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גרדיאנט – ויקיפדיה

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href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&amp;returnto=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o" class=""><span>כניסה לחשבון</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="אפשרויות נוספות" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="כלים אישיים" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">כלים אישיים</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="תפריט משתמש" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&amp;utm_medium=sidebar&amp;utm_campaign=C13_he.wikipedia.org&amp;uselang=he"><span>תרומה לוויקיפדיה</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&amp;returnto=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>יצירת חשבון</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&amp;returnto=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>כניסה לחשבון</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> דפים לעורכים שלא נכנסו לחשבון <a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D" aria-label="מידע נוסף על עריכה"><span>מידע נוסף</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%AA%D7%A8%D7%95%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C%D7%99" title="רשימת העריכות שנעשו מכתובת IP זו [y]" accesskey="y"><span>תרומות</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A9%D7%99%D7%97%D7%94_%D7%A9%D7%9C%D7%99" title="דיון על העריכות שנעשו מכתובת IP זו [n]" accesskey="n"><span>שיחה</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eהסתרה\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"anonnotice\" lang=\"he\" dir=\"rtl\"\u003E\u003Cp\u003E\u003Cb\u003Eתמיד רציתם לכתוב בוויקיפדיה אבל לא ידעתם איך? אתם מוזמנים לסדנת עריכה בוויקיפדיה. הסדנה תתקיים בספרייה הלאומית (בבניינה החדש) בירושלים ביום שישי, 06.12.24, בשעה 09:00. להרשמה לחצו \u003Ca href=\"/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%9E%D7%99%D7%96%D7%9E%D7%99_%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94/%D7%92%D7%9C%D7%90%D7%9D/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA/%D7%90%D7%99%D7%A8%D7%95%D7%A2%D7%99%D7%9D/%D7%A1%D7%93%D7%A0%D7%AA_%D7%A2%D7%A8%D7%99%D7%9B%D7%94_%D7%93%D7%A6%D7%9E%D7%91%D7%A8_2024\" title=\"ויקיפדיה:מיזמי ויקיפדיה/גלאם/הספרייה הלאומית/אירועים/סדנת עריכה דצמבר 2024\"\u003Eכאן\u003C/a\u003E.\u003C/b\u003E\n\u003C/p\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="אתר"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="תוכן עניינים" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">תוכן עניינים</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">הסתרה</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">התחלה</div> </a> </li> <li id="toc-אינטואיציה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#אינטואיציה"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>אינטואיציה</span> </div> </a> <ul id="toc-אינטואיציה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-סימון" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#סימון"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>סימון</span> </div> </a> <ul id="toc-סימון-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הגדרה_פורמלית_במרחב_האוקלידי_התלת־ממדי" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הגדרה_פורמלית_במרחב_האוקלידי_התלת־ממדי"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>הגדרה פורמלית במרחב האוקלידי התלת־ממדי</span> </div> </a> <ul id="toc-הגדרה_פורמלית_במרחב_האוקלידי_התלת־ממדי-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-למה_וקטור_גרדיאנט_נותן_את_הכיוון_בו_השינוי_הוא_מקסימלי?" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#למה_וקטור_גרדיאנט_נותן_את_הכיוון_בו_השינוי_הוא_מקסימלי?"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>למה וקטור גרדיאנט נותן את הכיוון בו השינוי הוא מקסימלי?</span> </div> </a> <ul id="toc-למה_וקטור_גרדיאנט_נותן_את_הכיוון_בו_השינוי_הוא_מקסימלי?-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-דוגמה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#דוגמה"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>דוגמה</span> </div> </a> <ul id="toc-דוגמה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-גרדיאנט_באנליזה_על_יריעות" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#גרדיאנט_באנליזה_על_יריעות"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>גרדיאנט באנליזה על יריעות</span> </div> </a> <ul id="toc-גרדיאנט_באנליזה_על_יריעות-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-גרדיאנט_במערכת_קואורדינטות_אורתוגונליות_כלשהי" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#גרדיאנט_במערכת_קואורדינטות_אורתוגונליות_כלשהי"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>גרדיאנט במערכת קואורדינטות אורתוגונליות כלשהי</span> </div> </a> <button aria-controls="toc-גרדיאנט_במערכת_קואורדינטות_אורתוגונליות_כלשהי-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק גרדיאנט במערכת קואורדינטות אורתוגונליות כלשהי</span> </button> <ul id="toc-גרדיאנט_במערכת_קואורדינטות_אורתוגונליות_כלשהי-sublist" class="vector-toc-list"> <li id="toc-הוכחה_המבוססת_על_גאומטריה_דיפרנציאלית" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#הוכחה_המבוססת_על_גאומטריה_דיפרנציאלית"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>הוכחה המבוססת על גאומטריה דיפרנציאלית</span> </div> </a> <ul id="toc-הוכחה_המבוססת_על_גאומטריה_דיפרנציאלית-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-קשרים_בין_אופרטורים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קשרים_בין_אופרטורים"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>קשרים בין אופרטורים</span> </div> </a> <ul id="toc-קשרים_בין_אופרטורים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-משפט_הגרדיאנט" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#משפט_הגרדיאנט"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>משפט הגרדיאנט</span> </div> </a> <ul id="toc-משפט_הגרדיאנט-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ראו_גם" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ראו_גם"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>ראו גם</span> </div> </a> <ul id="toc-ראו_גם-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-לקריאה_נוספת" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#לקריאה_נוספת"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>לקריאה נוספת</span> </div> </a> <ul id="toc-לקריאה_נוספת-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קישורים_חיצוניים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קישורים_חיצוניים"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>קישורים חיצוניים</span> </div> </a> <ul id="toc-קישורים_חיצוניים-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="תוכן עניינים" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="מצב תוכן העניינים" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">מצב תוכן העניינים</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">גרדיאנט</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="מעבר לערך בשפה אחרת. זמין ב־63 שפות" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-63" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">63 שפות</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Gradient" title="Gradient – אנגלית" lang="en" hreflang="en" data-title="Gradient" data-language-autonym="English" data-language-local-name="אנגלית" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%89%80%E1%89%A0%E1%89%B5" title="አቀበት – אמהרית" lang="am" hreflang="am" data-title="አቀበት" data-language-autonym="አማርኛ" data-language-local-name="אמהרית" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AF%D8%B1%D8%AC_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="تدرج (رياضيات) – ערבית" lang="ar" hreflang="ar" data-title="تدرج (رياضيات)" data-language-autonym="العربية" data-language-local-name="ערבית" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Qradiyent" title="Qradiyent – אזרית" lang="az" hreflang="az" data-title="Qradiyent" data-language-autonym="Azərbaycanca" data-language-local-name="אזרית" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D1%8B%D0%B5%D0%BD%D1%82" title="Градыент – בלארוסית" lang="be" hreflang="be" data-title="Градыент" data-language-autonym="Беларуская" data-language-local-name="בלארוסית" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D1%8B%D0%B5%D0%BD%D1%82" title="Градыент – בלארוסית טרשקביץ׳" lang="be-tarask" hreflang="be-tarask" data-title="Градыент" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="בלארוסית טרשקביץ׳" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – בולגרית" lang="bg" hreflang="bg" data-title="Градиент" data-language-autonym="Български" data-language-local-name="בולגרית" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8D%E0%A6%B0%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%A1%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A7%87%E0%A6%A8%E0%A7%8D%E0%A6%9F" title="গ্র্যাডিয়েন্ট – בנגלית" lang="bn" hreflang="bn" data-title="গ্র্যাডিয়েন্ট" data-language-autonym="বাংলা" data-language-local-name="בנגלית" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Gradijent" title="Gradijent – בוסנית" lang="bs" hreflang="bs" data-title="Gradijent" data-language-autonym="Bosanski" data-language-local-name="בוסנית" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Gradient_(matem%C3%A0tiques)" title="Gradient (matemàtiques) – קטלאנית" lang="ca" hreflang="ca" data-title="Gradient (matemàtiques)" data-language-autonym="Català" data-language-local-name="קטלאנית" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Gradient_(matematika)" title="Gradient (matematika) – צ׳כית" lang="cs" hreflang="cs" data-title="Gradient (matematika)" data-language-autonym="Čeština" data-language-local-name="צ׳כית" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – צ׳ובשית" lang="cv" hreflang="cv" data-title="Градиент" data-language-autonym="Чӑвашла" data-language-local-name="צ׳ובשית" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Gradient" title="Gradient – דנית" lang="da" hreflang="da" data-title="Gradient" data-language-autonym="Dansk" data-language-local-name="דנית" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gradient_(Mathematik)" title="Gradient (Mathematik) – גרמנית" lang="de" hreflang="de" data-title="Gradient (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="גרמנית" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Gradiento_(matematiko)" title="Gradiento (matematiko) – אספרנטו" lang="eo" hreflang="eo" data-title="Gradiento (matematiko)" data-language-autonym="Esperanto" data-language-local-name="אספרנטו" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Gradiente" title="Gradiente – ספרדית" lang="es" hreflang="es" data-title="Gradiente" data-language-autonym="Español" data-language-local-name="ספרדית" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Gradient" title="Gradient – אסטונית" lang="et" hreflang="et" data-title="Gradient" data-language-autonym="Eesti" data-language-local-name="אסטונית" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gradiente" title="Gradiente – בסקית" lang="eu" hreflang="eu" data-title="Gradiente" data-language-autonym="Euskara" data-language-local-name="בסקית" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D8%A7%D8%AF%DB%8C%D8%A7%D9%86" title="گرادیان – פרסית" lang="fa" hreflang="fa" data-title="گرادیان" data-language-autonym="فارسی" data-language-local-name="פרסית" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Gradientti" title="Gradientti – פינית" lang="fi" hreflang="fi" data-title="Gradientti" data-language-autonym="Suomi" data-language-local-name="פינית" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Gradient" title="Gradient – צרפתית" lang="fr" hreflang="fr" data-title="Gradient" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Gr%C3%A1d%C3%A1n" title="Grádán – אירית" lang="ga" hreflang="ga" data-title="Grádán" data-language-autonym="Gaeilge" data-language-local-name="אירית" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Gradiente" title="Gradiente – גליסית" lang="gl" hreflang="gl" data-title="Gradiente" data-language-autonym="Galego" data-language-local-name="גליסית" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gradiens" title="Gradiens – הונגרית" lang="hu" hreflang="hu" data-title="Gradiens" data-language-autonym="Magyar" data-language-local-name="הונגרית" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D6%80%D5%A1%D5%A4%D5%AB%D5%A5%D5%B6%D5%BF" title="Գրադիենտ – ארמנית" lang="hy" hreflang="hy" data-title="Գրադիենտ" data-language-autonym="Հայերեն" data-language-local-name="ארמנית" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gradien" title="Gradien – אינדונזית" lang="id" hreflang="id" data-title="Gradien" data-language-autonym="Bahasa Indonesia" data-language-local-name="אינדונזית" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Gradiento" title="Gradiento – אידו" lang="io" hreflang="io" data-title="Gradiento" data-language-autonym="Ido" data-language-local-name="אידו" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Stigull" title="Stigull – איסלנדית" lang="is" hreflang="is" data-title="Stigull" data-language-autonym="Íslenska" data-language-local-name="איסלנדית" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gradiente" title="Gradiente – איטלקית" lang="it" hreflang="it" data-title="Gradiente" data-language-autonym="Italiano" data-language-local-name="איטלקית" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8B%BE%E9%85%8D_(%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E8%A7%A3%E6%9E%90)" title="勾配 (ベクトル解析) – יפנית" lang="ja" hreflang="ja" data-title="勾配 (ベクトル解析)" data-language-autonym="日本語" data-language-local-name="יפנית" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%A0%E1%83%90%E1%83%93%E1%83%98%E1%83%94%E1%83%9C%E1%83%A2%E1%83%98" title="გრადიენტი – גאורגית" lang="ka" hreflang="ka" data-title="გრადიენტი" data-language-autonym="ქართული" data-language-local-name="גאורגית" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – קזחית" lang="kk" hreflang="kk" data-title="Градиент" data-language-autonym="Қазақша" data-language-local-name="קזחית" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B8%B0%EC%9A%B8%EA%B8%B0_(%EB%B2%A1%ED%84%B0)" title="기울기 (벡터) – קוריאנית" lang="ko" hreflang="ko" data-title="기울기 (벡터)" data-language-autonym="한국어" data-language-local-name="קוריאנית" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – קירגיזית" lang="ky" hreflang="ky" data-title="Градиент" data-language-autonym="Кыргызча" data-language-local-name="קירגיזית" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Gradientas" title="Gradientas – ליטאית" lang="lt" hreflang="lt" data-title="Gradientas" data-language-autonym="Lietuvių" data-language-local-name="ליטאית" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Gradients" title="Gradients – לטבית" lang="lv" hreflang="lv" data-title="Gradients" data-language-autonym="Latviešu" data-language-local-name="לטבית" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B5%87%E0%B4%A1%E0%B4%BF%E0%B4%AF%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%8D" title="ഗ്രേഡിയന്റ് – מליאלאם" lang="ml" hreflang="ml" data-title="ഗ്രേഡിയന്റ്" data-language-autonym="മലയാളം" data-language-local-name="מליאלאם" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Gradi%C3%ABnt_(wiskunde)" title="Gradiënt (wiskunde) – הולנדית" lang="nl" hreflang="nl" data-title="Gradiënt (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="הולנדית" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Gradient" title="Gradient – נורווגית חדשה" lang="nn" hreflang="nn" data-title="Gradient" data-language-autonym="Norsk nynorsk" data-language-local-name="נורווגית חדשה" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gradient" title="Gradient – נורווגית ספרותית" lang="nb" hreflang="nb" data-title="Gradient" data-language-autonym="Norsk bokmål" data-language-local-name="נורווגית ספרותית" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%8D%E0%A8%B0%E0%A9%87%E0%A8%A1%E0%A9%80%E0%A8%85%E0%A9%B0%E0%A8%9F" title="ਗ੍ਰੇਡੀਅੰਟ – פנג׳אבי" lang="pa" hreflang="pa" data-title="ਗ੍ਰੇਡੀਅੰਟ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="פנג׳אבי" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Gradient_(matematyka)" title="Gradient (matematyka) – פולנית" lang="pl" hreflang="pl" data-title="Gradient (matematyka)" data-language-autonym="Polski" data-language-local-name="פולנית" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Gradiente" title="Gradiente – פורטוגזית" lang="pt" hreflang="pt" data-title="Gradiente" data-language-autonym="Português" data-language-local-name="פורטוגזית" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Gradient" title="Gradient – רומנית" lang="ro" hreflang="ro" data-title="Gradient" data-language-autonym="Română" data-language-local-name="רומנית" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – רוסית" lang="ru" hreflang="ru" data-title="Градиент" data-language-autonym="Русский" data-language-local-name="רוסית" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Gradijent" title="Gradijent – סרבו-קרואטית" lang="sh" hreflang="sh" data-title="Gradijent" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="סרבו-קרואטית" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Gradient" title="Gradient – אנגלית פשוטה" lang="en-simple" hreflang="en-simple" data-title="Gradient" data-language-autonym="Simple English" data-language-local-name="אנגלית פשוטה" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Gradient_(matematika)" title="Gradient (matematika) – סלובקית" lang="sk" hreflang="sk" data-title="Gradient (matematika)" data-language-autonym="Slovenčina" data-language-local-name="סלובקית" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Gradient" title="Gradient – סלובנית" lang="sl" hreflang="sl" data-title="Gradient" data-language-autonym="Slovenščina" data-language-local-name="סלובנית" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Gradient" title="Gradient – אלבנית" lang="sq" hreflang="sq" data-title="Gradient" data-language-autonym="Shqip" data-language-local-name="אלבנית" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Gradijent" title="Gradijent – סרבית" lang="sr" hreflang="sr" data-title="Gradijent" data-language-autonym="Српски / srpski" data-language-local-name="סרבית" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gradient_(matematik)" title="Gradient (matematik) – שוודית" lang="sv" hreflang="sv" data-title="Gradient (matematik)" data-language-autonym="Svenska" data-language-local-name="שוודית" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – טג׳יקית" lang="tg" hreflang="tg" data-title="Градиент" data-language-autonym="Тоҷикӣ" data-language-local-name="טג׳יקית" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%81%E0%B8%A3%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2%E0%B8%99%E0%B8%95%E0%B9%8C" title="เกรเดียนต์ – תאית" lang="th" hreflang="th" data-title="เกรเดียนต์" data-language-autonym="ไทย" data-language-local-name="תאית" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Gradient_(kalkulong_bektor)" title="Gradient (kalkulong bektor) – טאגאלוג" lang="tl" hreflang="tl" data-title="Gradient (kalkulong bektor)" data-language-autonym="Tagalog" data-language-local-name="טאגאלוג" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Gradyan" title="Gradyan – טורקית" lang="tr" hreflang="tr" data-title="Gradyan" data-language-autonym="Türkçe" data-language-local-name="טורקית" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Градиент (математика) – טטרית" lang="tt" hreflang="tt" data-title="Градиент (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="טטרית" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D1%96%D1%94%D0%BD%D1%82" title="Градієнт – אוקראינית" lang="uk" hreflang="uk" data-title="Градієнт" data-language-autonym="Українська" data-language-local-name="אוקראינית" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Gradiyent" title="Gradiyent – אוזבקית" lang="uz" hreflang="uz" data-title="Gradiyent" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="אוזבקית" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gradient" title="Gradient – וייטנאמית" lang="vi" hreflang="vi" data-title="Gradient" data-language-autonym="Tiếng Việt" data-language-local-name="וייטנאמית" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%A2%AF%E5%BA%A6" title="梯度 – סינית וו" lang="wuu" hreflang="wuu" data-title="梯度" data-language-autonym="吴语" data-language-local-name="סינית וו" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A2%AF%E5%BA%A6" title="梯度 – סינית" lang="zh" hreflang="zh" data-title="梯度" data-language-autonym="中文" data-language-local-name="סינית" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%A2%AF%E5%BA%A6" title="梯度 – קנטונזית" lang="yue" hreflang="yue" data-title="梯度" data-language-autonym="粵語" data-language-local-name="קנטונזית" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q173582#sitelinks-wikipedia" title="עריכת קישורים בין־לשוניים" class="wbc-editpage">עריכת הקישורים</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="מרחבי שם"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" title="צפייה בדף התוכן [c]" accesskey="c"><span>ערך</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%D7%A9%D7%99%D7%97%D7%94:%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" rel="discussion" title="שיחה על דף התוכן [t]" accesskey="t"><span>שיחה</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="שינוי הגוון השפה" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" 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for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">כלים</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">כלים</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">הסתרה</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="אפשרויות נוספות" > <div class="vector-menu-heading"> פעולות </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98"><span>קריאה</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit" title="עריכת קוד המקור של הדף הזה [e]" accesskey="e"><span>עריכת קוד מקור</span></a></li><li id="ca-more-ve-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit" title="עריכת הדף הזה [v]" accesskey="v"><span>עריכה</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=history"><span>גרסאות קודמות</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> כללי </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%93%D7%A4%D7%99%D7%9D_%D7%94%D7%9E%D7%A7%D7%95%D7%A9%D7%A8%D7%99%D7%9D_%D7%9C%D7%9B%D7%90%D7%9F/%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" title="רשימה של כל דפי הוויקי שמקשרים לדף הזה [j]" accesskey="j"><span>דפים המקושרים לכאן</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A9%D7%99%D7%A0%D7%95%D7%99%D7%99%D7%9D_%D7%91%D7%93%D7%A4%D7%99%D7%9D_%D7%94%D7%9E%D7%A7%D7%95%D7%A9%D7%A8%D7%99%D7%9D/%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" rel="nofollow" title="השינויים האחרונים בדפים המקושרים מהדף הזה [k]" accesskey="k"><span>שינויים בדפים המקושרים</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%93%D7%A4%D7%99%D7%9D_%D7%9E%D7%99%D7%95%D7%97%D7%93%D7%99%D7%9D" title="רשימה של כל הדפים המיוחדים [q]" accesskey="q"><span>דפים מיוחדים</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;oldid=38679546" title="קישור קבוע לגרסה הזאת של הדף הזה"><span>קישור קבוע</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=info" title="מידע נוסף על הדף הזה"><span>מידע על הדף</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A6%D7%99%D7%98%D7%95%D7%98_%D7%93%D7%A3_%D7%96%D7%94&amp;page=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;id=38679546&amp;wpFormIdentifier=titleform" title="מידע איך לצטט את הדף הזה"><span>ציטוט הדף הזה</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9E%D7%A7%D7%A6%D7%A8_%D7%9B%D7%AA%D7%95%D7%91%D7%95%D7%AA&amp;url=https%3A%2F%2Fhe.wikipedia.org%2Fwiki%2F%25D7%2592%25D7%25A8%25D7%2593%25D7%2599%25D7%2590%25D7%25A0%25D7%2598"><span>קבלת כתובת מקוצרת</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:QrCode&amp;url=https%3A%2F%2Fhe.wikipedia.org%2Fwiki%2F%25D7%2592%25D7%25A8%25D7%2593%25D7%2599%25D7%2590%25D7%25A0%25D7%2598"><span>הורדת קוד QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> הדפסה/יצוא </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A1%D7%A4%D7%A8&amp;bookcmd=book_creator&amp;referer=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98"><span>יצירת ספר</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:DownloadAsPdf&amp;page=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=show-download-screen"><span>הורדה כ־PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;printable=yes" title="גרסה להדפסה של הדף הזה [p]" accesskey="p"><span>גרסה להדפסה</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> במיזמים אחרים </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Gradient_fields" hreflang="en"><span>ויקישיתוף</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q173582" title="קישור לפריט המשויך במאגר הנתונים [g]" accesskey="g"><span>פריט ויקינתונים</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="כלי דף"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="מראה"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">מראה</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">הסתרה</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">מתוך ויקיפדיה, האנציקלופדיה החופשית</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-rtl mw-parser-output" lang="he" dir="rtl"><table cellpadding="1" style="float: left; clear:both; border: 1px solid #8888aa; background: #f7f8ff; padding: 5px; font-size: 95%; margin: 0.5em 1em 0.5em 0.5em;"> <tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/60px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/90px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/120px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <p>בערך זה<br />נעשה שימוש<br />בסימנים מוסכמים<br />מתחום המתמטיקה.<br />להבהרת הסימנים<br />ראו <a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי">סימון מתמטי</a>. </p> </td></tr></tbody></table> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Gradient2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Gradient2.svg/220px-Gradient2.svg.png" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Gradient2.svg/330px-Gradient2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Gradient2.svg/440px-Gradient2.svg.png 2x" data-file-width="512" data-file-height="255" /></a><figcaption>המחשה של גרדיאנט. באיורים האלה, השדה הסקלרי מתואר באמצעות שינוי הצבע, כאשר אזורים כהים יותר הם ערכים גדולים יותר של הפונקציה. החצים הכחולים מתארים את הגרדיאנט הנגזר מהשדה הסקלרי. החצים פונים אל עבר האזורים הגבוהים יותר.</figcaption></figure> <p><b>גְּרַדִיאֶנְט</b> הוא <a href="/wiki/%D7%94%D7%9B%D7%9C%D7%9C%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="הכללה (מתמטיקה)">הכללה</a> של מושג ה<a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">נגזרת</a> בעבור <a href="/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" title="חשבון אינפיניטסימלי">חשבון אינפיניטסימלי</a> של מספר משתנים. הגרדיאנט הוא <a href="/wiki/%D7%90%D7%95%D7%A4%D7%A8%D7%98%D7%95%D7%A8" title="אופרטור">אופרטור</a> <a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" class="mw-redirect" title="וקטור (אלגברה)">וקטורי</a> המופעל על <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%A1%D7%A7%D7%9C%D7%A8%D7%99" title="שדה סקלרי">שדה סקלרי</a>. הגרדיאנט של שדה סקלרי הוא <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="שדה וקטורי">שדה וקטורי</a> המשייך לכל נקודה במרחב וקטור. </p><p>כיוון וקטור הגרדיאנט מצביע אל הכיוון בו השינוי בשדה הסקלרי מקסימלי (חיובי). גודל וקטור הגרדיאנט כשיעור השינוי המקסימלי. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="אינטואיציה"><span id=".D7.90.D7.99.D7.A0.D7.98.D7.95.D7.90.D7.99.D7.A6.D7.99.D7.94"></span>אינטואיציה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=1" title="עריכת קוד המקור של הפרק: אינטואיציה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=1" title="עריכת פסקה: &quot;אינטואיציה&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>נתבונן לדוגמה, באזור הררי. הגובה של כל נקודה ברכס ההרים יתואר על ידי פונקציה (שדה סקלרי). בכל נקודה, ניתן להסתכל על הכיוון בו השיפוע לפונקציה שהותאמה לה הוא הגדול ביותר. זהו, למעשה, הכיוון בו הגובה משתנה בצורה המהירה (החזקה) ביותר. אם נשחרר כדור בנקודה זו, הוא יתגלגל בדיוק לכיוון ההפוך (שיפוע חזק ביותר לכיוון השלילי). באופן זה, ניתן להתאים לכל נקודה וקטור בכיוון השיפוע הגדול ביותר, וגודלו נקבע על פי גודל השיפוע. וקטור זה הוא וקטור הגרדיאנט. </p><p>דוגמה נוספת היא בניית כבישים וגגות כך שהמים יתנקזו באופן יעיל. יש ליצור שיפוע קטן, על מנת שהמים יזרמו אל פתח הניקוז. המסלול שהמים מבצעים בדרכם אל המרזב, הוא הכיוון בו הגובה משתנה בצורה הכי מהירה, שהוא גם וקטור הגרדיאנט. </p> <div class="mw-heading mw-heading2"><h2 id="סימון"><span id=".D7.A1.D7.99.D7.9E.D7.95.D7.9F"></span>סימון</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=2" title="עריכת קוד המקור של הפרק: סימון"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=2" title="עריכת פסקה: &quot;סימון&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>הגרדיאנט של פונקציה סקלרית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span>מסומן על ידי&#160;: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {grad} \ f=\nabla \ f={\vec {\nabla }}\ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>grad</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mtext>&#xA0;</mtext> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mtext>&#xA0;</mtext> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> <mtext>&#xA0;</mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {grad} \ f=\nabla \ f={\vec {\nabla }}\ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22f185e186c71ed6b2c66007cbbccaa72b27d92" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.563ex; height:3.176ex;" alt="{\displaystyle \operatorname {grad} \ f=\nabla \ f={\vec {\nabla }}\ f}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\nabla }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\nabla }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/415e835f2d447e778b31d4f4d98d71fb7349880e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle {\nabla }}"></span> מסמל את אופרטור הגזירה דֶל (סמל המשולש עצמו נקרא בשם "<a href="/wiki/%D7%A0%D7%91%D7%9C%D7%94_(%D7%A1%D7%99%D7%9E%D7%9F)" title="נבלה (סימן)">נבּלה</a>"). כאשר משתמשים במערכת קואורדינטות, ווקטורי הבסיס אינם פונקציות של המיקום במרחב, הגרדיאנט נתון כ<a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" class="mw-redirect" title="וקטור (אלגברה)">ווקטור</a> של <a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA_%D7%97%D7%9C%D7%A7%D7%99%D7%AA" title="נגזרת חלקית">נגזרות חלקיות</a>. למשל ב<a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%90%D7%95%D7%A7%D7%9C%D7%99%D7%93%D7%99" title="מרחב אוקלידי">מרחב אוקלידי</a> תלת־ממדי <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\nabla }\equiv {\hat {x}}{\frac {\partial }{\partial x}}+{\hat {y}}{\frac {\partial }{\partial y}}+{\hat {z}}{\frac {\partial }{\partial z}}\equiv \left(\ \partial _{x},\ \ \partial _{y},\ \ \partial _{z}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> </mrow> <mo>&#x2261;<!-- ≡ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>&#x2261;<!-- ≡ --></mo> <mrow> <mo>(</mo> <mrow> <mtext>&#xA0;</mtext> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mtext>&#xA0;</mtext> <mtext>&#xA0;</mtext> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mtext>&#xA0;</mtext> <mtext>&#xA0;</mtext> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\nabla }\equiv {\hat {x}}{\frac {\partial }{\partial x}}+{\hat {y}}{\frac {\partial }{\partial y}}+{\hat {z}}{\frac {\partial }{\partial z}}\equiv \left(\ \partial _{x},\ \ \partial _{y},\ \ \partial _{z}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eabb29b3d32aa24c13e56f96f727bd1239e43ab3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.485ex; height:6.009ex;" alt="{\displaystyle {\nabla }\equiv {\hat {x}}{\frac {\partial }{\partial x}}+{\hat {y}}{\frac {\partial }{\partial y}}+{\hat {z}}{\frac {\partial }{\partial z}}\equiv \left(\ \partial _{x},\ \ \partial _{y},\ \ \partial _{z}\right)}"></span> </p><p>מתוך ההגדרה נובע שהגרדיאנט הוא וקטור, כיוון שהפעלת וקטור הנגזרות החלקיות (אופרטור הגזירה דֶל) על הפונקציה סקלרית מחזירה וקטור (הגרדיאנט). </p> <div class="mw-heading mw-heading2"><h2 id="הגדרה_פורמלית_במרחב_האוקלידי_התלת־ממדי"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.94_.D7.A4.D7.95.D7.A8.D7.9E.D7.9C.D7.99.D7.AA_.D7.91.D7.9E.D7.A8.D7.97.D7.91_.D7.94.D7.90.D7.95.D7.A7.D7.9C.D7.99.D7.93.D7.99_.D7.94.D7.AA.D7.9C.D7.AA.D6.BE.D7.9E.D7.9E.D7.93.D7.99"></span>הגדרה פורמלית במרחב האוקלידי התלת־ממדי</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=3" title="עריכת קוד המקור של הפרק: הגדרה פורמלית במרחב האוקלידי התלת־ממדי"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=3" title="עריכת פסקה: &quot;הגדרה פורמלית במרחב האוקלידי התלת־ממדי&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>במרחב אוקלידי תלת־ממדי, הגרדיאנט של שדה סקלרי כלשהו שמתואר על ידי <a href="/wiki/%D7%9E%D7%A2%D7%A8%D7%9B%D7%AA_%D7%A6%D7%99%D7%A8%D7%99%D7%9D_%D7%A7%D7%A8%D7%98%D7%96%D7%99%D7%AA" title="מערכת צירים קרטזית">קואורדינטות קרטזיות</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \psi (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>&#x03C8;<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \psi (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/536d68e0eb305ebe1274d746ee18902d6f7a3399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.544ex; height:2.843ex;" alt="{\displaystyle \ \psi (x,y,z)}"></span> מוגדר כך: <br /> </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\nabla }\psi (x,y,z)\equiv {\frac {\partial \psi (x,y,z)}{\partial x}}{\hat {x}}+{\frac {\partial \psi (x,y,z)}{\partial y}}{\hat {y}}+{\frac {\partial \psi (x,y,z)}{\partial z}}{\hat {z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> </mrow> <mi>&#x03C8;<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>&#x2261;<!-- ≡ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>&#x03C8;<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>&#x03C8;<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>&#x03C8;<!-- ψ --></mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\nabla }\psi (x,y,z)\equiv {\frac {\partial \psi (x,y,z)}{\partial x}}{\hat {x}}+{\frac {\partial \psi (x,y,z)}{\partial y}}{\hat {y}}+{\frac {\partial \psi (x,y,z)}{\partial z}}{\hat {z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0760dc0a4c653c375fbe9b16675b0c48a96bf8c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.96ex; height:6.343ex;" alt="{\displaystyle {\nabla }\psi (x,y,z)\equiv {\frac {\partial \psi (x,y,z)}{\partial x}}{\hat {x}}+{\frac {\partial \psi (x,y,z)}{\partial y}}{\hat {y}}+{\frac {\partial \psi (x,y,z)}{\partial z}}{\hat {z}}}"></span></dd></dl></dd></dl> <p>כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\hat {x}},{\hat {y}},{\hat {z}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>z</mi> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\hat {x}},{\hat {y}},{\hat {z}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f6be6e49f5162dd1abcbde62d9f5ce43daaab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.321ex; height:2.843ex;" alt="{\displaystyle \{{\hat {x}},{\hat {y}},{\hat {z}}\}}"></span> הם <a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_%D7%99%D7%97%D7%99%D7%93%D7%94" title="וקטור יחידה">וקטורי היחידה</a> המקבילים לצירים. </p><p>באופן כללי, עבור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d5ff7312a01506eee6ecea7dca662763a101c9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.859ex; height:2.509ex;" alt="{\displaystyle \ f}"></span> <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="פונקציה">פונקציה</a> סקלרית כלשהי מעל <a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">מרחב וקטורי</a> בעל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf8b0f621a23f81aa20d63b5cd59d3dcad83ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle \ n}"></span>ממדים, ניתן להגדיר את הגרדיאנט כך: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {grad} f(a)\equiv \left({\frac {\partial f}{\partial x_{1}}}(a),{\frac {\partial f}{\partial x_{2}}}(a),\dots ,{\frac {\partial f}{\partial x_{n}}}(a)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>grad</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>&#x2261;<!-- ≡ --></mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>&#x2026;<!-- … --></mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {grad} f(a)\equiv \left({\frac {\partial f}{\partial x_{1}}}(a),{\frac {\partial f}{\partial x_{2}}}(a),\dots ,{\frac {\partial f}{\partial x_{n}}}(a)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f92c220bfb9cb7ed45cc9ce85b7ef6f1e83bffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.862ex; height:6.176ex;" alt="{\displaystyle \operatorname {grad} f(a)\equiv \left({\frac {\partial f}{\partial x_{1}}}(a),{\frac {\partial f}{\partial x_{2}}}(a),\dots ,{\frac {\partial f}{\partial x_{n}}}(a)\right)}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="למה_וקטור_גרדיאנט_נותן_את_הכיוון_בו_השינוי_הוא_מקסימלי?"><span id=".D7.9C.D7.9E.D7.94_.D7.95.D7.A7.D7.98.D7.95.D7.A8_.D7.92.D7.A8.D7.93.D7.99.D7.90.D7.A0.D7.98_.D7.A0.D7.95.D7.AA.D7.9F_.D7.90.D7.AA_.D7.94.D7.9B.D7.99.D7.95.D7.95.D7.9F_.D7.91.D7.95_.D7.94.D7.A9.D7.99.D7.A0.D7.95.D7.99_.D7.94.D7.95.D7.90_.D7.9E.D7.A7.D7.A1.D7.99.D7.9E.D7.9C.D7.99.3F"></span>למה וקטור גרדיאנט נותן את הכיוון בו השינוי הוא מקסימלי?</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=4" title="עריכת קוד המקור של הפרק: למה וקטור גרדיאנט נותן את הכיוון בו השינוי הוא מקסימלי?"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=4" title="עריכת פסקה: &quot;למה וקטור גרדיאנט נותן את הכיוון בו השינוי הוא מקסימלי?&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>נסתכל על ההגדרה של <a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA_%D7%9B%D7%99%D7%95%D7%95%D7%A0%D7%99%D7%AA" title="נגזרת כיוונית">נגזרת כיוונית</a>, ונשאל מהו הווקטור שבו ה<a href="/wiki/%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%A1%D7%A7%D7%9C%D7%A8%D7%99%D7%AA" title="מכפלה סקלרית">מכפלה הסקלרית</a> עם וקטור הגרדיאנט תהיה מקסימלית? מהגדרת המכפלה הסקלרית, ברור שהמקסימום יתקבל כאשר הזווית בין הווקטורים תהיה אפס (<a href="/wiki/%D7%A7%D7%95%D7%A1%D7%99%D7%A0%D7%95%D7%A1" title="קוסינוס">קוסינוס</a> מקבל ערך מקסימלי באפס), ומכאן נובע שהווקטור שבו המכפלה הסקלרית תהיה מקסימלית היא וקטור הגרדיאנט עצמו. </p> <div class="mw-heading mw-heading2"><h2 id="דוגמה"><span id=".D7.93.D7.95.D7.92.D7.9E.D7.94"></span>דוגמה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=5" title="עריכת קוד המקור של הפרק: דוגמה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=5" title="עריכת פסקה: &quot;דוגמה&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%D7%90%D7%A0%D7%A8%D7%92%D7%99%D7%94_%D7%A4%D7%95%D7%98%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="אנרגיה פוטנציאלית">הפוטנציאל</a> הוא שדה סקלרי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ U({\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ U({\vec {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f62f3f774d6293e97b4c964833645a0dd62693d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.396ex; height:2.843ex;" alt="{\displaystyle \ U({\vec {r}})}"></span>, והכוח שפועל על <a href="/wiki/%D7%97%D7%9C%D7%A7%D7%99%D7%A7" title="חלקיק">חלקיק</a> שנמצא בהשראת אנרגיה פוטנציאלית הוא הגרדיאנט של האנרגיה הפוטנציאלית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\vec {F}}({\vec {r}})=-{\nabla }U({\vec {r}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>&#x2212;<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> </mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\vec {F}}({\vec {r}})=-{\nabla }U({\vec {r}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db994377568dca26a940b0d80a9ab9175fa5b822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.041ex; height:3.343ex;" alt="{\displaystyle \ {\vec {F}}({\vec {r}})=-{\nabla }U({\vec {r}})}"></span> (ראה דוגמאות ספציפיות בערך <a href="/wiki/%D7%90%D7%A0%D7%A8%D7%92%D7%99%D7%94_%D7%A4%D7%95%D7%98%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="אנרגיה פוטנציאלית">אנרגיה פוטנציאלית</a>). לכן, הכוח שיפעל על הגוף ימשוך אותו לכיוון בו הפוטנציאל יקטן הכי הרבה. </p><p><b>דוגמה נוספת (חישובית):</b> </p><p>יהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {r} )=|\mathbf {r} |=r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {r} )=|\mathbf {r} |=r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47230a7f94a29f359827ca2345a1a11577ce831b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.831ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {r} )=|\mathbf {r} |=r}"></span> <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%A1%D7%A7%D7%9C%D7%A8%D7%99" title="שדה סקלרי">שדה סקלרי</a> (ה<a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">גזיר</a> בכל מקום פרט ל-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea91abce79b986f8283514f440f89893d27b8a02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.363ex; height:2.176ex;" alt="{\displaystyle \mathbf {r} =0}"></span>). אזי הגרדיאנט שלו (ב<a href="/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA_%D7%A7%D7%A8%D7%98%D7%96%D7%99%D7%95%D7%AA" class="mw-redirect" title="קואורדינטות קרטזיות">קואורדינטות קרטזיות</a>) הוא: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\nabla }f={\frac {\partial r}{\partial x}}{\hat {\mathbf {e} }}_{x}+{\frac {\partial r}{\partial y}}{\hat {\mathbf {e} }}_{y}+{\frac {\partial r}{\partial z}}{\hat {\mathbf {e} }}_{z}={\frac {x{\hat {\mathbf {e} }}_{x}+y{\hat {\mathbf {e} }}_{y}+z{\hat {\mathbf {e} }}_{z}}{\sqrt {x^{2}+y^{2}+z^{2}}}}={\frac {\mathbf {r} }{r}}={\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> </mrow> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>y</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>z</mi> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <mi>z</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\nabla }f={\frac {\partial r}{\partial x}}{\hat {\mathbf {e} }}_{x}+{\frac {\partial r}{\partial y}}{\hat {\mathbf {e} }}_{y}+{\frac {\partial r}{\partial z}}{\hat {\mathbf {e} }}_{z}={\frac {x{\hat {\mathbf {e} }}_{x}+y{\hat {\mathbf {e} }}_{y}+z{\hat {\mathbf {e} }}_{z}}{\sqrt {x^{2}+y^{2}+z^{2}}}}={\frac {\mathbf {r} }{r}}={\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca995e6630f65218396d8ebd06cd87f3319a58b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:58.314ex; height:7.009ex;" alt="{\displaystyle {\nabla }f={\frac {\partial r}{\partial x}}{\hat {\mathbf {e} }}_{x}+{\frac {\partial r}{\partial y}}{\hat {\mathbf {e} }}_{y}+{\frac {\partial r}{\partial z}}{\hat {\mathbf {e} }}_{z}={\frac {x{\hat {\mathbf {e} }}_{x}+y{\hat {\mathbf {e} }}_{y}+z{\hat {\mathbf {e} }}_{z}}{\sqrt {x^{2}+y^{2}+z^{2}}}}={\frac {\mathbf {r} }{r}}={\hat {\mathbf {r} }}}"></span></dd></dl> <p>שכן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial x}}r={\frac {\partial }{\partial x}}{\sqrt {x^{2}+y^{2}+z^{2}}}={\frac {2x}{2{\sqrt {x^{2}+y^{2}+z^{2}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial x}}r={\frac {\partial }{\partial x}}{\sqrt {x^{2}+y^{2}+z^{2}}}={\frac {2x}{2{\sqrt {x^{2}+y^{2}+z^{2}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f62e6ab6aa4156a1e6995084de51174fb68fe21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.707ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial }{\partial x}}r={\frac {\partial }{\partial x}}{\sqrt {x^{2}+y^{2}+z^{2}}}={\frac {2x}{2{\sqrt {x^{2}+y^{2}+z^{2}}}}}}"></span>.<br />כדאי לשים לב שהחישוב הוא מיידי ב<a href="/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA_%D7%9B%D7%93%D7%95%D7%A8%D7%99%D7%95%D7%AA" title="קואורדינטות כדוריות">קואורדינטות כדוריות</a>, שם הגרדיאנט נתון על ידי הנוסחה: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f={\frac {\partial f}{\partial r}}{\hat {\mathbf {r} }}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\hat {\mathbf {\theta } }}+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \phi }}{\hat {\mathbf {\phi } }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>&#x03B8;<!-- θ --></mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03B8;<!-- θ --></mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>r</mi> <mi>sin</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mi>&#x03B8;<!-- θ --></mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>&#x03D5;<!-- ϕ --></mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03D5;<!-- ϕ --></mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f={\frac {\partial f}{\partial r}}{\hat {\mathbf {r} }}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\hat {\mathbf {\theta } }}+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \phi }}{\hat {\mathbf {\phi } }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c11b1e783707ce30bb50825eb901cff70f657af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.988ex; height:6.176ex;" alt="{\displaystyle \nabla f={\frac {\partial f}{\partial r}}{\hat {\mathbf {r} }}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\hat {\mathbf {\theta } }}+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \phi }}{\hat {\mathbf {\phi } }}}"></span></dd></dl> <p>ומאחר ש <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial r}{\partial \theta }}=0={\frac {\partial r}{\partial \phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>&#x03B8;<!-- θ --></mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>&#x03D5;<!-- ϕ --></mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial r}{\partial \theta }}=0={\frac {\partial r}{\partial \phi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67da123eb35a8e92d96d7eb9839126a7ea3a135a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.144ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial r}{\partial \theta }}=0={\frac {\partial r}{\partial \phi }}}"></span> מקבלים ש- </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f={\frac {\partial f}{\partial r}}{\hat {\mathbf {r} }}={\frac {\partial r}{\partial r}}{\hat {\mathbf {r} }}={\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>r</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f={\frac {\partial f}{\partial r}}{\hat {\mathbf {r} }}={\frac {\partial r}{\partial r}}{\hat {\mathbf {r} }}={\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c27c7c3247abeaad2ee62046807ad74a0671898f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.633ex; height:5.676ex;" alt="{\displaystyle \nabla f={\frac {\partial f}{\partial r}}{\hat {\mathbf {r} }}={\frac {\partial r}{\partial r}}{\hat {\mathbf {r} }}={\hat {\mathbf {r} }}}"></span>.</dd></dl> <p>כצפוי, בשתי הדרכים קיבלנו תוצאה זהה. </p> <div class="mw-heading mw-heading2"><h2 id="גרדיאנט_באנליזה_על_יריעות"><span id=".D7.92.D7.A8.D7.93.D7.99.D7.90.D7.A0.D7.98_.D7.91.D7.90.D7.A0.D7.9C.D7.99.D7.96.D7.94_.D7.A2.D7.9C_.D7.99.D7.A8.D7.99.D7.A2.D7.95.D7.AA"></span>גרדיאנט באנליזה על יריעות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=6" title="עריכת קוד המקור של הפרק: גרדיאנט באנליזה על יריעות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=6" title="עריכת פסקה: &quot;גרדיאנט באנליזה על יריעות&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>את הגרדיאנט יותר טבעי להגדיר דווקא כ<a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%A0%D7%9C" title="פונקציונל">פונקציונל</a> הנקרא "<a href="/wiki/%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="דיפרנציאל (מתמטיקה)">דיפרנציאל</a>": </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ df=\sum _{\mu }{\frac {\partial f}{\partial x^{\mu }}}dx^{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>d</mi> <mi>f</mi> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ df=\sum _{\mu }{\frac {\partial f}{\partial x^{\mu }}}dx^{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f41819d9b99b8f6d01aeb2b7f93b6d623c86197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.392ex; height:7.009ex;" alt="{\displaystyle \ df=\sum _{\mu }{\frac {\partial f}{\partial x^{\mu }}}dx^{\mu }}"></span></dd></dl> <p>פונקציונל זה מקבל וקטור v ומחזיר את הסקלר: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (df)({\vec {v}})=\langle df,{\vec {v}}\rangle =\sum _{\mu }v^{\mu }{\frac {\partial f}{\partial x^{\mu }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mo stretchy="false">(</mo> <mi>d</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">&#x27E8;<!-- ⟨ --></mo> <mi>d</mi> <mi>f</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> <mo fence="false" stretchy="false">&#x27E9;<!-- ⟩ --></mo> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </munder> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (df)({\vec {v}})=\langle df,{\vec {v}}\rangle =\sum _{\mu }v^{\mu }{\frac {\partial f}{\partial x^{\mu }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53d326ebe1b173af1cbd7707932b40428dde9e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.379ex; height:7.009ex;" alt="{\displaystyle \ (df)({\vec {v}})=\langle df,{\vec {v}}\rangle =\sum _{\mu }v^{\mu }{\frac {\partial f}{\partial x^{\mu }}}}"></span></dd></dl> <p>האות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x03BC;<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> הכתובה בכתיב עילי מייצגת <a href="/wiki/%D7%90%D7%99%D7%A0%D7%93%D7%A7%D7%A1_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="אינדקס (מתמטיקה)">אינדקס</a> רץ שמייצג <a href="/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%94" class="mw-redirect" title="קואורדינטה">קואורדינטה</a> של וקטור ולא <a href="/wiki/%D7%97%D7%96%D7%A7%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="חזקה (מתמטיקה)">חזקה</a>. </p><p>במרחב עם <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A7%D7%94" title="מטריקה">מטריקה</a>, אפשר להגדיר את הגרדיאנט כווקטור על ידי "<a href="/wiki/%D7%94%D7%95%D7%A8%D7%93%D7%94_%D7%95%D7%94%D7%A2%D7%9C%D7%90%D7%94_%D7%A9%D7%9C_%D7%90%D7%99%D7%A0%D7%93%D7%A7%D7%A1%D7%99%D7%9D" title="הורדה והעלאה של אינדקסים">העלאת אינדקסים</a>", כלומר, על ידי התאמת וקטור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\nabla }f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\nabla }f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50f63d21957d549f5b6af4b980dad6040f41e32b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle {\nabla }f}"></span> ל-df כך ש- </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ df({\vec {v}})=g({\vec {v}},{\nabla }f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>d</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> </mrow> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ df({\vec {v}})=g({\vec {v}},{\nabla }f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9e4fdc0482fbe95480ebc7a52d3c26dac0c8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.507ex; height:2.843ex;" alt="{\displaystyle \ df({\vec {v}})=g({\vec {v}},{\nabla }f)}"></span></dd></dl> <p>כאשר g היא המטריקה: <a href="/wiki/%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%91%D7%99%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="תבנית ביליניארית">תבנית ביליניארית</a> סימטרית וחיובית בהחלט. </p><p>ב<a href="/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA" title="קואורדינטות">קואורדינטות</a> אפשר לכתוב את וקטור הגרדיאנט כך: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\nabla }f=\sum _{\mu ,\nu }g^{\mu \nu }(df)_{\nu }\partial _{\mu }=\sum _{\mu }\left(\sum _{\nu }g^{\mu \nu }{\frac {\partial f}{\partial x^{\nu }}}\right)\partial _{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> </mrow> <mi>f</mi> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> <mo>,</mo> <mi>&#x03BD;<!-- ν --></mi> </mrow> </munder> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> <mi>&#x03BD;<!-- ν --></mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>d</mi> <mi>f</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BD;<!-- ν --></mi> </mrow> </msub> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msub> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BD;<!-- ν --></mi> </mrow> </munder> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> <mi>&#x03BD;<!-- ν --></mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BD;<!-- ν --></mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\nabla }f=\sum _{\mu ,\nu }g^{\mu \nu }(df)_{\nu }\partial _{\mu }=\sum _{\mu }\left(\sum _{\nu }g^{\mu \nu }{\frac {\partial f}{\partial x^{\nu }}}\right)\partial _{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec345fab7470650713c65d52836faf81bbee673c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.623ex; height:7.676ex;" alt="{\displaystyle \ {\nabla }f=\sum _{\mu ,\nu }g^{\mu \nu }(df)_{\nu }\partial _{\mu }=\sum _{\mu }\left(\sum _{\nu }g^{\mu \nu }{\frac {\partial f}{\partial x^{\nu }}}\right)\partial _{\mu }}"></span></dd></dl> <p>כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{\mu \nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> <mi>&#x03BD;<!-- ν --></mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{\mu \nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a5070ab85fb1f8797a669a98003bcfcc62182d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.213ex; height:2.676ex;" alt="{\displaystyle g^{\mu \nu }}"></span> הוא האיבר בשורה ה-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x03BC;<!-- μ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> והעמודה ה--<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x03BD;<!-- ν --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }"></span> של המטריקה ההופכית (כלומר: ה<a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94" title="מטריצה">מטריצה</a> ההופכית למטריקה, g<sup>-1</sup>) ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\mu }={\frac {\partial {\vec {r}}}{\partial x^{\mu }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>r</mi> <mo stretchy="false">&#x2192;<!-- → --></mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>&#x03BC;<!-- μ --></mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }={\frac {\partial {\vec {r}}}{\partial x^{\mu }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504673665c1c6b27e5582912a47e01b2c8ef9b74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.264ex; height:5.509ex;" alt="{\displaystyle \partial _{\mu }={\frac {\partial {\vec {r}}}{\partial x^{\mu }}}}"></span> הם הווקטורים המשיקים הפורשים את <a href="/wiki/%D7%94%D7%9E%D7%A8%D7%97%D7%91_%D7%94%D7%9E%D7%A9%D7%99%D7%A7" title="המרחב המשיק">המרחב המשיק</a> בנקודה. </p> <div class="mw-heading mw-heading2"><h2 id="גרדיאנט_במערכת_קואורדינטות_אורתוגונליות_כלשהי"><span id=".D7.92.D7.A8.D7.93.D7.99.D7.90.D7.A0.D7.98_.D7.91.D7.9E.D7.A2.D7.A8.D7.9B.D7.AA_.D7.A7.D7.95.D7.90.D7.95.D7.A8.D7.93.D7.99.D7.A0.D7.98.D7.95.D7.AA_.D7.90.D7.95.D7.A8.D7.AA.D7.95.D7.92.D7.95.D7.A0.D7.9C.D7.99.D7.95.D7.AA_.D7.9B.D7.9C.D7.A9.D7.94.D7.99"></span>גרדיאנט במערכת קואורדינטות אורתוגונליות כלשהי</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=7" title="עריכת קוד המקור של הפרק: גרדיאנט במערכת קואורדינטות אורתוגונליות כלשהי"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=7" title="עריכת פסקה: &quot;גרדיאנט במערכת קואורדינטות אורתוגונליות כלשהי&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>נניח מערכת <a href="/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA" title="קואורדינטות">קואורדינטות</a> <a href="/wiki/%D7%90%D7%95%D7%A8%D7%AA%D7%95%D7%92%D7%95%D7%A0%D7%9C%D7%99%D7%95%D7%AA" title="אורתוגונליות">אורתוגונלית</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q_{1},q_{2},q_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q_{1},q_{2},q_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9bdab1698fc799d7114bcdeef602cc1b108ca1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.151ex; height:2.843ex;" alt="{\displaystyle (q_{1},q_{2},q_{3})}"></span>. כלומר, וקטורי היחידה המתאימים מקיימים: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {e} }}_{i}\cdot {\hat {\mathbf {e} }}_{j}=\delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>&#x22C5;<!-- ⋅ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>&#x03B4;<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {e} }}_{i}\cdot {\hat {\mathbf {e} }}_{j}=\delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b73e884f971623dcddba4ee3e89762e142d4566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.447ex; height:3.009ex;" alt="{\displaystyle {\hat {\mathbf {e} }}_{i}\cdot {\hat {\mathbf {e} }}_{j}=\delta _{ij}}"></span></dd></dl> <p>כאשר </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}={\begin{cases}1&amp;{\textrm {If}}\ i=j\\0&amp;{\textrm {If}}\ i\neq j\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>&#x03B4;<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>If</mtext> </mrow> </mrow> <mtext>&#xA0;</mtext> <mi>i</mi> <mo>=</mo> <mi>j</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>If</mtext> </mrow> </mrow> <mtext>&#xA0;</mtext> <mi>i</mi> <mo>&#x2260;<!-- ≠ --></mo> <mi>j</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}={\begin{cases}1&amp;{\textrm {If}}\ i=j\\0&amp;{\textrm {If}}\ i\neq j\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640e08c726264777b1e6f2a2085b57f93f84175b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.579ex; height:6.176ex;" alt="{\displaystyle \delta _{ij}={\begin{cases}1&amp;{\textrm {If}}\ i=j\\0&amp;{\textrm {If}}\ i\neq j\end{cases}}}"></span></dd></dl> <p>הוא ה<a href="/wiki/%D7%93%D7%9C%D7%AA%D7%90_%D7%A9%D7%9C_%D7%A7%D7%A8%D7%95%D7%A0%D7%A7%D7%A8" class="mw-redirect" title="דלתא של קרונקר">דלתא של קרונקר</a>. </p><p>הווקטורים המשיקים הם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {r} }{\partial q_{i}}}=h_{i}{\hat {\mathbf {e} }}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {r} }{\partial q_{i}}}=h_{i}{\hat {\mathbf {e} }}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d842e261fd8efc13de083f7b806c5f65e9d5a4b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.253ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {r} }{\partial q_{i}}}=h_{i}{\hat {\mathbf {e} }}_{i}}"></span>. </p><p>נרשום צורה כללית לגרדיאנט של פונקציה סקלרית כלשהי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f=\sum _{i=1}^{3}f_{i}{\hat {\mathbf {e} }}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>=</mo> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f=\sum _{i=1}^{3}f_{i}{\hat {\mathbf {e} }}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df1a45542f39a71b818f13df92ef66304d45f811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.019ex; height:7.176ex;" alt="{\displaystyle \nabla f=\sum _{i=1}^{3}f_{i}{\hat {\mathbf {e} }}_{i}}"></span></dd></dl> <p>ונמצא את המקדמים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1},f_{2},f_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1},f_{2},f_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8005d01ebb187d915834307d071c2ccb3aa4a852" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.648ex; height:2.509ex;" alt="{\displaystyle f_{1},f_{2},f_{3}}"></span>. </p><p>לשם כך נחשב את ה<a href="/wiki/%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="דיפרנציאל (מתמטיקה)">דיפרנציאל</a> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df=\sum _{i=1}^{3}{\frac {\partial f}{\partial q_{i}}}dq_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>f</mi> <mo>=</mo> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df=\sum _{i=1}^{3}{\frac {\partial f}{\partial q_{i}}}dq_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7adb849d9bf463135e0fe32ec093dd817a170b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.379ex; height:7.176ex;" alt="{\displaystyle df=\sum _{i=1}^{3}{\frac {\partial f}{\partial q_{i}}}dq_{i}}"></span></dd></dl> <p>מתקיים ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle df=\nabla f\cdot d\mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle df=\nabla f\cdot d\mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a3a80fcf642620d60ec5d82e0618fac56a13d0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.804ex; height:2.509ex;" alt="{\displaystyle df=\nabla f\cdot d\mathbf {r} }"></span> אבל </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\mathbf {r} =\sum _{i=1}^{3}{\frac {\partial \mathbf {r} }{\partial q_{i}}}dq_{i}=\sum _{i=1}^{3}h_{i}{\hat {\mathbf {e} }}_{i}dq_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {r} =\sum _{i=1}^{3}{\frac {\partial \mathbf {r} }{\partial q_{i}}}dq_{i}=\sum _{i=1}^{3}h_{i}{\hat {\mathbf {e} }}_{i}dq_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26f8be29d1cc7a275fb9a997e74fca1f0e9f4501" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.259ex; height:7.176ex;" alt="{\displaystyle d\mathbf {r} =\sum _{i=1}^{3}{\frac {\partial \mathbf {r} }{\partial q_{i}}}dq_{i}=\sum _{i=1}^{3}h_{i}{\hat {\mathbf {e} }}_{i}dq_{i}}"></span></dd></dl> <p>ולכן, כיוון שווקטורי היחידה המשיקים הם אורתוגונליים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {e} }}_{i}\cdot {\hat {\mathbf {e} }}_{j}=\delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>&#x22C5;<!-- ⋅ --></mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>&#x03B4;<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {e} }}_{i}\cdot {\hat {\mathbf {e} }}_{j}=\delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b73e884f971623dcddba4ee3e89762e142d4566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.447ex; height:3.009ex;" alt="{\displaystyle {\hat {\mathbf {e} }}_{i}\cdot {\hat {\mathbf {e} }}_{j}=\delta _{ij}}"></span> מתקיים </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{3}{\frac {\partial f}{\partial q_{i}}}dq_{i}=df=\nabla f\cdot d\mathbf {r} =\sum _{i=1}^{3}f_{i}h_{i}dq_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>d</mi> <mi>f</mi> <mo>=</mo> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{3}{\frac {\partial f}{\partial q_{i}}}dq_{i}=df=\nabla f\cdot d\mathbf {r} =\sum _{i=1}^{3}f_{i}h_{i}dq_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba57a03ba3dfe28954b44c0cf2145299ceb3dd7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:40.659ex; height:7.176ex;" alt="{\displaystyle \sum _{i=1}^{3}{\frac {\partial f}{\partial q_{i}}}dq_{i}=df=\nabla f\cdot d\mathbf {r} =\sum _{i=1}^{3}f_{i}h_{i}dq_{i}}"></span></dd></dl> <p>מכיוון ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dq_{1},dq_{2},dq_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dq_{1},dq_{2},dq_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852308688a193b9fbb4d06a8216d5299583bf1c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.989ex; height:2.509ex;" alt="{\displaystyle dq_{1},dq_{2},dq_{3}}"></span> בלתי תלויים נשווה מקדמים איבר איבר ונקבל </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}={\frac {1}{h_{i}}}{\frac {\partial f}{\partial q_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}={\frac {1}{h_{i}}}{\frac {\partial f}{\partial q_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fd3dbda6ffd572066eaf7d3606da8381142bf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.003ex; height:6.009ex;" alt="{\displaystyle f_{i}={\frac {1}{h_{i}}}{\frac {\partial f}{\partial q_{i}}}}"></span></dd></dl> <p>ומכאן נקבל </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f=\sum _{i=1}^{3}{\frac {1}{h_{i}}}{\frac {\partial f}{\partial q_{i}}}{\hat {\mathbf {e} }}_{i}={\frac {1}{h_{1}}}{\frac {\partial f}{\partial q^{1}}}{\hat {\mathbf {e} }}_{1}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial q^{2}}}{\hat {\mathbf {e} }}_{2}+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial q^{3}}}{\hat {\mathbf {e} }}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>=</mo> <munderover> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f=\sum _{i=1}^{3}{\frac {1}{h_{i}}}{\frac {\partial f}{\partial q_{i}}}{\hat {\mathbf {e} }}_{i}={\frac {1}{h_{1}}}{\frac {\partial f}{\partial q^{1}}}{\hat {\mathbf {e} }}_{1}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial q^{2}}}{\hat {\mathbf {e} }}_{2}+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial q^{3}}}{\hat {\mathbf {e} }}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d1d53ac866e8916b90c618ae0c39605991ad69a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:57.215ex; height:7.176ex;" alt="{\displaystyle \nabla f=\sum _{i=1}^{3}{\frac {1}{h_{i}}}{\frac {\partial f}{\partial q_{i}}}{\hat {\mathbf {e} }}_{i}={\frac {1}{h_{1}}}{\frac {\partial f}{\partial q^{1}}}{\hat {\mathbf {e} }}_{1}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial q^{2}}}{\hat {\mathbf {e} }}_{2}+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial q^{3}}}{\hat {\mathbf {e} }}_{3}}"></span></dd></dl> <p>כנדרש. </p><p>הערה: ההכללה ל<a href="/wiki/%D7%9E%D7%9E%D7%93_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA)" title="ממד (אלגברה ליניארית)">ממד</a> כלשהו n מיידית. עובדים עם הצגת הסכומים כ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x03A3;<!-- Σ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e1f558f53cda207614abdf90162266c70bc5c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }"></span> ונותנים לכל אינדקס לרוץ מ-1 עד n. </p> <div class="mw-heading mw-heading3"><h3 id="הוכחה_המבוססת_על_גאומטריה_דיפרנציאלית"><span id=".D7.94.D7.95.D7.9B.D7.97.D7.94_.D7.94.D7.9E.D7.91.D7.95.D7.A1.D7.A1.D7.AA_.D7.A2.D7.9C_.D7.92.D7.90.D7.95.D7.9E.D7.98.D7.A8.D7.99.D7.94_.D7.93.D7.99.D7.A4.D7.A8.D7.A0.D7.A6.D7.99.D7.90.D7.9C.D7.99.D7.AA"></span>הוכחה המבוססת על <a href="/wiki/%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="גאומטריה דיפרנציאלית">גאומטריה דיפרנציאלית</a></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=8" title="עריכת קוד המקור של הפרק: הוכחה המבוססת על גאומטריה דיפרנציאלית"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=8" title="עריכת פסקה: &quot;הוכחה המבוססת על גאומטריה דיפרנציאלית&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>נניח מערכת <a href="/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA" title="קואורדינטות">קואורדינטות</a> <a href="/wiki/%D7%90%D7%95%D7%A8%D7%AA%D7%95%D7%92%D7%95%D7%A0%D7%9C%D7%99%D7%95%D7%AA" title="אורתוגונליות">אורתוגונלית</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q^{1},q^{2},q^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q^{1},q^{2},q^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd050e8679bb81579db593363e56f09c0da9b4c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.278ex; height:3.176ex;" alt="{\displaystyle (q^{1},q^{2},q^{3})}"></span>. </p><p>הווקטורים המשיקים הם: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{q^{i}}={\frac {\partial \mathbf {r} }{\partial q^{i}}}=h_{i}{\hat {\mathbf {e} }}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{q^{i}}={\frac {\partial \mathbf {r} }{\partial q^{i}}}=h_{i}{\hat {\mathbf {e} }}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/133af8e91f7ea26dca853011d297eae676cd60cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.249ex; height:6.009ex;" alt="{\displaystyle \partial _{q^{i}}={\frac {\partial \mathbf {r} }{\partial q^{i}}}=h_{i}{\hat {\mathbf {e} }}_{i}}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {e} }}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {e} }}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38435968475058bd50296f545cc5a8ae9c22b883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.025ex; height:2.676ex;" alt="{\displaystyle {\hat {\mathbf {e} }}_{i}}"></span> הם וקטורים משיקים <a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_%D7%99%D7%97%D7%99%D7%93%D7%94" title="וקטור יחידה">מנורמלים</a> ו-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d535f210cbd9b9fe6689e61427b3e213e5b2d547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.139ex; height:2.509ex;" alt="{\displaystyle h_{i}}"></span> הם ה-<a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ScaleFactor.html">scale factors</a>. </p><p>ה<a href="/w/index.php?title=%D7%98%D7%A0%D7%96%D7%95%D7%A8_%D7%9E%D7%98%D7%A8%D7%99&amp;action=edit&amp;redlink=1" class="new" title="טנזור מטרי (הדף אינו קיים)">טנזור המטרי</a> במקרה של קואורדינטות אורתוגונליות </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g={\mbox{diag}}(h_{1}^{2},h_{2}^{2},h_{3}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>diag</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g={\mbox{diag}}(h_{1}^{2},h_{2}^{2},h_{3}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d567562358461bcfd34645eab4089b23ec95d5be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.536ex; height:3.176ex;" alt="{\displaystyle g={\mbox{diag}}(h_{1}^{2},h_{2}^{2},h_{3}^{2})}"></span></dd></dl> <p>הוא <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%90%D7%9C%D7%9B%D7%A1%D7%95%D7%A0%D7%99%D7%AA" title="מטריצה אלכסונית">מטריצה אלכסונית</a>, ונזכור ש- </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ dq^{i}=\sum _{j}g^{ij}\partial _{q^{j}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext>&#xA0;</mtext> <mi>d</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ dq^{i}=\sum _{j}g^{ij}\partial _{q^{j}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff1cff63663c07a5dd2b4d7ed8181b71205f329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.056ex; height:5.843ex;" alt="{\displaystyle \ dq^{i}=\sum _{j}g^{ij}\partial _{q^{j}}}"></span></dd></dl> <p>כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{ij}=g^{-1}={\mbox{diag}}(1/h_{1}^{2},1/h_{2}^{2},1/h_{3}^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>&#x2212;<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>diag</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{ij}=g^{-1}={\mbox{diag}}(1/h_{1}^{2},1/h_{2}^{2},1/h_{3}^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81ab4fe27fe19076abbed5a9b3f38648edd0348e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.539ex; height:3.343ex;" alt="{\displaystyle g^{ij}=g^{-1}={\mbox{diag}}(1/h_{1}^{2},1/h_{2}^{2},1/h_{3}^{2})}"></span> היא ה<a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94" title="מטריצה">מטריצה</a> ההופכית ל-g. </p><p>נציב בהגדרת הגרדיאנט, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f=\sum _{i}{\frac {\partial f}{\partial q^{i}}}dq^{i}=\sum _{ij}g^{ij}{\frac {\partial f}{\partial q^{i}}}\partial _{q^{j}}=\sum _{ij}{\frac {1}{h_{i}^{2}}}\delta ^{ij}{\frac {\partial f}{\partial q^{i}}}\partial _{q^{j}}=\sum _{i}{\frac {1}{h_{i}^{2}}}{\frac {\partial f}{\partial q^{i}}}h_{i}{\hat {\mathbf {e} }}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </munder> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </msub> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <msup> <mi>&#x03B4;<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> </msub> <mo>=</mo> <munder> <mo>&#x2211;<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f=\sum _{i}{\frac {\partial f}{\partial q^{i}}}dq^{i}=\sum _{ij}g^{ij}{\frac {\partial f}{\partial q^{i}}}\partial _{q^{j}}=\sum _{ij}{\frac {1}{h_{i}^{2}}}\delta ^{ij}{\frac {\partial f}{\partial q^{i}}}\partial _{q^{j}}=\sum _{i}{\frac {1}{h_{i}^{2}}}{\frac {\partial f}{\partial q^{i}}}h_{i}{\hat {\mathbf {e} }}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c53f3225a236bddbe55f68846666f5e44113f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:71.442ex; height:7.009ex;" alt="{\displaystyle \nabla f=\sum _{i}{\frac {\partial f}{\partial q^{i}}}dq^{i}=\sum _{ij}g^{ij}{\frac {\partial f}{\partial q^{i}}}\partial _{q^{j}}=\sum _{ij}{\frac {1}{h_{i}^{2}}}\delta ^{ij}{\frac {\partial f}{\partial q^{i}}}\partial _{q^{j}}=\sum _{i}{\frac {1}{h_{i}^{2}}}{\frac {\partial f}{\partial q^{i}}}h_{i}{\hat {\mathbf {e} }}_{i}}"></span></dd></dl> <p>(המעבר השלישי נעשה כי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b14a4aa3b277a89268fd9026b8f16a749199cb10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.595ex; height:3.009ex;" alt="{\displaystyle g^{ij}}"></span> <a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%90%D7%9C%D7%9B%D7%A1%D7%95%D7%A0%D7%99%D7%AA" title="מטריצה אלכסונית">מטריצה אלכסונית</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{ij}=(1/h_{i}^{2})\delta ^{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <msup> <mi>&#x03B4;<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{ij}=(1/h_{i}^{2})\delta ^{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fccc8e72dc1ab468bf7b8e77ccb82bcd3a825cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.752ex; height:3.343ex;" alt="{\displaystyle g^{ij}=(1/h_{i}^{2})\delta ^{ij}}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ^{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>&#x03B4;<!-- δ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ^{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742f3fcc9c517aab2ce39e9ff3a4e07e696ce381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.531ex; height:2.676ex;" alt="{\displaystyle \delta ^{ij}}"></span> היא <a href="/wiki/%D7%94%D7%93%D7%9C%D7%AA%D7%90_%D7%A9%D7%9C_%D7%A7%D7%A8%D7%95%D7%A0%D7%A7%D7%A8" title="הדלתא של קרונקר">הדלתא של קרונקר</a>) <br /> ובסך הכול נקבל </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f={\frac {1}{h_{1}}}{\frac {\partial f}{\partial q^{1}}}{\hat {\mathbf {e} }}_{1}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial q^{2}}}{\hat {\mathbf {e} }}_{2}+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial q^{3}}}{\hat {\mathbf {e} }}_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mo stretchy="false">&#x005E;<!-- ^ --></mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f={\frac {1}{h_{1}}}{\frac {\partial f}{\partial q^{1}}}{\hat {\mathbf {e} }}_{1}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial q^{2}}}{\hat {\mathbf {e} }}_{2}+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial q^{3}}}{\hat {\mathbf {e} }}_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f26c09c132ff11c329142271b72612ae5ba1cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.384ex; height:6.176ex;" alt="{\displaystyle \nabla f={\frac {1}{h_{1}}}{\frac {\partial f}{\partial q^{1}}}{\hat {\mathbf {e} }}_{1}+{\frac {1}{h_{2}}}{\frac {\partial f}{\partial q^{2}}}{\hat {\mathbf {e} }}_{2}+{\frac {1}{h_{3}}}{\frac {\partial f}{\partial q^{3}}}{\hat {\mathbf {e} }}_{3}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="קשרים_בין_אופרטורים"><span id=".D7.A7.D7.A9.D7.A8.D7.99.D7.9D_.D7.91.D7.99.D7.9F_.D7.90.D7.95.D7.A4.D7.A8.D7.98.D7.95.D7.A8.D7.99.D7.9D"></span>קשרים בין אופרטורים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=9" title="עריכת קוד המקור של הפרק: קשרים בין אופרטורים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=9" title="עריכת פסקה: &quot;קשרים בין אופרטורים&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><ul><li><a href="/wiki/%D7%93%D7%99%D7%91%D7%A8%D7%92%D7%A0%D7%A5" title="דיברגנץ">דיברגנץ</a>, גרדיאנט ו<a href="/wiki/%D7%9C%D7%A4%D7%9C%D7%A1%D7%99%D7%90%D7%9F" title="לפלסיאן">לפלסיאן</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} \ \operatorname {grad} =\Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>div</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mtext>&#xA0;</mtext> <mi>grad</mi> <mo>=</mo> <mi mathvariant="normal">&#x0394;<!-- Δ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {div} \ \operatorname {grad} =\Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a4b7cd868ca7aea634277f13eb91fb288678bde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.085ex; height:2.509ex;" alt="{\displaystyle \operatorname {div} \ \operatorname {grad} =\Delta }"></span></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="משפט_הגרדיאנט"><span id=".D7.9E.D7.A9.D7.A4.D7.98_.D7.94.D7.92.D7.A8.D7.93.D7.99.D7.90.D7.A0.D7.98"></span>משפט הגרדיאנט</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=10" title="עריכת קוד המקור של הפרק: משפט הגרדיאנט"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=10" title="עריכת פסקה: &quot;משפט הגרדיאנט&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" title="משפט הגרדיאנט">משפט הגרדיאנט</a> אומר שאם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {r} )=f(x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathbf {r} )=f(x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29845d391efa18b39b3c127696694cdb9db26ed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.017ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {r} )=f(x,y,z)}"></span> היא <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%A1%D7%A7%D7%9C%D7%A8%D7%99" title="שדה סקלרי">שדה סקלרי</a> (פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">&#x2192;<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaacd09d802018f742872a07cffb088d594218d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.24ex; height:3.009ex;" alt="{\displaystyle f:\mathbb {R} ^{3}\to \mathbb {R} }"></span> ) חלק מספיק (גזיר ברציפות), אזי לכל <a href="/wiki/%D7%A2%D7%A7%D7%95%D7%9E%D7%94" title="עקומה">מסילה</a> שמתחילה בנקודה כלשהי <b>A</b> ומסתיימת בנקודה כלשהי <b>B</b> ה<a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A7%D7%95%D7%95%D7%99" title="אינטגרל קווי">אינטגרל הקווי</a> על <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b4d6de89b52c5a5e6e1583cb63eaee263e307b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \nabla f}"></span> לאורך המסילה לא תלוי במסילה עצמה אלא רק בנקודות הקצה ומתקיים </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbf {A} }^{\mathbf {B} }\nabla f\cdot d\mathbf {r} =f(\mathbf {B} )-f(\mathbf {A} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>&#x222B;<!-- ∫ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> </msubsup> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>&#x22C5;<!-- ⋅ --></mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>&#x2212;<!-- − --></mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbf {A} }^{\mathbf {B} }\nabla f\cdot d\mathbf {r} =f(\mathbf {B} )-f(\mathbf {A} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265ceed641a61615c3d550662983bb7d7b4111bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.674ex; height:6.176ex;" alt="{\displaystyle \int _{\mathbf {A} }^{\mathbf {B} }\nabla f\cdot d\mathbf {r} =f(\mathbf {B} )-f(\mathbf {A} )}"></span></dd></dl> <p>לשדה הווקטורי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b4d6de89b52c5a5e6e1583cb63eaee263e307b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \nabla f}"></span> קוראים "<a href="/wiki/%D7%A9%D7%93%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99_%D7%9E%D7%A9%D7%9E%D7%A8" title="שדה וקטורי משמר">שדה וקטורי משמר</a>" או בהקשר של <a href="/wiki/%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94" title="פיזיקה">פיזיקה</a> "<a href="/wiki/%D7%9B%D7%95%D7%97_%D7%9E%D7%A9%D7%9E%D7%A8" title="כוח משמר">כוח משמר</a>" ומתקיים </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \nabla f=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mo>&#x00D7;<!-- × --></mo> <mi mathvariant="normal">&#x2207;<!-- ∇ --></mi> <mi>f</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \nabla f=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7657f22df2c178fdab95b8f35f766f91143e13be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.252ex; height:2.509ex;" alt="{\displaystyle \nabla \times \nabla f=0}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="ראו_גם"><span id=".D7.A8.D7.90.D7.95_.D7.92.D7.9D"></span>ראו גם</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=11" title="עריכת קוד המקור של הפרק: ראו גם"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=11" title="עריכת פסקה: &quot;ראו גם&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%A0%D7%9C" title="פונקציונל">פונקציונל</a></li> <li><a href="/wiki/%D7%98%D7%A0%D7%96%D7%95%D7%A8" title="טנזור">טנזור</a></li> <li><a href="/wiki/%D7%A7%D7%95%D7%90%D7%95%D7%A8%D7%93%D7%99%D7%A0%D7%98%D7%95%D7%AA" title="קואורדינטות">קואורדינטות</a></li> <li><a href="/wiki/%D7%93%D7%99%D7%91%D7%A8%D7%92%D7%A0%D7%A5" title="דיברגנץ">דיברגנץ</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="לקריאה_נוספת"><span id=".D7.9C.D7.A7.D7.A8.D7.99.D7.90.D7.94_.D7.A0.D7.95.D7.A1.D7.A4.D7.AA"></span>לקריאה נוספת</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=12" title="עריכת קוד המקור של הפרק: לקריאה נוספת"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=12" title="עריכת פסקה: &quot;לקריאה נוספת&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>מורי ר. שפיגל, <b>אנליזה וקטורית</b>, <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%A9%D7%90%D7%95%D7%9D" title="סדרת שאום">סדרת שאום</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="קישורים_חיצוניים"><span id=".D7.A7.D7.99.D7.A9.D7.95.D7.A8.D7.99.D7.9D_.D7.97.D7.99.D7.A6.D7.95.D7.A0.D7.99.D7.99.D7.9D"></span>קישורים חיצוניים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;action=edit&amp;section=13" title="עריכת קוד המקור של הפרק: קישורים חיצוניים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98&amp;veaction=edit&amp;section=13" title="עריכת פסקה: &quot;קישורים חיצוניים&quot;" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterwikilinkT"><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/" title="ויקישיתוף"><img alt="ויקישיתוף" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> מדיה וקבצים בנושא <b><a href="https://commons.wikimedia.org/wiki/Category:Gradient_fields" class="extiw" title="commons:Category:Gradient fields">גרדיאנט</a></b> ב<a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A9%D7%99%D7%AA%D7%95%D7%A3" title="ויקישיתוף">וויקישיתוף</a></div> <ul><li><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Gradient">גרדיאנט</a>, באתר <a href="/wiki/%D7%90%D7%A0%D7%A6%D7%99%D7%A7%D7%9C%D7%95%D7%A4%D7%93%D7%99%D7%94_%D7%9C%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="אנציקלופדיה למתמטיקה">אנציקלופדיה למתמטיקה</a> <span dir="rtl" class="languageicon">(באנגלית)</span><style data-mw-deduplicate="TemplateStyles:r36549940">.mw-parser-output .languageicon{font-size:0.95em;font-weight:bold;color:#555}</style></li> <li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Gradient.html">גרדיאנט</a>, באתר <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> <span dir="rtl" class="languageicon">(באנגלית)</span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36549940"></li></ul> <p><br /> </p> <table class="navbox nowraplinks mw-collapsible autocollapse" style="width: 90%; clear: both; margin: 0.5em auto; margin-top: 0.5em; margin-bottom: 0.5em; padding: 0.2em; text-align: right;"> <tbody><tr> <th colspan="3" style="text-align: center; padding-top: 0.1em; padding-bottom: 0.1em; color: black; background:#d1eeee; font-weight: bold;"><a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="אנליזה וקטורית">אנליזה וקטורית</a> </th></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">מושגים </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA_-_%D7%9E%D7%95%D7%A0%D7%97%D7%99%D7%9D" title="אנליזה מתמטית - מונחים">אנליזה מתמטית - מונחים</a> • <a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">מרחב וקטורי</a> • <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%A1%D7%A7%D7%9C%D7%A8%D7%99" title="שדה סקלרי">שדה סקלרי</a> • <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="שדה וקטורי">שדה וקטורי</a> • <a class="mw-selflink selflink">גרדיאנט</a> • <a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA_%D7%9B%D7%99%D7%95%D7%95%D7%A0%D7%99%D7%AA" title="נגזרת כיוונית">נגזרת כיוונית</a> • <a href="/wiki/%D7%93%D7%99%D7%91%D7%A8%D7%92%D7%A0%D7%A5" title="דיברגנץ">דיברגנץ</a> • <a href="/wiki/%D7%A8%D7%95%D7%98%D7%95%D7%A8_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="רוטור (מתמטיקה)">רוטור</a> • <a href="/wiki/%D7%9C%D7%A4%D7%9C%D7%A1%D7%99%D7%90%D7%9F" title="לפלסיאן">לפלסיאן</a> • <a href="/wiki/%D7%93%D7%9C_%D7%91%D7%9E%D7%A2%D7%A8%D7%9B%D7%95%D7%AA_%D7%A6%D7%99%D7%A8%D7%99%D7%9D_%D7%A9%D7%95%D7%A0%D7%95%D7%AA" title="דל במערכות צירים שונות">דל במערכות צירים שונות</a> • <a href="/wiki/%D7%93%27%D7%90%D7%9C%D7%9E%D7%91%D7%A8%D7%98%D7%99%D7%90%D7%9F" title="ד&#39;אלמברטיאן">ד'אלמברטיאן</a> • <a href="/wiki/%D7%A4%D7%95%D7%98%D7%A0%D7%A6%D7%99%D7%90%D7%9C_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="פוטנציאל וקטורי (מתמטיקה)">פוטנציאל וקטורי</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">משפטים </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%90%D7%95%D7%A1" title="משפט גאוס">משפט גאוס</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F" title="משפט גרין">משפט גרין</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" title="משפט הגרדיאנט">משפט הגרדיאנט</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A1%D7%98%D7%95%D7%A7%D7%A1" title="משפט סטוקס">משפט סטוקס</a> </td></tr> <tr> <td colspan="3" style="background-color: #F2F3F4; text-align: center; font-weight: bold;"><a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA" title="אנליזה מתמטית">אנליזה מתמטית</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="אנליזה וקטורית">אנליזה וקטורית</a> • <a href="/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94" title="טופולוגיה">טופולוגיה</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA" title="אנליזה מרוכבת">אנליזה מרוכבת</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%A0%D7%9C%D7%99%D7%AA" title="אנליזה פונקציונלית">אנליזה פונקציונלית</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%99%D7%93%D7%94" title="תורת המידה">תורת המידה</a> • <a href="/wiki/%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="גאומטריה דיפרנציאלית">גאומטריה דיפרנציאלית</a> </td></tr> </tbody></table> <div role="navigation" class="navbox authority-control" aria-labelledby="בקרת_זהויות_15px&amp;#124;link=https&amp;#58;//www.wikidata.org/wiki/Q173582?uselang=he&amp;#124;עריכת_הנתון_בוויקינתונים" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="בקרת_זהויות_15px&amp;#124;link=https&amp;#58;//www.wikidata.org/wiki/Q173582?uselang=he&amp;#124;עריכת_הנתון_בוויקינתונים" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A7%D7%A8%D7%AA_%D7%96%D7%94%D7%95%D7%99%D7%95%D7%AA" title="עזרה:בקרת זהויות">בקרת זהויות</a> <span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q173582?uselang=he" title="עריכת הנתון בוויקינתונים"><img alt="עריכת הנתון בוויקינתונים" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/15px-Blue_pencil_RTL.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/23px-Blue_pencil_RTL.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/30px-Blue_pencil_RTL.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></th><td class="navbox-list navbox-odd" style="text-align:left;border-right-width:2px;border-right-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><div dir="ltr"> <ul><li><span class="nowrap"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5_%D7%91%D7%A7%D7%A8%D7%94_%D7%9E%D7%A9%D7%95%D7%9C%D7%91" title="קובץ בקרה משולב">GND</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4323954-7">4323954-7</a></span></span></li></ul> </div></div></td></tr></tbody></table></div> <!-- NewPP limit report Parsed by mw‐web.eqiad.main‐64476968cd‐t9q9j Cached time: 20241101213557 Cache expiry: 2592000 Reduced expiry: false Complications: [show‐toc] CPU time usage: 0.185 seconds 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