinite-wire-with-a-current/' title='Symmetry Arguments and the Infinite Wire with a Current'>Symmetry Arguments and the Infinite Wire with a Current</a></h3><span class="av-vertical-delimiter"></span></header><div class="slide-meta"><time class='slide-meta-time updated' itemprop="datePublished" datetime="2022-05-17T07:56:44-05:00" >May 17, 2022</time><div class="slide-meta-del">/</div><div class="slide-meta-comments"><a href='https://www.physicsforums.com/insights/symmetry-arguments-and-the-infinite-wire-with-a-current/#respond'>0 Comments</a></div></div><div class='slide-entry-excerpt entry-content' itemprop="text" >Many people reading this will be familiar with symmetry arguments related to the use of Gauss law. Finding the electric field around a spherically symmetric…</div></div><footer class="entry-footer"></footer><span class='hidden'> <span class='av-structured-data' itemprop="image" itemscope="itemscope" itemtype="https://schema.org/ImageObject" > <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2022/05/magnetism-current.png</span> <span itemprop='height'>135</span> <span itemprop='width'>240</span> </span> <span class='av-structured-data' itemprop="publisher" itemtype="https://schema.org/Organization" itemscope="itemscope" > <span itemprop='name'>Orodruin</span> <span itemprop='logo' itemscope itemtype='https://schema.org/ImageObject'> <span itemprop='url'>https://www.physicsforums.com/insights/wp-content/uploads/2019/02/Physics_Forums_Insights_logo.png</span> </span> </span><span class='av-structured-data' itemprop="author" itemscope="itemscope" itemtype="https://schema.org/Person" ><span itemprop='name'>Orodruin</span></span><span class='av-structured-data' itemprop="datePublished" datetime="2019-09-27T11:29:49-05:00" >2022-05-17 07:56:44</span><span class='av-structured-data' itemprop="dateModified" itemtype="https://schema.org/dateModified" >2022-05-17 08:06:59</span><span class='av-structured-data' itemprop="mainEntityOfPage" itemtype="https://schema.org/mainEntityOfPage" ><span itemprop='name'>Symmetry Arguments and the Infinite Wire with a Current</span></span></span></article></div></div></div></div></div></div></main><aside class='sidebar sidebar_right smartphones_sidebar_active alpha units' aria-label="Sidebar" role="complementary" itemscope="itemscope" itemtype="https://schema.org/WPSideBar" ><div class="inner_sidebar extralight-border"><section id="custom_html-2" class="widget_text widget clearfix widget_custom_html"><div class="textwidget custom-html-widget"><h2 class="widgettitle">Trending Articles</h2><ul class="wpp-list wpp-sunset"><li class="" style="--item-position: 1;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/can-see-atom/" class="wpp-post-title" target="_self">Can We See an Atom?</a></div></li><li class="" style="--item-position: 2;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/hand-wavy-discussion-planck-length/" class="wpp-post-title" target="_self">What Planck Length Is and It’s Common Misconceptions</a></div></li><li class="" style="--item-position: 3;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/self-study-algebra-part-ii-abstract-algebra/" class="wpp-post-title" target="_self">How to Self Study Abstract Algebra</a></div></li><li class="" style="--item-position: 4;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/self-study-basic-high-school-mathematics/" class="wpp-post-title" target="_self">How to Self-Study Basic High School Mathematics</a></div></li><li class="" style="--item-position: 5;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/the-7-basic-rules-of-quantum-mechanics/" class="wpp-post-title" target="_self">The 7 Basic Rules of Quantum Mechanics</a></div></li><li class="" style="--item-position: 6;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/visualizing-2-d-particle-box/" class="wpp-post-title" target="_self">How to Visualize the 2-D Particle in a Box</a></div></li><li class="" style="--item-position: 7;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/what-is-a-tensor/" class="wpp-post-title" target="_self">What Is a Tensor? The mathematical point of view.</a></div></li><li class="" style="--item-position: 8;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/parallel-programming-on-an-nvidia-gpu/" class="wpp-post-title" target="_self">Parallel Programming on an NVIDIA GPU</a></div></li><li class="" style="--item-position: 9;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/relativity-using-bondi-k-calculus/" class="wpp-post-title" target="_self">Learn Relativity Using the Bondi K-calculus</a></div></li><li class="" style="--item-position: 10;--total-items: 10"><div class="item-position"></div><div class="item-data"><a href="https://www.physicsforums.com/insights/schrodingers-cat-and-the-qbit/" class="wpp-post-title" target="_self">Schrödinger’s Cat and the Qbit</a></div></li></ul></div><span class="seperator extralight-border"></span></section><section id="nav_menu-3" class="widget clearfix widget_nav_menu"><h3 class="widgettitle">Physics Forums</h3><div class="menu-sidebar-forums-container"><ul id="menu-sidebar-forums" class="menu"><li id="menu-item-26724" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26724"><a href="https://www.physicsforums.com/forums/classical-physics.61/">Classical Physics</a></li><li id="menu-item-26725" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26725"><a href="https://www.physicsforums.com/forums/atomic-and-condensed-matter.64/">Atomic and Condensed Matter</a></li><li id="menu-item-26726" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26726"><a href="https://www.physicsforums.com/forums/quantum-physics.62/">Quantum Physics</a></li><li id="menu-item-26727" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26727"><a href="https://www.physicsforums.com/forums/special-and-general-relativity.70/">Special and General Relativity</a></li><li id="menu-item-26728" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26728"><a href="https://www.physicsforums.com/forums/beyond-the-standard-model.66/">Beyond the Standard Model</a></li><li id="menu-item-26729" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26729"><a href="https://www.physicsforums.com/forums/high-energy-nuclear-particle-physics.65/">High Energy, Nuclear, Particle Physics</a></li><li id="menu-item-26731" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26731"><a href="https://www.physicsforums.com/forums/astronomy-and-astrophysics.71/">Astronomy and Astrophysics</a></li><li id="menu-item-26732" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26732"><a href="https://www.physicsforums.com/forums/cosmology.69/">Cosmology</a></li><li id="menu-item-26730" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-26730"><a href="https://www.physicsforums.com/forums/other-physics-topics.111/">Other Physics Topics</a></li></ul></div><span class="seperator extralight-border"></span></section><section id="text-4" class="widget clearfix widget_text"><h3 class="widgettitle">Receive Insights Articles to Your Inbox</h3><div class="textwidget"><form style="border:0px solid #ccc;text-align:left;" action="https://feedburner.google.com/fb/a/mailverify" method="post" target="popupwindow" onsubmit="window.open('https://feedburner.google.com/fb/a/mailverify?uri=PhysicsForumsInsights', 'popupwindow', 'scrollbars=yes,width=550,height=520');return true"><p>Enter your email address:</p><p><input type="text" style="width:200px;border:2px solid #ccc;" name="email"/></p><input type="hidden" value="PhysicsForumsInsights" name="uri"/><input type="hidden" name="loc" value="en_US"/><input type="submit" value="Subscribe" /></p></form></div> <span class="seperator extralight-border"></span></section><section id="nav_menu-2" class="widget clearfix widget_nav_menu"><h3 class="widgettitle">Blog Information</h3><div class="menu-sidebar-container"><ul id="menu-sidebar" class="menu"><li id="menu-item-14015" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-14015"><a href="https://www.physicsforums.com/register">Become a Member!</a></li><li id="menu-item-14012" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-14012"><a href="https://www.physicsforums.com/insights/application/">Write for Us!</a></li><li id="menu-item-18348" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-18348"><a href="https://www.physicsforums.com/insights/postindex/">Table of Contents</a></li><li id="menu-item-18353" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-18353"><a href="https://www.physicsforums.com/insights/authors/">Blog Author List</a></li></ul></div><span class="seperator extralight-border"></span></section><section id="tag_cloud-3" class="widget clearfix widget_tag_cloud"><h3 class="widgettitle">Popular Topics</h3><div class="tagcloud"><a href="https://www.physicsforums.com/insights/tag/astronomy/" class="tag-cloud-link tag-link-307 tag-link-position-1" style="font-size: 9.1666666666667pt;" aria-label="astronomy (17 items)">astronomy<span class="tag-link-count"> (17)</span></a> <a href="https://www.physicsforums.com/insights/tag/black-holes/" class="tag-cloud-link tag-link-28 tag-link-position-2" style="font-size: 9.1666666666667pt;" aria-label="black holes (17 items)">black holes<span class="tag-link-count"> (17)</span></a> <a href="https://www.physicsforums.com/insights/tag/classical-physics/" class="tag-cloud-link tag-link-852 tag-link-position-3" style="font-size: 20.833333333333pt;" aria-label="classical physics (35 items)">classical physics<span class="tag-link-count"> (35)</span></a> <a href="https://www.physicsforums.com/insights/tag/cosmology/" class="tag-cloud-link tag-link-224 tag-link-position-4" style="font-size: 8pt;" aria-label="cosmology (16 items)">cosmology<span class="tag-link-count"> (16)</span></a> <a href="https://www.physicsforums.com/insights/tag/education/" class="tag-cloud-link tag-link-104 tag-link-position-5" style="font-size: 13.833333333333pt;" aria-label="education (23 items)">education<span class="tag-link-count"> (23)</span></a> <a href="https://www.physicsforums.com/insights/tag/electromagnetism/" class="tag-cloud-link tag-link-21 tag-link-position-6" style="font-size: 10.722222222222pt;" aria-label="electromagnetism (19 items)">electromagnetism<span class="tag-link-count"> (19)</span></a> <a href="https://www.physicsforums.com/insights/tag/general-relativity/" class="tag-cloud-link tag-link-333 tag-link-position-7" style="font-size: 10.722222222222pt;" aria-label="general relativity (19 items)">general relativity<span class="tag-link-count"> (19)</span></a> <a href="https://www.physicsforums.com/insights/tag/gravity/" class="tag-cloud-link tag-link-40 tag-link-position-8" style="font-size: 14.611111111111pt;" aria-label="gravity (24 items)">gravity<span class="tag-link-count"> (24)</span></a> <a href="https://www.physicsforums.com/insights/tag/interview/" class="tag-cloud-link tag-link-858 tag-link-position-9" style="font-size: 13.055555555556pt;" aria-label="interview (22 items)">interview<span class="tag-link-count"> (22)</span></a> <a href="https://www.physicsforums.com/insights/tag/mathematics/" class="tag-cloud-link tag-link-69 tag-link-position-10" style="font-size: 22pt;" aria-label="mathematics (38 items)">mathematics<span class="tag-link-count"> (38)</span></a> <a href="https://www.physicsforums.com/insights/tag/mathematics-self-study/" class="tag-cloud-link tag-link-484 tag-link-position-11" style="font-size: 9.9444444444444pt;" aria-label="mathematics self-study (18 items)">mathematics self-study<span class="tag-link-count"> (18)</span></a> <a href="https://www.physicsforums.com/insights/tag/physicist/" class="tag-cloud-link tag-link-11 tag-link-position-12" style="font-size: 15.777777777778pt;" aria-label="Physicist (26 items)">Physicist<span class="tag-link-count"> (26)</span></a> <a href="https://www.physicsforums.com/insights/tag/programming/" class="tag-cloud-link tag-link-42 tag-link-position-13" style="font-size: 9.9444444444444pt;" aria-label="programming (18 items)">programming<span class="tag-link-count"> (18)</span></a> <a href="https://www.physicsforums.com/insights/tag/quantum-field-theory/" class="tag-cloud-link tag-link-87 tag-link-position-14" style="font-size: 18.888888888889pt;" aria-label="Quantum Field Theory (31 items)">Quantum Field Theory<span class="tag-link-count"> (31)</span></a> <a href="https://www.physicsforums.com/insights/tag/quantum-mechanics/" class="tag-cloud-link tag-link-62 tag-link-position-15" style="font-size: 21.222222222222pt;" aria-label="quantum mechanics (36 items)">quantum mechanics<span class="tag-link-count"> (36)</span></a> <a href="https://www.physicsforums.com/insights/tag/quantum-physics/" class="tag-cloud-link tag-link-314 tag-link-position-16" style="font-size: 13.055555555556pt;" aria-label="quantum physics (22 items)">quantum physics<span class="tag-link-count"> (22)</span></a> <a href="https://www.physicsforums.com/insights/tag/relativity/" class="tag-cloud-link tag-link-114 tag-link-position-17" style="font-size: 21.611111111111pt;" aria-label="relativity (37 items)">relativity<span class="tag-link-count"> (37)</span></a> <a href="https://www.physicsforums.com/insights/tag/special-relativity/" class="tag-cloud-link tag-link-89 tag-link-position-18" style="font-size: 8pt;" aria-label="Special Relativity (16 items)">Special Relativity<span class="tag-link-count"> (16)</span></a> <a href="https://www.physicsforums.com/insights/tag/technology/" class="tag-cloud-link tag-link-861 tag-link-position-19" style="font-size: 10.722222222222pt;" aria-label="technology (19 items)">technology<span class="tag-link-count"> (19)</span></a> <a href="https://www.physicsforums.com/insights/tag/universe/" class="tag-cloud-link tag-link-265 tag-link-position-20" style="font-size: 12.277777777778pt;" aria-label="universe (21 items)">universe<span class="tag-link-count"> (21)</span></a></div> <span class="seperator extralight-border"></span></section></div></aside></div></div><footer class='container_wrap socket_color' id='socket' role="contentinfo" itemscope="itemscope" itemtype="https://schema.org/WPFooter" aria-label="Copyright and company info" ><div class='container'><span class='copyright'> 2024 © PHYSICS FORUMS, ALL RIGHTS RESERVED - <a href="https://www.physicsforums.com/misc/contact">Contact Us</a> - <a href="https://www.physicsforums.com/threads/physics-forums-privacy-policy.797829/">Privacy Policy</a> - <a href="https://www.physicsforums.com/insights/about-pf-insights/">About PF Insights</a></span><ul class='noLightbox social_bookmarks icon_count_4'><li class='social_bookmarks_twitter av-social-link-twitter social_icon_1'><a target="_blank" aria-label="Link to X" href="https://twitter.com/physicsforums" aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='X' rel="noopener"><span class='avia_hidden_link_text'>X</span></a></li><li class='social_bookmarks_facebook av-social-link-facebook social_icon_2'><a target="_blank" aria-label="Link to Facebook" href='https://www.facebook.com/physicsforums' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='Facebook' rel="noopener"><span class='avia_hidden_link_text'>Facebook</span></a></li><li class='social_bookmarks_linkedin av-social-link-linkedin social_icon_3'><a target="_blank" aria-label="Link to LinkedIn" href='https://www.linkedin.com/company/physics-forums/' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='LinkedIn' rel="noopener"><span class='avia_hidden_link_text'>LinkedIn</span></a></li><li class='social_bookmarks_youtube av-social-link-youtube social_icon_4'><a target="_blank" aria-label="Link to Youtube" href='https://www.youtube.com/@physicsforum' aria-hidden='false' data-av_icon='' data-av_iconfont='entypo-fontello' title='Youtube' rel="noopener"><span class='avia_hidden_link_text'>Youtube</span></a></li></ul></div></footer></div></div><a href='#top' title='Scroll to top' id='scroll-top-link' aria-hidden='true' data-av_icon='' data-av_iconfont='entypo-fontello' tabindex='-1'><span class="avia_hidden_link_text">Scroll to top</span></a><div id="fb-root"></div> <script data-optimized="1" type="text/javascript" src="https://www.physicsforums.com/insights/wp-content/litespeed/js/ffe2445567cebbe851874e864ab606aa.js?ver=606aa" defer></script> <script type='text/javascript'>/* */ var avia_framework_globals = avia_framework_globals || {}; avia_framework_globals.frameworkUrl = 'https://www.physicsforums.com/insights/wp-content/themes/enfold/framework/'; avia_framework_globals.installedAt = 'https://www.physicsforums.com/insights/wp-content/themes/enfold/'; avia_framework_globals.ajaxurl = 'https://www.physicsforums.com/insights/wp-admin/admin-ajax.php'; /* */</script> <script type="text/javascript" async='async' src="https://www.physicsforums.com/insights/wp-includes/js/jquery/jquery.min.js?ver=3.7.1" id="jquery-core-js"></script> <script type="text/javascript" id="wpil-frontend-script-js-extra">/* */ var wpilFrontend = {"ajaxUrl":"\/insights\/wp-admin\/admin-ajax.php","postId":"25754","postType":"post","openInternalInNewTab":"0","openExternalInNewTab":"1","disableClicks":"0","openLinksWithJS":"0","trackAllElementClicks":"0","clicksI18n":{"imageNoText":"Image in link: No Text","imageText":"Image Title: ","noText":"No Anchor Text Found"}}; /* */</script> <script data-optimized="1" type="text/javascript" async='async' src="https://www.physicsforums.com/insights/wp-content/litespeed/js/32fbbb44e22394d8c0f1ada21cfadab5.js?ver=e0bd3" id="wpil-frontend-script-js"></script> <script data-optimized="1" type="text/javascript" async='async' src="https://www.physicsforums.com/insights/wp-content/litespeed/js/7433b54a869d4261e4332aa625ab7215.js?ver=b7215" id="avia-footer-scripts-js"></script> <script data-no-optimize="1">!function(t,e){"object"==typeof exports&&"undefined"!=typeof module?module.exports=e():"function"==typeof define&&define.amd?define(e):(t="undefined"!=typeof globalThis?globalThis:t||self).LazyLoad=e()}(this,function(){"use strict";function e(){return(e=Object.assign||function(t){for(var e=1;e<arguments.length;e++){var n,a=arguments[e];for(n in a)Object.prototype.hasOwnProperty.call(a,n)&&(t[n]=a[n])}return t}).apply(this,arguments)}function i(t){return e({},it,t)}function o(t,e){var n,a="LazyLoad::Initialized",i=new t(e);try{n=new CustomEvent(a,{detail:{instance:i}})}catch(t){(n=document.createEvent("CustomEvent")).initCustomEvent(a,!1,!1,{instance:i})}window.dispatchEvent(n)}function l(t,e){return t.getAttribute(gt+e)}function c(t){return l(t,bt)}function s(t,e){return function(t,e,n){e=gt+e;null!==n?t.setAttribute(e,n):t.removeAttribute(e)}(t,bt,e)}function r(t){return s(t,null),0}function u(t){return null===c(t)}function d(t){return c(t)===vt}function f(t,e,n,a){t&&(void 0===a?void 0===n?t(e):t(e,n):t(e,n,a))}function _(t,e){nt?t.classList.add(e):t.className+=(t.className?" ":"")+e}function v(t,e){nt?t.classList.remove(e):t.className=t.className.replace(new RegExp("(^|\\s+)"+e+"(\\s+|$)")," ").replace(/^\s+/,"").replace(/\s+$/,"")}function g(t){return t.llTempImage}function b(t,e){!e||(e=e._observer)&&e.unobserve(t)}function p(t,e){t&&(t.loadingCount+=e)}function h(t,e){t&&(t.toLoadCount=e)}function n(t){for(var e,n=[],a=0;e=t.children[a];a+=1)"SOURCE"===e.tagName&&n.push(e);return n}function m(t,e){(t=t.parentNode)&&"PICTURE"===t.tagName&&n(t).forEach(e)}function a(t,e){n(t).forEach(e)}function E(t){return!!t[st]}function I(t){return t[st]}function y(t){return delete t[st]}function A(e,t){var n;E(e)||(n={},t.forEach(function(t){n[t]=e.getAttribute(t)}),e[st]=n)}function k(a,t){var i;E(a)&&(i=I(a),t.forEach(function(t){var e,n;e=a,(t=i[n=t])?e.setAttribute(n,t):e.removeAttribute(n)}))}function L(t,e,n){_(t,e.class_loading),s(t,ut),n&&(p(n,1),f(e.callback_loading,t,n))}function w(t,e,n){n&&t.setAttribute(e,n)}function x(t,e){w(t,ct,l(t,e.data_sizes)),w(t,rt,l(t,e.data_srcset)),w(t,ot,l(t,e.data_src))}function O(t,e,n){var a=l(t,e.data_bg_multi),i=l(t,e.data_bg_multi_hidpi);(a=at&&i?i:a)&&(t.style.backgroundImage=a,n=n,_(t=t,(e=e).class_applied),s(t,ft),n&&(e.unobserve_completed&&b(t,e),f(e.callback_applied,t,n)))}function N(t,e){!e||0<e.loadingCount||0<e.toLoadCount||f(t.callback_finish,e)}function C(t,e,n){t.addEventListener(e,n),t.llEvLisnrs[e]=n}function M(t){return!!t.llEvLisnrs}function z(t){if(M(t)){var e,n,a=t.llEvLisnrs;for(e in a){var i=a[e];n=e,i=i,t.removeEventListener(n,i)}delete t.llEvLisnrs}}function R(t,e,n){var a;delete t.llTempImage,p(n,-1),(a=n)&&--a.toLoadCount,v(t,e.class_loading),e.unobserve_completed&&b(t,n)}function T(o,r,c){var l=g(o)||o;M(l)||function(t,e,n){M(t)||(t.llEvLisnrs={});var a="VIDEO"===t.tagName?"loadeddata":"load";C(t,a,e),C(t,"error",n)}(l,function(t){var e,n,a,i;n=r,a=c,i=d(e=o),R(e,n,a),_(e,n.class_loaded),s(e,dt),f(n.callback_loaded,e,a),i||N(n,a),z(l)},function(t){var e,n,a,i;n=r,a=c,i=d(e=o),R(e,n,a),_(e,n.class_error),s(e,_t),f(n.callback_error,e,a),i||N(n,a),z(l)})}function G(t,e,n){var a,i,o,r,c;t.llTempImage=document.createElement("IMG"),T(t,e,n),E(c=t)||(c[st]={backgroundImage:c.style.backgroundImage}),o=n,r=l(a=t,(i=e).data_bg),c=l(a,i.data_bg_hidpi),(r=at&&c?c:r)&&(a.style.backgroundImage='url("'.concat(r,'")'),g(a).setAttribute(ot,r),L(a,i,o)),O(t,e,n)}function D(t,e,n){var a;T(t,e,n),a=e,e=n,(t=It[(n=t).tagName])&&(t(n,a),L(n,a,e))}function V(t,e,n){var a;a=t,(-1<yt.indexOf(a.tagName)?D:G)(t,e,n)}function F(t,e,n){var a;t.setAttribute("loading","lazy"),T(t,e,n),a=e,(e=It[(n=t).tagName])&&e(n,a),s(t,vt)}function j(t){t.removeAttribute(ot),t.removeAttribute(rt),t.removeAttribute(ct)}function P(t){m(t,function(t){k(t,Et)}),k(t,Et)}function S(t){var e;(e=At[t.tagName])?e(t):E(e=t)&&(t=I(e),e.style.backgroundImage=t.backgroundImage)}function U(t,e){var n;S(t),n=e,u(e=t)||d(e)||(v(e,n.class_entered),v(e,n.class_exited),v(e,n.class_applied),v(e,n.class_loading),v(e,n.class_loaded),v(e,n.class_error)),r(t),y(t)}function $(t,e,n,a){var i;n.cancel_on_exit&&(c(t)!==ut||"IMG"===t.tagName&&(z(t),m(i=t,function(t){j(t)}),j(i),P(t),v(t,n.class_loading),p(a,-1),r(t),f(n.callback_cancel,t,e,a)))}function q(t,e,n,a){var i,o,r=(o=t,0<=pt.indexOf(c(o)));s(t,"entered"),_(t,n.class_entered),v(t,n.class_exited),i=t,o=a,n.unobserve_entered&&b(i,o),f(n.callback_enter,t,e,a),r||V(t,n,a)}function H(t){return t.use_native&&"loading"in HTMLImageElement.prototype}function B(t,i,o){t.forEach(function(t){return(a=t).isIntersecting||0<a.intersectionRatio?q(t.target,t,i,o):(e=t.target,n=t,a=i,t=o,void(u(e)||(_(e,a.class_exited),$(e,n,a,t),f(a.callback_exit,e,n,t))));var e,n,a})}function J(e,n){var t;et&&!H(e)&&(n._observer=new IntersectionObserver(function(t){B(t,e,n)},{root:(t=e).container===document?null:t.container,rootMargin:t.thresholds||t.threshold+"px"}))}function K(t){return Array.prototype.slice.call(t)}function Q(t){return t.container.querySelectorAll(t.elements_selector)}function W(t){return c(t)===_t}function X(t,e){return e=t||Q(e),K(e).filter(u)}function Y(e,t){var n;(n=Q(e),K(n).filter(W)).forEach(function(t){v(t,e.class_error),r(t)}),t.update()}function t(t,e){var n,a,t=i(t);this._settings=t,this.loadingCount=0,J(t,this),n=t,a=this,Z&&window.addEventListener("online",function(){Y(n,a)}),this.update(e)}var Z="undefined"!=typeof window,tt=Z&&!("onscroll"in window)||"undefined"!=typeof navigator&&/(gle|ing|ro)bot|crawl|spider/i.test(navigator.userAgent),et=Z&&"IntersectionObserver"in window,nt=Z&&"classList"in document.createElement("p"),at=Z&&1<window.devicePixelRatio,it={elements_selector:".lazy",container:tt||Z?document:null,threshold:300,thresholds:null,data_src:"src",data_srcset:"srcset",data_sizes:"sizes",data_bg:"bg",data_bg_hidpi:"bg-hidpi",data_bg_multi:"bg-multi",data_bg_multi_hidpi:"bg-multi-hidpi",data_poster:"poster",class_applied:"applied",class_loading:"litespeed-loading",class_loaded:"litespeed-loaded",class_error:"error",class_entered:"entered",class_exited:"exited",unobserve_completed:!0,unobserve_entered:!1,cancel_on_exit:!0,callback_enter:null,callback_exit:null,callback_applied:null,callback_loading:null,callback_loaded:null,callback_error:null,callback_finish:null,callback_cancel:null,use_native:!1},ot="src",rt="srcset",ct="sizes",lt="poster",st="llOriginalAttrs",ut="loading",dt="loaded",ft="applied",_t="error",vt="native",gt="data-",bt="ll-status",pt=[ut,dt,ft,_t],ht=[ot],mt=[ot,lt],Et=[ot,rt,ct],It={IMG:function(t,e){m(t,function(t){A(t,Et),x(t,e)}),A(t,Et),x(t,e)},IFRAME:function(t,e){A(t,ht),w(t,ot,l(t,e.data_src))},VIDEO:function(t,e){a(t,function(t){A(t,ht),w(t,ot,l(t,e.data_src))}),A(t,mt),w(t,lt,l(t,e.data_poster)),w(t,ot,l(t,e.data_src)),t.load()}},yt=["IMG","IFRAME","VIDEO"],At={IMG:P,IFRAME:function(t){k(t,ht)},VIDEO:function(t){a(t,function(t){k(t,ht)}),k(t,mt),t.load()}},kt=["IMG","IFRAME","VIDEO"];return t.prototype={update:function(t){var e,n,a,i=this._settings,o=X(t,i);{if(h(this,o.length),!tt&&et)return H(i)?(e=i,n=this,o.forEach(function(t){-1!==kt.indexOf(t.tagName)&&F(t,e,n)}),void h(n,0)):(t=this._observer,i=o,t.disconnect(),a=t,void i.forEach(function(t){a.observe(t)}));this.loadAll(o)}},destroy:function(){this._observer&&this._observer.disconnect(),Q(this._settings).forEach(function(t){y(t)}),delete this._observer,delete this._settings,delete this.loadingCount,delete this.toLoadCount},loadAll:function(t){var e=this,n=this._settings;X(t,n).forEach(function(t){b(t,e),V(t,n,e)})},restoreAll:function(){var e=this._settings;Q(e).forEach(function(t){U(t,e)})}},t.load=function(t,e){e=i(e);V(t,e)},t.resetStatus=function(t){r(t)},Z&&function(t,e){if(e)if(e.length)for(var n,a=0;n=e[a];a+=1)o(t,n);else o(t,e)}(t,window.lazyLoadOptions),t});!function(e,t){"use strict";function a(){t.body.classList.add("litespeed_lazyloaded")}function n(){console.log("[LiteSpeed] Start Lazy Load Images"),d=new LazyLoad({elements_selector:"[data-lazyloaded]",callback_finish:a}),o=function(){d.update()},e.MutationObserver&&new MutationObserver(o).observe(t.documentElement,{childList:!0,subtree:!0,attributes:!0})}var d,o;e.addEventListener?e.addEventListener("load",n,!1):e.attachEvent("onload",n)}(window,document);</script></body></html>