משפט גרין – ויקיפדיה

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mw-list-item user-links-collapsible-item"><a data-mw="interface" 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&returnto=%D7%9E%D7%A9%D7%A4%D7%98+%D7%92%D7%A8%D7%99%D7%9F" 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&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&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&returnto=%D7%9E%D7%A9%D7%A4%D7%98+%D7%92%D7%A8%D7%99%D7%9F" 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&returnto=%D7%9E%D7%A9%D7%A4%D7%98+%D7%92%D7%A8%D7%99%D7%9F" 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 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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-סימון_מקובל_בפיזיקה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#סימון_מקובל_בפיזיקה"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>סימון מקובל בפיזיקה</span> </div> </a> <ul id="toc-סימון_מקובל_בפיזיקה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-שימוש_לדוגמה:_שדה_מגנטי_מסביב_לתיל_נושא_זרם" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#שימוש_לדוגמה:_שדה_מגנטי_מסביב_לתיל_נושא_זרם"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>שימוש לדוגמה: שדה מגנטי מסביב לתיל נושא זרם</span> </div> </a> <ul id="toc-שימוש_לדוגמה:_שדה_מגנטי_מסביב_לתיל_נושא_זרם-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הוכחה_עבור_תחום_מלבני" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הוכחה_עבור_תחום_מלבני"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>הוכחה עבור תחום מלבני</span> </div> </a> <ul id="toc-הוכחה_עבור_תחום_מלבני-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">4</span> <span>ראו גם</span> </div> </a> <ul id="toc-ראו_גם-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">5</span> <span>קישורים חיצוניים</span> </div> </a> <ul id="toc-קישורים_חיצוניים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הערות_שוליים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הערות_שוליים"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>הערות שוליים</span> </div> </a> <ul id="toc-הערות_שוליים-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">משפט גרין</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־39 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-39" 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">39 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/Green%27s_theorem" title="Green's theorem – אנגלית" lang="en" hreflang="en" data-title="Green's theorem" data-language-autonym="English" data-language-local-name="אנגלית" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D8%BA%D8%B1%D9%8A%D9%86" title="مبرهنة غرين – ערבית" lang="ar" hreflang="ar" data-title="مبرهنة غرين" data-language-autonym="العربية" data-language-local-name="ערבית" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teorema_de_Green" title="Teorema de Green – אסטורית" lang="ast" hreflang="ast" data-title="Teorema de Green" data-language-autonym="Asturianu" data-language-local-name="אסטורית" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B8_%D0%BD%D0%B0_%D0%93%D1%80%D0%B8%D0%B9%D0%BD" title="Теореми на Грийн – בולגרית" lang="bg" hreflang="bg" data-title="Теореми на Грийн" data-language-autonym="Български" data-language-local-name="בולגרית" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Greenova_teorema" title="Greenova teorema – בוסנית" lang="bs" hreflang="bs" data-title="Greenova teorema" 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/Teorema_de_Green" title="Teorema de Green – קטלאנית" lang="ca" hreflang="ca" data-title="Teorema de Green" 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/Greenova_v%C4%9Bta" title="Greenova věta – צ׳כית" lang="cs" hreflang="cs" data-title="Greenova věta" 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%B8%D0%BD_%D1%84%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B8" title="Грин формули – צ׳ובשית" lang="cv" hreflang="cv" data-title="Грин формули" data-language-autonym="Чӑвашла" data-language-local-name="צ׳ובשית" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_von_Green" title="Satz von Green – גרמנית" lang="de" hreflang="de" data-title="Satz von Green" 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/Teoremo_de_Green" title="Teoremo de Green – אספרנטו" lang="eo" hreflang="eo" data-title="Teoremo de Green" 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/Teorema_de_Green" title="Teorema de Green – ספרדית" lang="es" hreflang="es" data-title="Teorema de Green" 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/Greeni_valem" title="Greeni valem – אסטונית" lang="et" hreflang="et" data-title="Greeni valem" data-language-autonym="Eesti" data-language-local-name="אסטונית" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%DA%AF%D8%B1%DB%8C%D9%86" title="قضیه گرین – פרסית" lang="fa" hreflang="fa" data-title="قضیه گرین" 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/Greenin_teoreema" title="Greenin teoreema – פינית" lang="fi" hreflang="fi" data-title="Greenin teoreema" 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/Th%C3%A9or%C3%A8me_de_Green" title="Théorème de Green – צרפתית" lang="fr" hreflang="fr" data-title="Théorème de Green" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teorema_de_Green" title="Teorema de Green – גליסית" lang="gl" hreflang="gl" data-title="Teorema de Green" data-language-autonym="Galego" data-language-local-name="גליסית" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A5%8D%E0%A4%B0%E0%A5%80%E0%A4%A8_%E0%A4%95%E0%A4%BE_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AE%E0%A5%87%E0%A4%AF" title="ग्रीन का प्रमेय – הינדי" lang="hi" hreflang="hi" data-title="ग्रीन का प्रमेय" data-language-autonym="हिन्दी" data-language-local-name="הינדי" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D6%80%D5%AB%D5%B6%D5%AB_%D5%A2%D5%A1%D5%B6%D5%A1%D5%B1%D6%87%D5%A5%D6%80" title="Գրինի բանաձևեր – ארמנית" lang="hy" hreflang="hy" data-title="Գրինի բանաձևեր" 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/Teorema_Green" title="Teorema Green – אינדונזית" lang="id" hreflang="id" data-title="Teorema Green" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/L%C3%B6gm%C3%A1l_Greens" title="Lögmál Greens – איסלנדית" lang="is" hreflang="is" data-title="Lögmál Greens" 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/Teorema_di_Green" title="Teorema di Green – איטלקית" lang="it" hreflang="it" data-title="Teorema di Green" 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/%E3%82%B0%E3%83%AA%E3%83%BC%E3%83%B3%E3%81%AE%E5%AE%9A%E7%90%86" title="グリーンの定理 – יפנית" lang="ja" hreflang="ja" data-title="グリーンの定理" 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-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%91%E1%9F%92%E1%9E%9A%E1%9E%B9%E1%9E%9F%E1%9F%92%E1%9E%8F%E1%9E%B8%E1%9E%94%E1%9E%91%E1%9E%82%E1%9F%92%E1%9E%9A%E1%9E%B8%E1%9E%93" title="ទ្រឹស្តីបទគ្រីន – קמרית" lang="km" hreflang="km" data-title="ទ្រឹស្តីបទគ្រីន" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="קמרית" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B7%B8%EB%A6%B0_%EC%A0%95%EB%A6%AC" title="그린 정리 – קוריאנית" lang="ko" hreflang="ko" data-title="그린 정리" data-language-autonym="한국어" data-language-local-name="קוריאנית" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Gryno_formul%C4%97" title="Gryno formulė – ליטאית" lang="lt" hreflang="lt" data-title="Gryno formulė" data-language-autonym="Lietuvių" data-language-local-name="ליטאית" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%93%D1%80%D0%B8%D0%BD%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Гринова теорема – מקדונית" lang="mk" hreflang="mk" data-title="Гринова теорема" data-language-autonym="Македонски" data-language-local-name="מקדונית" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Green" title="Stelling van Green – הולנדית" lang="nl" hreflang="nl" data-title="Stelling van Green" data-language-autonym="Nederlands" data-language-local-name="הולנדית" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Greens_teorem" title="Greens teorem – נורווגית ספרותית" lang="nb" hreflang="nb" data-title="Greens teorem" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Greena" title="Twierdzenie Greena – פולנית" lang="pl" hreflang="pl" data-title="Twierdzenie Greena" 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/Teorema_de_Green" title="Teorema de Green – פורטוגזית" lang="pt" hreflang="pt" data-title="Teorema de Green" data-language-autonym="Português" data-language-local-name="פורטוגזית" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%93%D1%80%D0%B8%D0%BD%D0%B0" title="Теорема Грина – רוסית" lang="ru" hreflang="ru" data-title="Теорема Грина" data-language-autonym="Русский" data-language-local-name="רוסית" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teorema_e_Grinit" title="Teorema e Grinit – אלבנית" lang="sq" hreflang="sq" data-title="Teorema e Grinit" 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/%D0%93%D1%80%D0%B8%D0%BD%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0" title="Гринова теорема – סרבית" lang="sr" hreflang="sr" data-title="Гринова теорема" 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/Greens_sats" title="Greens sats – שוודית" lang="sv" hreflang="sv" data-title="Greens sats" data-language-autonym="Svenska" data-language-local-name="שוודית" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%9A%E0%B8%97%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%81%E0%B8%A3%E0%B8%B5%E0%B8%99" title="ทฤษฎีบทของกรีน – תאית" lang="th" hreflang="th" data-title="ทฤษฎีบทของกรีน" data-language-autonym="ไทย" data-language-local-name="תאית" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Green_teoremi" title="Green teoremi – טורקית" lang="tr" hreflang="tr" data-title="Green teoremi" data-language-autonym="Türkçe" data-language-local-name="טורקית" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%93%D1%80%D1%96%D0%BD%D0%B0" title="Теорема Гріна – אוקראינית" lang="uk" hreflang="uk" data-title="Теорема Гріна" data-language-autonym="Українська" data-language-local-name="אוקראינית" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Green" title="Định lý Green – וייטנאמית" lang="vi" hreflang="vi" data-title="Định lý Green" 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A0%BC%E6%9E%97%E5%85%AC%E5%BC%8F" title="格林公式 – סינית" lang="zh" hreflang="zh" data-title="格林公式" data-language-autonym="中文" data-language-local-name="סינית" class="interlanguage-link-target"><span>中文</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/Q321237#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%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F" title="צפייה בדף התוכן [c]" accesskey="c"><span>ערך</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a 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href="https://commons.wikimedia.org/wiki/Category:Green%27s_theorem" 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/Q321237" 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"><p><b>משפט גרין</b> הוא משפט ב<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> המגדיר קשר בין <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 קווי</a> של <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="פונקציה">פונקציה</a> על עקום סגור ופשוט לבין ה<a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9B%D7%A4%D7%95%D7%9C" class="mw-redirect" title="אינטגרל כפול">אינטגרל הכפול</a> על השטח החסום על ידי העקום. משפט גרין הוא <a href="/wiki/%D7%9E%D7%A7%D7%A8%D7%94_%D7%A4%D7%A8%D7%98%D7%99" title="מקרה פרטי">מקרה פרטי</a> דו-ממדי של <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>. הוא נקרא על שם ה<a href="/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%90%D7%99" title="מתמטיקאי">מתמטיקאי</a> ה<a href="/wiki/%D7%90%D7%A0%D7%92%D7%9C%D7%99%D7%94" title="אנגליה">אנגלי</a> <a href="/wiki/%D7%92%27%D7%95%D7%A8%D7%92%27_%D7%92%D7%A8%D7%99%D7%9F" title="ג'ורג' גרין">ג'ורג' גרין</a>. </p><p>למשפט שימושים רבים במתמטיקה ובהנדסה. לדוגמה, הבסיס המתמטי לפעולת ה<a href="/wiki/%D7%A4%D7%9C%D7%A0%D7%99%D7%9E%D7%98%D7%A8" title="פלנימטר">פלנימטר</a>, שהוא מכשיר המודד שטח של צורה מישורית כלשהי, הוא משפט גרין, או <a href="/w/index.php?title=%D7%A0%D7%95%D7%A1%D7%97%D7%AA_%D7%94%D7%A1%D7%A7%D7%98%D7%95%D7%A8_%D7%A9%D7%9C_%D7%9C%D7%99%D7%99%D7%91%D7%A0%D7%99%D7%A5&action=edit&redlink=1" class="new" title="נוסחת הסקטור של לייבניץ (הדף אינו קיים)">נוסחת הסקטור</a> של <a href="/wiki/%D7%9C%D7%99%D7%99%D7%91%D7%A0%D7%99%D7%A5" class="mw-redirect" title="לייבניץ">לייבניץ</a> כמקרה פרטי שלו. ממשפט גרין נובעת <a href="/wiki/%D7%A0%D7%95%D7%A1%D7%97%D7%AA_%D7%94%D7%A9%D7%A8%D7%95%D7%9A" title="נוסחת השרוך">נוסחת השרוך</a>. </p><p>המשפט: תהי <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> <a href="/wiki/%D7%A2%D7%A7%D7%95%D7%9E%D7%94" title="עקומה">מסילה פשוטה סגורה</a>, מכוונת חיובית (נגד <a href="/wiki/%D7%9B%D7%99%D7%95%D7%95%D7%9F_%D7%94%D7%A9%D7%A2%D7%95%D7%9F" title="כיוון השעון">כיוון השעון</a>) ו<a href="/wiki/%D7%92%D7%96%D7%99%D7%A8%D7%95%D7%AA_%D7%9C%D7%9E%D7%A7%D7%95%D7%98%D7%A2%D7%99%D7%9F" class="mw-redirect" title="גזירות למקוטעין">גזירה למקוטעין</a> החוסמת שטח ב-<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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>, ונסמן ב-<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> את השטח החסום על ידי המסילה <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. אם <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 P(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b3d8f37f5458c22b61eaf26e5af0523acb63e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.074ex; height:2.843ex;" alt="{\displaystyle P(x,y)}"></span> ,<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(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x,y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97a151e980598f776e01ae247354ba03cf8e8143" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.167ex; height:2.843ex;" alt="{\displaystyle Q(x,y)}"></span> פונקציות בעלות <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> רציפות עד סדר ראשון בסביבה המכילה את <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>, אזי: </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 \oint _{C}(P\,\mathrm {d} x+Q\,\mathrm {d} y)=\iint _{D}\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)\,\mathrm {d} x\mathrm {d} y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>+</mo> <mi>Q</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>Q</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}(P\,\mathrm {d} x+Q\,\mathrm {d} y)=\iint _{D}\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)\,\mathrm {d} x\mathrm {d} y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713d9f96ab9205da8aaf93c1689f0e28dde2d80c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.447ex; height:6.176ex;" alt="{\displaystyle \oint _{C}(P\,\mathrm {d} x+Q\,\mathrm {d} y)=\iint _{D}\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)\,\mathrm {d} x\mathrm {d} y}"></span>.</dd></dl> <p>כאשר הביטוי משמאל מגדיר אינטגרל קווי על עקום סגור (ולכן סימון העיגול על סימן האינטגרל) ומימין מבוטא האינטגרל הכפול עבור שטח התחום הסגור <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="סימון_מקובל_בפיזיקה"><span id=".D7.A1.D7.99.D7.9E.D7.95.D7.9F_.D7.9E.D7.A7.D7.95.D7.91.D7.9C_.D7.91.D7.A4.D7.99.D7.96.D7.99.D7.A7.D7.94"></span>סימון מקובל בפיזיקה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F&action=edit&section=1" title="עריכת קוד המקור של הפרק: סימון מקובל בפיזיקה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F&veaction=edit&section=1" title="עריכת פסקה: "סימון מקובל בפיזיקה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>מקרה שימושי במיוחד הוא כאשר הפונקציות <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 P,Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>,</mo> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P,Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94b3c5b21ee2d8a12253a952e4c4b1733ceb0735" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.618ex; height:2.509ex;" alt="{\displaystyle P,Q}"></span> הן רכיבים של <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>: <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}})=(P,Q)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</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">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>Q</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {f}}({\vec {r}})=(P,Q)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc5254ff7d8b8a7a46cdb2a806e67f435ef2b48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.223ex; height:3.509ex;" alt="{\displaystyle {\vec {f}}({\vec {r}})=(P,Q)}"></span>. הסימון המקובל הוא <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 P=f_{x},Q=f_{y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mi>Q</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=f_{x},Q=f_{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb80825d10dcdf84fc9d7896055b904666718fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.315ex; height:2.843ex;" alt="{\displaystyle P=f_{x},Q=f_{y}}"></span>. במקרה זה המשפט מקשר בין האינטגרל המשטחי על <a href="/wiki/%D7%A9%D7%98%D7%A3" title="שטף">שטף</a> ה<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> של השדה הווקטורי, לבין ה<a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9E%D7%A1%D7%99%D7%9C%D7%AA%D7%99" class="mw-redirect" title="אינטגרל מסילתי">אינטגרל המסילתי</a> של השדה. אם נסמן ב-<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 {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d125dccc556f5c8b0bf98a4f3847590b3f353bd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:2.176ex;" alt="{\displaystyle {\hat {n}}}"></span> את הווקטור הניצב למשטח, וב-<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 {dl}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>d</mi> <mi>l</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {dl}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f743a7156ec3b5b5c8a6340c8f531773dc0cc293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:3.676ex;" alt="{\displaystyle {\vec {dl}}}"></span> את אלמנט המסילה המקיפה את המשטח, ניסוח המשפט יהיה </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 \oint _{C}{\vec {f}}\cdot {\vec {dl}}=\iint _{D}\left({\vec {\nabla }}\times {\vec {f}}\right)\cdot {\hat {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>d</mi> <mi>l</mi> </mrow> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}{\vec {f}}\cdot {\vec {dl}}=\iint _{D}\left({\vec {\nabla }}\times {\vec {f}}\right)\cdot {\hat {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e74cd4a9d297aa863f7692b5c6db57aac237c445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.716ex; height:5.676ex;" alt="{\displaystyle \oint _{C}{\vec {f}}\cdot {\vec {dl}}=\iint _{D}\left({\vec {\nabla }}\times {\vec {f}}\right)\cdot {\hat {n}}}"></span></dd></dl> <p>בניסוח הזה, המשפט הוא כללי יותר. הוא תקף לא רק כאשר התחום <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> מוכל במישור, אלא גם במקרה כללי שבו <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> הוא <a href="/wiki/%D7%99%D7%A8%D7%99%D7%A2%D7%94" title="יריעה">יריעה</a> חלקה דו-ממדית <a href="/wiki/%D7%AA%D7%97%D7%95%D7%9D_%D7%A4%D7%A9%D7%95%D7%98_%D7%A7%D7%A9%D7%A8" class="mw-redirect" title="תחום פשוט קשר">פשוטת קשר</a>, והעקום <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> הוא <a href="/wiki/%D7%A9%D7%A4%D7%94_(%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94)" title="שפה (טופולוגיה)">שפת</a> היריעה (שצריכה להיות גזירה למקוטעין). </p> <div class="mw-heading mw-heading2"><h2 id="שימוש_לדוגמה:_שדה_מגנטי_מסביב_לתיל_נושא_זרם"><span id=".D7.A9.D7.99.D7.9E.D7.95.D7.A9_.D7.9C.D7.93.D7.95.D7.92.D7.9E.D7.94:_.D7.A9.D7.93.D7.94_.D7.9E.D7.92.D7.A0.D7.98.D7.99_.D7.9E.D7.A1.D7.91.D7.99.D7.91_.D7.9C.D7.AA.D7.99.D7.9C_.D7.A0.D7.95.D7.A9.D7.90_.D7.96.D7.A8.D7.9D"></span>שימוש לדוגמה: <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%9E%D7%92%D7%A0%D7%98%D7%99" title="שדה מגנטי">שדה מגנטי</a> מסביב לתיל נושא <a href="/wiki/%D7%96%D7%A8%D7%9D_%D7%97%D7%A9%D7%9E%D7%9C%D7%99" title="זרם חשמלי">זרם</a></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F&action=edit&section=2" title="עריכת קוד המקור של הפרק: שימוש לדוגמה: שדה מגנטי מסביב לתיל נושא זרם"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F&veaction=edit&section=2" title="עריכת פסקה: "שימוש לדוגמה: שדה מגנטי מסביב לתיל נושא זרם"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Electromagnetism.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Electromagnetism.svg/220px-Electromagnetism.svg.png" decoding="async" width="220" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Electromagnetism.svg/330px-Electromagnetism.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Electromagnetism.svg/440px-Electromagnetism.svg.png 2x" data-file-width="651" data-file-height="709" /></a><figcaption>זרם חשמלי מיצר שדה מגנטי.</figcaption></figure> <p>נניח כי נתון לנו תיל ישר, אינסופי, הנושא זרם <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 I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>. הקשר בין הזרם לשדה המגנטי נתון (במקרה הסטטי) על ידי <a href="/wiki/%D7%97%D7%95%D7%A7_%D7%90%D7%9E%D7%A4%D7%A8" title="חוק אמפר">חוק אמפר</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 {\vec {\nabla }}\times {\vec {B}}=\mu _{0}{\vec {J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {\nabla }}\times {\vec {B}}=\mu _{0}{\vec {J}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d62e7209e87b269f2755572d3decb948f3f5b52e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.883ex; height:3.343ex;" alt="{\displaystyle {\vec {\nabla }}\times {\vec {B}}=\mu _{0}{\vec {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 {\vec {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ae7d80cab55b606de217162280b2279142bbb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.843ex;" alt="{\displaystyle {\vec {B}}}"></span> הוא השדה המגנטי, <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 {J}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {J}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f64ab02528d2b80e1df79bc2a8762489a986afa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.788ex; height:2.843ex;" alt="{\displaystyle {\vec {J}}}"></span> הוא <a href="/wiki/%D7%A6%D7%A4%D7%99%D7%A4%D7%95%D7%AA_%D7%96%D7%A8%D7%9D" title="צפיפות זרם">צפיפות הזרם</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 \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}"></span> הוא <a href="/wiki/%D7%A4%D7%A8%D7%9E%D7%99%D7%90%D7%91%D7%99%D7%9C%D7%99%D7%95%D7%AA" class="mw-redirect" title="פרמיאביליות">קבוע הפרמיאביליות</a> של ה<a href="/wiki/%D7%A8%D7%99%D7%A7" title="ריק">ריק</a>. נבנה משטח מעגלי מאונך לתיל ברדיוס <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>, שמרכזו נמצא על התיל (כמו <a href="/wiki/%D7%A7%D7%95%D7%95%D7%99_%D7%A9%D7%93%D7%94" title="קווי שדה">קווי השדה</a> האדומים בתמונה). נסמן את המשטח ב-<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> ואת העקום שמקיף אותו ב-<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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. מכיוון שהמערכת <a href="/wiki/%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%94" title="סימטריה">סימטרית</a> לסיבוב סביב התיל, השדה המגנטי לאורך <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> קבוע. משיקולים אחרים<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> ניתן לקבל שכיוון השדה המגנטי <a href="/wiki/%D7%9E%D7%A9%D7%99%D7%A7" title="משיק">משיק</a> 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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> בכל נקודה. לכן, האינטגרל המסלולי על השדה המגנטי שווה פשוט ל: </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 \oint _{C}{\vec {B}}\cdot {\vec {d}}l=\int _{C}B=2\pi rB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>l</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>B</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}{\vec {B}}\cdot {\vec {d}}l=\int _{C}B=2\pi rB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8898a3dba3ffcf48814b949e70a0fc0b3282bfc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.274ex; height:5.676ex;" alt="{\displaystyle \oint _{C}{\vec {B}}\cdot {\vec {d}}l=\int _{C}B=2\pi rB}"></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 B=\left|{\vec {B}}\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\left|{\vec {B}}\right|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c3a584d8c8193cc4971340a399958178b88d99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.92ex; height:3.843ex;" alt="{\displaystyle B=\left|{\vec {B}}\right|}"></span>. כעת נשתמש במשפט גרין ובחוק אמפר כדי לחשב את גודל השדה: </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 2\pi rB=\oint _{C}{\vec {B}}\cdot {\vec {d}}l=\iint _{D}{\vec {\nabla }}\times {\vec {B}}\cdot {\hat {n}}=\iint _{D}\mu _{0}{\vec {J}}\cdot {\hat {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>B</mi> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>d</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>l</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>B</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>J</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi rB=\oint _{C}{\vec {B}}\cdot {\vec {d}}l=\iint _{D}{\vec {\nabla }}\times {\vec {B}}\cdot {\hat {n}}=\iint _{D}\mu _{0}{\vec {J}}\cdot {\hat {n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f10de539a9ec9a7a6f2dc312264af7f6d139b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.377ex; height:5.676ex;" alt="{\displaystyle 2\pi rB=\oint _{C}{\vec {B}}\cdot {\vec {d}}l=\iint _{D}{\vec {\nabla }}\times {\vec {B}}\cdot {\hat {n}}=\iint _{D}\mu _{0}{\vec {J}}\cdot {\hat {n}}}"></span></dd></dl> <p>לפי הגדרה, האינטגרל על צפיפות הזרם הוא הזרם, ולכן קיבלנו <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 2\pi rB=\mu _{0}I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>B</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi rB=\mu _{0}I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd88ca868642086d0abd81714757bc9a023e7bf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.033ex; height:2.676ex;" alt="{\displaystyle 2\pi rB=\mu _{0}I}"></span>, או בניסוח המקובל יותר: </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 B(r)={\frac {\mu _{0}I}{2\pi r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>I</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(r)={\frac {\mu _{0}I}{2\pi r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7b9e5c6c1413f37610c39f9e5223f2fd4ab79d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.184ex; height:5.343ex;" alt="{\displaystyle B(r)={\frac {\mu _{0}I}{2\pi r}}}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="הוכחה_עבור_תחום_מלבני"><span id=".D7.94.D7.95.D7.9B.D7.97.D7.94_.D7.A2.D7.91.D7.95.D7.A8_.D7.AA.D7.97.D7.95.D7.9D_.D7.9E.D7.9C.D7.91.D7.A0.D7.99"></span>הוכחה עבור תחום מלבני</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F&action=edit&section=3" title="עריכת קוד המקור של הפרק: הוכחה עבור תחום מלבני"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F&veaction=edit&section=3" title="עריכת פסקה: "הוכחה עבור תחום מלבני"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Green%27s-theorem-simple-region.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Green%27s-theorem-simple-region.svg/300px-Green%27s-theorem-simple-region.svg.png" decoding="async" width="300" height="297" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Green%27s-theorem-simple-region.svg/450px-Green%27s-theorem-simple-region.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Green%27s-theorem-simple-region.svg/600px-Green%27s-theorem-simple-region.svg.png 2x" data-file-width="429" data-file-height="425" /></a><figcaption>D הוא תחום החסום על ידי העקום הגזיר למקוטעין הבנוי מ: <i>C</i><sub>1</sub>, <i>C</i><sub>2</sub>, <i>C</i><sub>3</sub>, <i>C</i><sub>4</sub></figcaption></figure> <p>נוכיח כי 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 \oint _{C}P\,dx=\iint _{D}\left(-{\frac {\partial P}{\partial y}}\right)\,dA\qquad \mathrm {(1)} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>A</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}P\,dx=\iint _{D}\left(-{\frac {\partial P}{\partial y}}\right)\,dA\qquad \mathrm {(1)} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00dd414181741d4fbdb24749f5917f9093ff4f90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.915ex; height:6.176ex;" alt="{\displaystyle \oint _{C}P\,dx=\iint _{D}\left(-{\frac {\partial P}{\partial y}}\right)\,dA\qquad \mathrm {(1)} }"></span></dd></dl> <p>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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> שהוא תחום פשוט מסוג 1 - תחום חסום על ידי שני ישרים מאונכים לציר x ושתי פונקציות גזירות 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> (כמתואר בתמונה). </p><p>נגדיר: </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=\{(x,y)|a\leq x\leq b,g_{1}(x)\leq y\leq g_{2}(x)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>≤</mo> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=\{(x,y)|a\leq x\leq b,g_{1}(x)\leq y\leq g_{2}(x)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c65b670fe127102c6957f70f93660fbb5a45a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.068ex; height:2.843ex;" alt="{\displaystyle D=\{(x,y)|a\leq x\leq b,g_{1}(x)\leq y\leq g_{2}(x)\}}"></span></dd></dl> <p><br /> 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_{1},g_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1},g_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e39b0b718144602bbf108d7ee37c3c65b1de72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.36ex; height:2.009ex;" alt="{\displaystyle g_{1},g_{2}}"></span> גזירות למקוטעין ב-<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 [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span>. נחשב את האינטגרל הכפול ב-(1):<br /> </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 \iint _{D}\left(-{\frac {\partial P}{\partial y}}\right)\,dxdy=\int _{a}^{b}\!\!\int _{g_{1}(x)}^{g_{2}(x)}\left(-{\frac {\partial P}{\partial y}}(x,y)\right)\,dy\,dx=\int _{a}^{b}{\big \{}-P(x,g_{2}(x))+P(x,g_{1}(x))\,{\big \}}\,dx\qquad \mathrm {(2)} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mi>d</mi> <mi>y</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">{</mo> </mrow> </mrow> <mo>−</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">}</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iint _{D}\left(-{\frac {\partial P}{\partial y}}\right)\,dxdy=\int _{a}^{b}\!\!\int _{g_{1}(x)}^{g_{2}(x)}\left(-{\frac {\partial P}{\partial y}}(x,y)\right)\,dy\,dx=\int _{a}^{b}{\big \{}-P(x,g_{2}(x))+P(x,g_{1}(x))\,{\big \}}\,dx\qquad \mathrm {(2)} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b4a44ab4c0556a9e77c0ed975eabf7c4304fbde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:101.543ex; height:6.676ex;" alt="{\displaystyle \iint _{D}\left(-{\frac {\partial P}{\partial y}}\right)\,dxdy=\int _{a}^{b}\!\!\int _{g_{1}(x)}^{g_{2}(x)}\left(-{\frac {\partial P}{\partial y}}(x,y)\right)\,dy\,dx=\int _{a}^{b}{\big \{}-P(x,g_{2}(x))+P(x,g_{1}(x))\,{\big \}}\,dx\qquad \mathrm {(2)} }"></span></dd></dl> <p>כעת נחשב את האינטגרל הקווי ב-(1): ניתן לרשום את <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 C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span> כאיחוד ארבעת העקומים: <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 C_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59cf1a8e46030a81fc175be95561e4f161241a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{4}}"></span>, <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 C_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e9abeb5057b7afbf88e3169101849354f13c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{3}}"></span>, <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 C_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ec545f7870665e1028b7492746848d149878808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{2}}"></span>, <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 C_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/babf569931f1a7b5182b9bec51873c2f5692fbb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{1}}"></span>. </p><p>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 C_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/babf569931f1a7b5182b9bec51873c2f5692fbb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{1}}"></span> נשתמש בהצגה הפרמטרית: <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 a<=x<=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><=</mo> <mi>x</mi> <mo><=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<=x<=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b39e719612dca92b7a7a4149d092783dcd55220a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.37ex; height:2.176ex;" alt="{\displaystyle a<=x<=b}"></span><i>, <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 y=g_{1}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=g_{1}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1dd7035b804184bd7ca1ddcef83acf1bf37aca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.556ex; height:2.843ex;" alt="{\displaystyle y=g_{1}(x)}"></span></i>, <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 x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d5b1d45c7aa8346eacace86bbb5cb0226f8fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.758ex; height:1.676ex;" alt="{\displaystyle x=x}"></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 \int _{C_{1}}P(x,y)\,dx=\int _{a}^{b}P(x,g_{1}(x))\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{C_{1}}P(x,y)\,dx=\int _{a}^{b}P(x,g_{1}(x))\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82f2bf94b2270d6b1df9d8945c13f4154114e44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.966ex; height:6.676ex;" alt="{\displaystyle \int _{C_{1}}P(x,y)\,dx=\int _{a}^{b}P(x,g_{1}(x))\,dx}"></span></dd></dl> <p>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 C_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e9abeb5057b7afbf88e3169101849354f13c65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.716ex; height:2.509ex;" alt="{\displaystyle C_{3}}"></span> נשתמש בהצגה הפרמטרית: <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 a<=x<=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><=</mo> <mi>x</mi> <mo><=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<=x<=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b39e719612dca92b7a7a4149d092783dcd55220a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.37ex; height:2.176ex;" alt="{\displaystyle a<=x<=b}"></span><i>, <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 y=g_{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=g_{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14309df59896395c876a062cfe80d59babef2350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.556ex; height:2.843ex;" alt="{\displaystyle y=g_{2}(x)}"></span></i>, <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 x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d5b1d45c7aa8346eacace86bbb5cb0226f8fb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.758ex; height:1.676ex;" alt="{\displaystyle x=x}"></span> ולכן: </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 _{C_{3}}P(x,y)\,dx=\int _{b}^{a}P(x,g_{2}(x))\,dx=-\int _{a}^{b}P(x,g_{2}(x))\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msubsup> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{C_{3}}P(x,y)\,dx=\int _{b}^{a}P(x,g_{2}(x))\,dx=-\int _{a}^{b}P(x,g_{2}(x))\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6be9a1eb8fcd0bdda49f8f392a94b56105b58416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:58.366ex; height:6.676ex;" alt="{\displaystyle \int _{C_{3}}P(x,y)\,dx=\int _{b}^{a}P(x,g_{2}(x))\,dx=-\int _{a}^{b}P(x,g_{2}(x))\,dx}"></span></dd></dl> <p><br /> 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 C_{2},C_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2},C_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86262dd0b8211b5116a0b29f39f944ea83a31f3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.466ex; height:2.509ex;" alt="{\displaystyle C_{2},C_{4}}"></span>, הערך של x זהה בשני גבולות האינטגרציה, ולכן </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 _{C_{4}}P(x,y)\,dx=\int _{C_{2}}P(x,y)\,dx=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </msub> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{C_{4}}P(x,y)\,dx=\int _{C_{2}}P(x,y)\,dx=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f64cdc86e831aa719a7b660c0f0e0a64b0d4f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:35.209ex; height:6.009ex;" alt="{\displaystyle \int _{C_{4}}P(x,y)\,dx=\int _{C_{2}}P(x,y)\,dx=0}"></span></dd></dl> <p>מסקנה: </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 \oint _{C}P\,dx=\int _{C_{1}}P(x,y)\,dx+\int _{C_{2}}P(x,y)\,dx+\int _{C_{3}}P(x,y)\,dx+\int _{C_{4}}P(x,y)\,dx=\int _{a}^{b}P(x,g_{1}(x))\,dx-\int _{a}^{b}P(x,g_{2}(x))\,dx\qquad \mathrm {(3)} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </msub> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}P\,dx=\int _{C_{1}}P(x,y)\,dx+\int _{C_{2}}P(x,y)\,dx+\int _{C_{3}}P(x,y)\,dx+\int _{C_{4}}P(x,y)\,dx=\int _{a}^{b}P(x,g_{1}(x))\,dx-\int _{a}^{b}P(x,g_{2}(x))\,dx\qquad \mathrm {(3)} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e11c5341bf7f60cce5fff960ac510cf7390ceb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:124.599ex; height:6.676ex;" alt="{\displaystyle \oint _{C}P\,dx=\int _{C_{1}}P(x,y)\,dx+\int _{C_{2}}P(x,y)\,dx+\int _{C_{3}}P(x,y)\,dx+\int _{C_{4}}P(x,y)\,dx=\int _{a}^{b}P(x,g_{1}(x))\,dx-\int _{a}^{b}P(x,g_{2}(x))\,dx\qquad \mathrm {(3)} }"></span></dd></dl> <p>התוצאות במשוואות (3) ו-(2) זהות, ובכך קיבלנו את השוויון ב-(1). </p><p>באופן דומה ניתן להוכיח כי 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 \oint _{C}Q\,dy=\iint _{D}\left({\frac {\partial Q}{\partial x}}\right)\,dA\qquad \mathrm {(4)} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mi>Q</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>Q</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>A</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}Q\,dy=\iint _{D}\left({\frac {\partial Q}{\partial x}}\right)\,dA\qquad \mathrm {(4)} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b81df05f1d98e766acd2241735c784770c47013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.119ex; height:6.176ex;" alt="{\displaystyle \oint _{C}Q\,dy=\iint _{D}\left({\frac {\partial Q}{\partial x}}\right)\,dA\qquad \mathrm {(4)} }"></span></dd></dl> <p>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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> שהוא תחום פשוט מסוג 2 - חסום על ידי שני ישרים מאונכים לציר y ושתי פונקציות גזירות של y. </p><p>משילוב (1) ו-(4) מתקבל כי עבור תחום מלבני (שהוא תחום פשוט מסוג 1 וגם תחום פשוט מסוג 2) 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 \oint _{C}(P\,dx+Q\,dy)=\iint _{D}\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)\,dA\qquad \mathrm {(5)} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>P</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>+</mo> <mi>Q</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>Q</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>P</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>A</mi> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{C}(P\,dx+Q\,dy)=\iint _{D}\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)\,dA\qquad \mathrm {(5)} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a5cc7c81ebed20db38909bd645b3a36d67ec043" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.799ex; height:6.176ex;" alt="{\displaystyle \oint _{C}(P\,dx+Q\,dy)=\iint _{D}\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right)\,dA\qquad \mathrm {(5)} }"></span></dd></dl> <p>על ידי שימוש בתכונות של אינטגרלים (אדיטיביות על תחומים זרים, והיפוך סימן בהיפוך כיוון) ניתן לראות כי המשפט תקף עבור כל שטח שניתן לחלוקה למספר סופי של תחומים מלבניים.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <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 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הערות שוליים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%9E%D7%A9%D7%A4%D7%98_%D7%92%D7%A8%D7%99%D7%9F&veaction=edit&section=6" title="עריכת פסקה: "הערות שוליים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist references-small"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">^</a></span> <span class="reference-text">עובדה זו נובעת מאי קיומם של <a href="/wiki/%D7%9E%D7%95%D7%A0%D7%95%D7%A4%D7%95%D7%9C_%D7%9E%D7%92%D7%A0%D7%98%D7%99" title="מונופול מגנטי">מונופולים מגנטים</a>, כלומר, מ<a href="/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%9E%D7%A7%D7%A1%D7%95%D7%95%D7%9C" title="משוואות מקסוול">משוואת מקסוול</a> הקובעת כי ה<a href="/wiki/%D7%93%D7%99%D7%91%D7%A8%D7%92%D7%A0%D7%A5" title="דיברגנץ">דיברגנץ</a> של השדה המגנטי מתאפס.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">^</a></span> <span class="reference-text">ההוכחה עבור מקרים כלליים יותר (<a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A7%D7%95%D7%9D_%D7%A9%D7%9C_%D7%96%27%D7%95%D7%A8%D7%93%D7%9F" title="משפט העקום של ז'ורדן">עקומי ז'ורדן</a> עם נגזרות רציפות למקוטעין) מורכבת מעט יותר, ומשתמשת בפירוק השטח לשטחים מלבניים עד כדי שגיאה קטנה כרצוננו.</span> </li> </ol></div></div> <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 href="/wiki/%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" title="גרדיאנט">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'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 class="mw-selflink selflink">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 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