Operator nabla – Wikipedia, wolna encyklopedia

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class="vector-toc-numb">2</span> <span>Zastosowania</span> </div> </a> <button aria-controls="toc-Zastosowania-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Przełącz podsekcję Zastosowania</span> </button> <ul id="toc-Zastosowania-sublist" class="vector-toc-list"> <li id="toc-Gradient_i_pochodna_kierunkowa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gradient_i_pochodna_kierunkowa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Gradient i pochodna kierunkowa</span> </div> </a> <ul id="toc-Gradient_i_pochodna_kierunkowa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dywergencja" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dywergencja"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Dywergencja</span> </div> </a> <ul id="toc-Dywergencja-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotacja" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotacja"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Rotacja</span> </div> </a> <ul id="toc-Rotacja-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Laplasjan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laplasjan"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Laplasjan</span> </div> </a> <ul id="toc-Laplasjan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pochodna_kowariantna" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pochodna_kowariantna"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Pochodna kowariantna</span> </div> </a> <ul id="toc-Pochodna_kowariantna-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Złożenia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Złożenia"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Złożenia</span> </div> </a> <ul id="toc-Złożenia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zobacz_też" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zobacz_też"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Zobacz też</span> </div> </a> <ul id="toc-Zobacz_też-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zastrzeżenia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zastrzeżenia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Zastrzeżenia</span> </div> </a> <ul id="toc-Zastrzeżenia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uwagi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Uwagi"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Uwagi</span> </div> </a> <ul id="toc-Uwagi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Przypisy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Przypisy"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Przypisy</span> </div> </a> <ul id="toc-Przypisy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linki_zewnętrzne" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Linki_zewnętrzne"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Linki zewnętrzne</span> </div> </a> <ul id="toc-Linki_zewnętrzne-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="Spis treści" 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="Przełącz stan spisu treści" > <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">Przełącz stan spisu treści</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">Operator nabla</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="Przejdź do artykułu w innym języku. Treść dostępna w 38 językach" > <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-38" 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">38 języków</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%A4%D8%AB%D8%B1_%D8%AF%D9%84" title="مؤثر دل – arabski" lang="ar" hreflang="ar" data-title="مؤثر دل" data-language-autonym="العربية" data-language-local-name="arabski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Operador_nabla" title="Operador nabla – kataloński" lang="ca" hreflang="ca" data-title="Operador nabla" data-language-autonym="Català" data-language-local-name="kataloński" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B1%D0%BB%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80" title="Набла оператор – czuwaski" lang="cv" hreflang="cv" data-title="Набла оператор" data-language-autonym="Чӑвашла" data-language-local-name="czuwaski" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Nabla" title="Nabla – czeski" lang="cs" hreflang="cs" data-title="Nabla" data-language-autonym="Čeština" data-language-local-name="czeski" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Nabla-operatoren" title="Nabla-operatoren – duński" lang="da" hreflang="da" data-title="Nabla-operatoren" data-language-autonym="Dansk" data-language-local-name="duński" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Nabla-Operator" title="Nabla-Operator – niemiecki" lang="de" hreflang="de" data-title="Nabla-Operator" data-language-autonym="Deutsch" data-language-local-name="niemiecki" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Nabla-operaator" title="Nabla-operaator – estoński" lang="et" hreflang="et" data-title="Nabla-operaator" data-language-autonym="Eesti" data-language-local-name="estoński" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CE%AC%CE%B4%CE%B5%CE%BB%CF%84%CE%B1" title="Ανάδελτα – grecki" lang="el" hreflang="el" data-title="Ανάδελτα" data-language-autonym="Ελληνικά" data-language-local-name="grecki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Del" title="Del – angielski" lang="en" hreflang="en" data-title="Del" data-language-autonym="English" data-language-local-name="angielski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Nabla" title="Nabla – hiszpański" lang="es" hreflang="es" data-title="Nabla" data-language-autonym="Español" data-language-local-name="hiszpański" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Nabla_operatoro" title="Nabla operatoro – esperanto" lang="eo" hreflang="eo" data-title="Nabla operatoro" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D9%85%D9%84%DA%AF%D8%B1_%D8%AF%D9%84" title="عملگر دل – perski" lang="fa" hreflang="fa" data-title="عملگر دل" data-language-autonym="فارسی" data-language-local-name="perski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nabla" title="Nabla – francuski" lang="fr" hreflang="fr" data-title="Nabla" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8D%B8_(%EC%97%B0%EC%82%B0%EC%9E%90)" title="델 (연산자) – koreański" lang="ko" hreflang="ko" data-title="델 (연산자)" data-language-autonym="한국어" data-language-local-name="koreański" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%86%D5%A1%D5%A2%D5%AC%D5%A1_%D6%85%D5%BA%D5%A5%D6%80%D5%A1%D5%BF%D5%B8%D6%80" title="Նաբլա օպերատոր – ormiański" lang="hy" hreflang="hy" data-title="Նաբլա օպերատոր" data-language-autonym="Հայերեն" data-language-local-name="ormiański" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Hamiltonov_operator" title="Hamiltonov operator – chorwacki" lang="hr" hreflang="hr" data-title="Hamiltonov operator" data-language-autonym="Hrvatski" data-language-local-name="chorwacki" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Operatore_nabla" title="Operatore nabla – włoski" lang="it" hreflang="it" data-title="Operatore nabla" data-language-autonym="Italiano" data-language-local-name="włoski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B1%D0%BB%D0%B0-%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80" title="Набла-оператор – kazachski" lang="kk" hreflang="kk" data-title="Набла-оператор" data-language-autonym="Қазақша" data-language-local-name="kazachski" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Nabla" title="Nabla – łotewski" lang="lv" hreflang="lv" data-title="Nabla" data-language-autonym="Latviešu" data-language-local-name="łotewski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Nabla_oper%C3%A1tor" title="Nabla operátor – węgierski" lang="hu" hreflang="hu" data-title="Nabla operátor" data-language-autonym="Magyar" data-language-local-name="węgierski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%92%E1%80%80%E1%80%BA%E1%80%9C%E1%80%BA_%E1%80%9C%E1%80%AF%E1%80%95%E1%80%BA%E1%80%86%E1%80%B1%E1%80%AC%E1%80%84%E1%80%BA%E1%80%81%E1%80%BB%E1%80%80%E1%80%BA" title="ဒက်လ် လုပ်ဆောင်ချက် – birmański" lang="my" hreflang="my" data-title="ဒက်လ် လုပ်ဆောင်ချက်" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmański" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Nabla" title="Nabla – niderlandzki" lang="nl" hreflang="nl" data-title="Nabla" data-language-autonym="Nederlands" data-language-local-name="niderlandzki" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%8A%E3%83%96%E3%83%A9" title="ナブラ – japoński" lang="ja" hreflang="ja" data-title="ナブラ" data-language-autonym="日本語" data-language-local-name="japoński" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Nabla-operator" title="Nabla-operator – norweski (bokmål)" lang="nb" hreflang="nb" data-title="Nabla-operator" data-language-autonym="Norsk bokmål" data-language-local-name="norweski (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Gamilton_operatori" title="Gamilton operatori – uzbecki" lang="uz" hreflang="uz" data-title="Gamilton operatori" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbecki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Del" title="Del – portugalski" lang="pt" hreflang="pt" data-title="Del" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Nabla" title="Nabla – rumuński" lang="ro" hreflang="ro" data-title="Nabla" data-language-autonym="Română" data-language-local-name="rumuński" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80_%D0%BD%D0%B0%D0%B1%D0%BB%D0%B0" title="Оператор набла – rosyjski" lang="ru" hreflang="ru" data-title="Оператор набла" data-language-autonym="Русский" data-language-local-name="rosyjski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Oper%C3%A1tor_nabla" title="Operátor nabla – słowacki" lang="sk" hreflang="sk" data-title="Operátor nabla" data-language-autonym="Slovenčina" data-language-local-name="słowacki" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Hamiltonov_operator" title="Hamiltonov operator – serbsko-chorwacki" lang="sh" hreflang="sh" data-title="Hamiltonov operator" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbsko-chorwacki" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Nabla" title="Nabla – fiński" lang="fi" hreflang="fi" data-title="Nabla" data-language-autonym="Suomi" data-language-local-name="fiński" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Nablaoperatorn" title="Nablaoperatorn – szwedzki" lang="sv" hreflang="sv" data-title="Nablaoperatorn" data-language-autonym="Svenska" data-language-local-name="szwedzki" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9F%E0%AF%86%E0%AE%B2%E0%AF%8D_%E0%AE%87%E0%AE%AF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AE%BF" title="டெல் இயக்கி – tamilski" lang="ta" hreflang="ta" data-title="டெல் இயக்கி" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B1%D0%BB%D0%B0_%D0%BE%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80%D1%8B" title="Набла операторы – tatarski" lang="tt" hreflang="tt" data-title="Набла операторы" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarski" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Del_i%C5%9Flemcisi" title="Del işlemcisi – turecki" lang="tr" hreflang="tr" data-title="Del işlemcisi" data-language-autonym="Türkçe" data-language-local-name="turecki" 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%9E%D0%BF%D0%B5%D1%80%D0%B0%D1%82%D0%BE%D1%80_%D0%93%D0%B0%D0%BC%D1%96%D0%BB%D1%8C%D1%82%D0%BE%D0%BD%D0%B0" title="Оператор Гамільтона – ukraiński" lang="uk" hreflang="uk" data-title="Оператор Гамільтона" data-language-autonym="Українська" data-language-local-name="ukraiński" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/Nabla_%E7%AE%97%E5%AD%90" title="Nabla 算子 – kantoński" lang="yue" hreflang="yue" data-title="Nabla 算子" data-language-autonym="粵語" data-language-local-name="kantoński" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%80%92%E4%B8%89%E8%A7%92%E7%AE%97%E7%AC%A6" title="倒三角算符 – chiński" lang="zh" hreflang="zh" data-title="倒三角算符" data-language-autonym="中文" data-language-local-name="chiński" 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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ukryj</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">Z Wikipedii, wolnej encyklopedii</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-ltr mw-parser-output" lang="pl" dir="ltr"><p><b>Nabla</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[a]</a></sup> – stosowana w <a href="/w/index.php?title=Rachunek_wektorowy&action=edit&redlink=1" class="new" title="Rachunek wektorowy (strona nie istnieje)">rachunku wektorowym</a> konwencja notacyjna z wykorzystaniem <a href="/wiki/Symbol_nabla" title="Symbol nabla">symbolu nabli</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 \nabla .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f06d4d98a887e6430b4b67aab1a7c3f191b9a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.583ex; height:2.176ex;" alt="{\displaystyle \nabla .}"></span> Ułatwia ona opis <a href="/wiki/Gradient_(matematyka)" title="Gradient (matematyka)">gradientu</a> (dla <a href="/wiki/Pole_skalarne" title="Pole skalarne">pola skalarnego</a>), czy też różnorodnych <a href="/wiki/Operator_r%C3%B3%C5%BCniczkowy" title="Operator różniczkowy">operatorów różniczkowych</a>, w tym <a href="/wiki/Pochodna_funkcji" title="Pochodna funkcji">pochodnej</a> (odpowiadającej gradientowi), <a href="/wiki/Dywergencja" title="Dywergencja">dywergencji</a>, <a href="/wiki/Rotacja" title="Rotacja">rotacji</a> (dla <a href="/wiki/Pole_wektorowe" title="Pole wektorowe">pola wektorowego</a>) czy <a href="/wiki/Operator_Laplace%E2%80%99a" title="Operator Laplace’a">laplasjanu</a> (dla pola wektorowego lub skalarnego). Siła notacji tkwi w tym, iż nabla traktowana jest w niej podobnie do wektora: można ją mnożyć <a href="/wiki/Iloczyn_skalarny" title="Iloczyn skalarny">skalarnie</a>, <a href="/wiki/Iloczyn_wektorowy" title="Iloczyn wektorowy">wektorowo</a>, a nawet <a href="/w/index.php?title=Iloczyn_tensorowy&action=edit&redlink=1" class="new" title="Iloczyn tensorowy (strona nie istnieje)">tensorowo</a> przez pola skalarne bądź wektorowe, uzyskując inne pola skalarne lub wektorowe (mnożenie lewostronne) albo kolejne operatory różniczkowe (mnożenie prawostronne – wynika to z <a href="/wiki/Przemienno%C5%9B%C4%87" title="Przemienność">nieprzemienności</a> „operatora”, zob. <i><a href="#Zastrzeżenia">Zastrzeżenia</a></i>). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicja">Definicja</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=1" title="Edytuj sekcję: Definicja" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Definicja"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>W trójwymiarowej <a href="/wiki/Przestrze%C5%84_euklidesowa" title="Przestrzeń euklidesowa">przestrzeni euklidesowej</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 \mathbb {R} ^{3}}"> <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>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" 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} ^{3}}"></span> z <a href="/wiki/Uk%C5%82ad_wsp%C3%B3%C5%82rz%C4%99dnych_kartezja%C5%84skich" title="Układ współrzędnych kartezjańskich">układem współrzędnych kartezjańskich</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 (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span> <b>nablę</b> definiuje się za pomocą <a href="/wiki/Pochodna_cz%C4%85stkowa" title="Pochodna cząstkowa">pochodnych cząstkowych</a> wzorem<sup id="cite_ref-epwn_2-0" class="reference"><a href="#cite_note-epwn-2">[1]</a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)=\mathbf {i} {\frac {\partial }{\partial x}}+\mathbf {j} {\frac {\partial }{\partial y}}+\mathbf {k} {\frac {\partial }{\partial z}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)=\mathbf {i} {\frac {\partial }{\partial x}}+\mathbf {j} {\frac {\partial }{\partial y}}+\mathbf {k} {\frac {\partial }{\partial z}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5919beecb138ef80a69bee17ad8f8177328d7611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.991ex; height:6.176ex;" alt="{\displaystyle \nabla =\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)=\mathbf {i} {\frac {\partial }{\partial x}}+\mathbf {j} {\frac {\partial }{\partial y}}+\mathbf {k} {\frac {\partial }{\partial z}},}"></span></dd></dl> <p>gdzie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {i} ,\;\mathbf {j} ,\;\mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>,</mo> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {i} ,\;\mathbf {j} ,\;\mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/357230501ec5df1888aaa9eb239ffd07dc735757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.328ex; height:2.509ex;" alt="{\displaystyle \mathbf {i} ,\;\mathbf {j} ,\;\mathbf {k} }"></span> oznaczają <a href="/wiki/Wektor_jednostkowy" title="Wektor jednostkowy">wektory jednostkowe osi</a> (wektory <a href="/wiki/Baza_standardowa" title="Baza standardowa">bazy standardowej</a>). </p><p>Nablę można uogólnić na przestrzeń <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} ^{n}}"> <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"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> z kartezjańskim układem współrzędnych <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_{1},\dots ,x_{n}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\dots ,x_{n}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a05a40411680be4e336351e7d752b937b19906f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.566ex; height:2.843ex;" alt="{\displaystyle (x_{1},\dots ,x_{n}),}"></span> definiując ją jako </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla =\left({\frac {\partial }{\partial x_{1}}},\dots ,{\frac {\partial }{\partial x_{n}}}\right)=\sum _{i=1}^{n}\mathbf {e} _{i}{\frac {\partial }{\partial x_{i}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla =\left({\frac {\partial }{\partial x_{1}}},\dots ,{\frac {\partial }{\partial x_{n}}}\right)=\sum _{i=1}^{n}\mathbf {e} _{i}{\frac {\partial }{\partial x_{i}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca29dcf185047d8e5df1201b8409ed26e4399726" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.67ex; height:6.843ex;" alt="{\displaystyle \nabla =\left({\frac {\partial }{\partial x_{1}}},\dots ,{\frac {\partial }{\partial x_{n}}}\right)=\sum _{i=1}^{n}\mathbf {e} _{i}{\frac {\partial }{\partial x_{i}}},}"></span></dd></dl> <p>gdzie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbf {e} _{i})_{i=1,\dots ,n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {e} _{i})_{i=1,\dots ,n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71fd7f7c1f09fd267ca1d2c7e80370188e36b8d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.561ex; height:3.009ex;" alt="{\displaystyle (\mathbf {e} _{i})_{i=1,\dots ,n}}"></span> oznacza <a href="/wiki/Baza_standardowa" title="Baza standardowa">bazę standardową</a>; w <a href="/wiki/Konwencja_sumacyjna_Einsteina" title="Konwencja sumacyjna Einsteina">konwencji sumacyjnej Einsteina</a> powyższy zapis ulega skróceniu do </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla =\mathbf {e} _{i}\partial _{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla =\mathbf {e} _{i}\partial _{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/239466e336a390907a3c659f1df87bf86af6e6f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.74ex; height:2.509ex;" alt="{\displaystyle \nabla =\mathbf {e} _{i}\partial _{i}.}"></span></dd></dl> <p>Postać w innych niż kartezjański układach współrzędnych jest bardziej złożona – postać w popularnych układach współrzędnych przedstawiono w <a href="/wiki/Operator_nabla_w_r%C3%B3%C5%BCnych_uk%C5%82adach_wsp%C3%B3%C5%82rz%C4%99dnych" title="Operator nabla w różnych układach współrzędnych">oddzielnym artykule</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Zastosowania">Zastosowania</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=2" title="Edytuj sekcję: Zastosowania" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Zastosowania"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><i>W dalszej części przestrzeń euklidesowa będzie miała trzy wymiary ze względu na użycie <a href="/wiki/Iloczyn_wektorowy" title="Iloczyn wektorowy">iloczynu wektorowego</a>.</i></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Gradient_i_pochodna_kierunkowa">Gradient i pochodna kierunkowa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=3" title="Edytuj sekcję: Gradient i pochodna kierunkowa" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=3" title="Edytuj kod źródłowy sekcji: Gradient i pochodna kierunkowa"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint relarticle mainarticle" style="margin:0.2em 0 0.5em 1.6em"><span class="nomobile navigation-not-searchable"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/16px-Information_icon4.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/24px-Information_icon4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/32px-Information_icon4.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span> </span><i>Osobne artykuły: <a href="/wiki/Gradient_(matematyka)" title="Gradient (matematyka)">gradient</a> i <a href="/wiki/Pochodna_kierunkowa" title="Pochodna kierunkowa">pochodna kierunkowa</a>.</i></div> <p>Jeśli <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 \varphi \colon \mathbb {R} ^{3}\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \colon \mathbb {R} ^{3}\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7e4dd6c087f836ce4a1dd0e49247b25ace4439b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.579ex; height:3.176ex;" alt="{\displaystyle \varphi \colon \mathbb {R} ^{3}\to \mathbb {R} }"></span> jest <a href="/wiki/Pole_skalarne" title="Pole skalarne">polem skalarnym</a>, to potraktowanie nabli jako funkcji pola skalarnego daje <a href="/wiki/Pole_wektorowe" title="Pole wektorowe">pole wektorowe</a> nazywane <i>gradientem</i>: </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 \mathrm {grad} \;\varphi =\left({\frac {\partial \varphi }{\partial x}},{\frac {\partial \varphi }{\partial y}},{\frac {\partial \varphi }{\partial z}}\right)=\mathbf {i} {\frac {\partial \varphi }{\partial x}}+\mathbf {j} {\frac {\partial \varphi }{\partial y}}+\mathbf {k} {\frac {\partial \varphi }{\partial z}}=\nabla \varphi ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thickmathspace" /> <mi>φ</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</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>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {grad} \;\varphi =\left({\frac {\partial \varphi }{\partial x}},{\frac {\partial \varphi }{\partial y}},{\frac {\partial \varphi }{\partial z}}\right)=\mathbf {i} {\frac {\partial \varphi }{\partial x}}+\mathbf {j} {\frac {\partial \varphi }{\partial y}}+\mathbf {k} {\frac {\partial \varphi }{\partial z}}=\nabla \varphi ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e4b049953753baa18f1efa6877dc19a7eaddae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.278ex; height:6.176ex;" alt="{\displaystyle \mathrm {grad} \;\varphi =\left({\frac {\partial \varphi }{\partial x}},{\frac {\partial \varphi }{\partial y}},{\frac {\partial \varphi }{\partial z}}\right)=\mathbf {i} {\frac {\partial \varphi }{\partial x}}+\mathbf {j} {\frac {\partial \varphi }{\partial y}}+\mathbf {k} {\frac {\partial \varphi }{\partial z}}=\nabla \varphi ;}"></span></dd></dl> <p>powyższy zapis można traktować jako mnożenie „wektora nabla” przez „skalar” (w tej właśnie kolejności – zob. <i><a href="#Zastrzeżenia">Zastrzeżenia</a></i>) dające w wyniku „wektor”. Stąd nablę można uważać za operator pochodnej wielowymiarowej, o ile tylko spełnione są pewne warunki regularności (zob. <a href="/wiki/Gradient_(matematyka)#Związek_z_pochodną_i_różniczką" title="Gradient (matematyka)">związek gradientu z pochodną i różniczką</a>). Przy ich założeniu <i>pochodna kierunkowa</i> wzdłuż wektora <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =(u_{x},u_{y},u_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =(u_{x},u_{y},u_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cbb47d56ee9442a0c7943bfdf250aea77020a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.673ex; height:3.009ex;" alt="{\displaystyle \mathbf {u} =(u_{x},u_{y},u_{z})}"></span> może być przedstawiona w postaci iloczynu skalarnego gradientu (w danym punkcie) przez wektor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f43009bb6396f43aeb31b8099f33d777790d1b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.132ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} ,}"></span> to </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 {\frac {\partial \varphi }{\partial \mathbf {u} }}=u_{x}{\frac {\partial \varphi }{\partial x}}+u_{y}{\frac {\partial \varphi }{\partial y}}+u_{z}{\frac {\partial \varphi }{\partial z}}=\mathbf {u} \cdot \nabla \varphi =(\mathbf {u} \cdot \nabla )\varphi ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial \mathbf {u} }}=u_{x}{\frac {\partial \varphi }{\partial x}}+u_{y}{\frac {\partial \varphi }{\partial y}}+u_{z}{\frac {\partial \varphi }{\partial z}}=\mathbf {u} \cdot \nabla \varphi =(\mathbf {u} \cdot \nabla )\varphi ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f4b3c96752c6a6019d8bf6fa242929f837e307b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.583ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \varphi }{\partial \mathbf {u} }}=u_{x}{\frac {\partial \varphi }{\partial x}}+u_{y}{\frac {\partial \varphi }{\partial y}}+u_{z}{\frac {\partial \varphi }{\partial z}}=\mathbf {u} \cdot \nabla \varphi =(\mathbf {u} \cdot \nabla )\varphi ;}"></span></dd></dl> <p>Symbol w nawiasie po ostatniej równości należy traktować jako całość; operatorem jest więc wektor (w ogólności również pole wektorowe) mnożony skalarnie przez „wektor nabla” (zob. <i><a href="#Zastrzeżenia">Zastrzeżenia</a></i>). Oznaczenia te wykorzystuje się również do zapisu <a href="/wiki/Operator_Stokesa" title="Operator Stokesa">pochodnej materialnej</a>. Innym spotykanym oznaczeniem pochodnej <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> w kierunku <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"></span> jest <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{\mathbf {u} }\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </msub> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla _{\mathbf {u} }\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4503bd351737bb51073b0b4f2cf88a001f1eb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.385ex; height:2.676ex;" alt="{\displaystyle \nabla _{\mathbf {u} }\varphi .}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Dywergencja">Dywergencja</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=4" title="Edytuj sekcję: Dywergencja" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=4" title="Edytuj kod źródłowy sekcji: Dywergencja"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint relarticle mainarticle" style="margin:0.2em 0 0.5em 1.6em"><span class="nomobile navigation-not-searchable"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/16px-Information_icon4.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/24px-Information_icon4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/32px-Information_icon4.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span> </span><i>Osobny artykuł: <a href="/wiki/Dywergencja" title="Dywergencja">dywergencja</a>.</i></div> <p>Jeżeli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} \colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} \colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f3cc174ac9abc388e6bf76611aa61f423a66e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.166ex; height:2.676ex;" alt="{\displaystyle \mathbf {f} \colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}"></span> jest <a href="/wiki/Pole_wektorowe" title="Pole wektorowe">polem wektorowym</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 (f_{x},f_{y},f_{z})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{x},f_{y},f_{z})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11820d59b238ab7a5b81e9d4e854c0f6e0e9a52c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.518ex; height:3.009ex;" alt="{\displaystyle (f_{x},f_{y},f_{z})}"></span> zmiennych <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,y,z),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccd88635b67420fb82863e226c5e6f772a58fd21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.097ex; height:2.843ex;" alt="{\displaystyle (x,y,z),}"></span> to <i>dywergencję</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 \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6194e680a4e7c521f2178c50eea302843a852d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }"></span> będącą <a href="/wiki/Pole_skalarne" title="Pole skalarne">polem skalarnym</a> można wyrazić za pomocą <a href="/wiki/Iloczyn_skalarny" title="Iloczyn skalarny">iloczynu skalarnego</a> nabli przez <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236315d3ceef9b508dabccbbd5879529d0c501b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.7ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} ,}"></span> tzn. </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 \mathrm {div} \;\mathbf {f} ={\frac {\partial f_{x}}{\partial x}}+{\frac {\partial f_{y}}{\partial y}}+{\frac {\partial f_{z}}{\partial z}}=\nabla \cdot \mathbf {f} ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {div} \;\mathbf {f} ={\frac {\partial f_{x}}{\partial x}}+{\frac {\partial f_{y}}{\partial y}}+{\frac {\partial f_{z}}{\partial z}}=\nabla \cdot \mathbf {f} ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f364fc26c582958f363ddd3da025b26206ed6bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.162ex; height:6.343ex;" alt="{\displaystyle \mathrm {div} \;\mathbf {f} ={\frac {\partial f_{x}}{\partial x}}+{\frac {\partial f_{y}}{\partial y}}+{\frac {\partial f_{z}}{\partial z}}=\nabla \cdot \mathbf {f} ;}"></span></dd></dl> <p>w ten sposób „wektor nabla” jest mnożony przez „wektor”, dając w wyniku „skalar” (znowu istotna jest kolejność – zob. <i><a href="#Zastrzeżenia">Zastrzeżenia</a></i>); innymi słowy <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 \mathrm {div} =\nabla \cdot .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {div} =\nabla \cdot .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b23d8589e9e0c9bb10a375ba0cf8a1778ee052ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.495ex; height:2.176ex;" alt="{\displaystyle \mathrm {div} =\nabla \cdot .}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Rotacja">Rotacja</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=5" title="Edytuj sekcję: Rotacja" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=5" title="Edytuj kod źródłowy sekcji: Rotacja"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint relarticle mainarticle" style="margin:0.2em 0 0.5em 1.6em"><span class="nomobile navigation-not-searchable"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/16px-Information_icon4.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/24px-Information_icon4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/32px-Information_icon4.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span> </span><i>Osobny artykuł: <a href="/wiki/Rotacja" title="Rotacja">rotacja</a>.</i></div> <p>Zamiana iloczynu skalarnego na <a href="/wiki/Iloczyn_wektorowy" title="Iloczyn wektorowy">iloczyn wektorowy</a> dla danego pola wektorowego <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6194e680a4e7c521f2178c50eea302843a852d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }"></span> w powyższym przypadku umożliwia zwarty sposób zapisu <i>rotacji</i>: </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 {\begin{aligned}\mathrm {rot} \;\mathbf {f} &=\left({\frac {\partial f_{z}}{\partial y}}-{\frac {\partial f_{y}}{\partial z}},\ {\frac {\partial f_{x}}{\partial z}}-{\frac {\partial f_{z}}{\partial x}},\ {\frac {\partial f_{y}}{\partial x}}-{\frac {\partial f_{x}}{\partial y}}\right)\\[2pt]&=\left({\frac {\partial f_{z}}{\partial y}}-{\frac {\partial f_{y}}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial f_{x}}{\partial z}}-{\frac {\partial f_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial f_{y}}{\partial x}}-{\frac {\partial f_{x}}{\partial y}}\right)\mathbf {k} \\[2pt]&=\nabla \times \mathbf {f} ;\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>;</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {rot} \;\mathbf {f} &=\left({\frac {\partial f_{z}}{\partial y}}-{\frac {\partial f_{y}}{\partial z}},\ {\frac {\partial f_{x}}{\partial z}}-{\frac {\partial f_{z}}{\partial x}},\ {\frac {\partial f_{y}}{\partial x}}-{\frac {\partial f_{x}}{\partial y}}\right)\\[2pt]&=\left({\frac {\partial f_{z}}{\partial y}}-{\frac {\partial f_{y}}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial f_{x}}{\partial z}}-{\frac {\partial f_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial f_{y}}{\partial x}}-{\frac {\partial f_{x}}{\partial y}}\right)\mathbf {k} \\[2pt]&=\nabla \times \mathbf {f} ;\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78ddd3d336b77393d50690b5e07a6b326e1726fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.838ex; width:63.331ex; height:16.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {rot} \;\mathbf {f} &=\left({\frac {\partial f_{z}}{\partial y}}-{\frac {\partial f_{y}}{\partial z}},\ {\frac {\partial f_{x}}{\partial z}}-{\frac {\partial f_{z}}{\partial x}},\ {\frac {\partial f_{y}}{\partial x}}-{\frac {\partial f_{x}}{\partial y}}\right)\\[2pt]&=\left({\frac {\partial f_{z}}{\partial y}}-{\frac {\partial f_{y}}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial f_{x}}{\partial z}}-{\frac {\partial f_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial f_{y}}{\partial x}}-{\frac {\partial f_{x}}{\partial y}}\right)\mathbf {k} \\[2pt]&=\nabla \times \mathbf {f} ;\end{aligned}}}"></span></dd></dl> <p>potwierdza to intuicję, iż „wektor nabla” mnożony wektorowo przez „wektor” daje inny „wektor” (z zachowaniem kolejności – zob. <i><a href="#Zastrzeżenia">Zastrzeżenia</a></i>); dlatego <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 \mathrm {rot} =\nabla \times .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {rot} =\nabla \times .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed4829d43316acbc9ca1df8a6bdfa479f2239868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.468ex; height:2.176ex;" alt="{\displaystyle \mathrm {rot} =\nabla \times .}"></span> Korzystając z <a href="/wiki/Mnemotechnika" title="Mnemotechnika">mnemoniku</a> <a href="/wiki/Wyznacznik" title="Wyznacznik">wyznacznikowego</a> dla iloczynu wektorowego rotację <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236315d3ceef9b508dabccbbd5879529d0c501b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.7ex; height:2.509ex;" alt="{\displaystyle \mathbf {f} ,}"></span> można wtedy zapisać w postaci </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \mathbf {f} =\det {\begin{bmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\f_{x}&f_{y}&f_{z}\\\mathbf {i} &\mathbf {j} &\mathbf {k} \end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {f} =\det {\begin{bmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\f_{x}&f_{y}&f_{z}\\\mathbf {i} &\mathbf {j} &\mathbf {k} \end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a984500583b4523c48184156e7ee5393536f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.599ex; margin-bottom: -0.239ex; width:29.52ex; height:10.843ex;" alt="{\displaystyle \nabla \times \mathbf {f} =\det {\begin{bmatrix}{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\f_{x}&f_{y}&f_{z}\\\mathbf {i} &\mathbf {j} &\mathbf {k} \end{bmatrix}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Laplasjan">Laplasjan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=6" title="Edytuj sekcję: Laplasjan" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=6" title="Edytuj kod źródłowy sekcji: Laplasjan"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint relarticle mainarticle" style="margin:0.2em 0 0.5em 1.6em"><span class="nomobile navigation-not-searchable"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/16px-Information_icon4.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/24px-Information_icon4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/32px-Information_icon4.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span> </span><i>Osobny artykuł: <a href="/wiki/Operator_Laplace%27a" class="mw-redirect" title="Operator Laplace'a">laplasjan</a>.</i></div> <p><i>Laplasjan</i>, nazywany również <i>operatorem Laplace’a</i>, jest operatorem skalarnym działającym na pole skalarne danym jako </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}=\nabla \cdot \nabla =\nabla ^{2};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo>=</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}=\nabla \cdot \nabla =\nabla ^{2};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56497204ff0143dcf94913875ecb0c2a8f44f133" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.306ex; height:6.343ex;" alt="{\displaystyle \Delta ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}=\nabla \cdot \nabla =\nabla ^{2};}"></span></dd></dl> <p>znajduje on zastosowanie w wielu działach współczesnej <a href="/wiki/Fizyka_matematyczna" title="Fizyka matematyczna">fizyki matematycznej</a>, pojawia się m.in. w <a href="/wiki/R%C3%B3wnanie_r%C3%B3%C5%BCniczkowe_Laplace%E2%80%99a" title="Równanie różniczkowe Laplace’a">równaniu Laplace’a</a>, <a href="/wiki/R%C3%B3wnanie_r%C3%B3%C5%BCniczkowe_Poissona" title="Równanie różniczkowe Poissona">równaniu Poissona</a>, <a href="/wiki/R%C3%B3wnanie_przewodnictwa_cieplnego" title="Równanie przewodnictwa cieplnego">równaniu przewodnictwa ciepła</a>, <a href="/wiki/R%C3%B3wnanie_falowe" title="Równanie falowe">równaniu falowym</a>, czy <a href="/wiki/R%C3%B3wnanie_Schr%C3%B6dingera" title="Równanie Schrödingera">równaniu Schrödingera</a>. </p><p>Stosuje się również <i>laplasjan wektorowy</i> będący operatorem wektorowym zwracającym pole wektorowe: jeżeli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6194e680a4e7c521f2178c50eea302843a852d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }"></span> jest polem wektorowym, to jest on zdefiniowany wzorem </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\mathbf {f} =\nabla (\nabla \cdot \mathbf {f} )-\nabla \times (\nabla \times \mathbf {f} );}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\mathbf {f} =\nabla (\nabla \cdot \mathbf {f} )-\nabla \times (\nabla \times \mathbf {f} );}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37e24b24927e25ee7c6d1bc4fb4b36692213bb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.458ex; height:3.176ex;" alt="{\displaystyle \nabla ^{2}\mathbf {f} =\nabla (\nabla \cdot \mathbf {f} )-\nabla \times (\nabla \times \mathbf {f} );}"></span></dd></dl> <p>we współrzędnych kartezjańskich przyjmuje on dużo prostszą postać (która może być postrzegana jako szczególny przypadek <a href="/wiki/Iloczyn_wektorowy#Wzór_Lagrange'a" title="Iloczyn wektorowy">wzoru Lagrange’a</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 \Delta \mathbf {f} =(\Delta f_{x},\Delta f_{y},\Delta f_{z}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {f} =(\Delta f_{x},\Delta f_{y},\Delta f_{z}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b232a4e8c72b8c130567709c7e28270ac39365" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.06ex; height:3.009ex;" alt="{\displaystyle \Delta \mathbf {f} =(\Delta f_{x},\Delta f_{y},\Delta f_{z}),}"></span></dd></dl> <p>gdzie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} =(f_{x},f_{y},f_{z}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} =(f_{x},f_{y},f_{z}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc1206cf367623c647151e7f07af93493440c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.317ex; height:3.009ex;" alt="{\displaystyle \mathbf {f} =(f_{x},f_{y},f_{z}).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Pochodna_kowariantna">Pochodna kowariantna</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=7" title="Edytuj sekcję: Pochodna kowariantna" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=7" title="Edytuj kod źródłowy sekcji: Pochodna kowariantna"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="noprint relarticle mainarticle" style="margin:0.2em 0 0.5em 1.6em"><span class="nomobile navigation-not-searchable"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/16px-Information_icon4.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/24px-Information_icon4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1d/Information_icon4.svg/32px-Information_icon4.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span> </span><i>Osobny artykuł: <a href="/wiki/Pochodna_kowariantna" title="Pochodna kowariantna">pochodna kowariantna</a>.</i></div> <p>Użycie <a href="/w/index.php?title=Iloczyn_tensorowy&action=edit&redlink=1" class="new" title="Iloczyn tensorowy (strona nie istnieje)">iloczynu tensorowego</a>, w tym przypadku <a href="/wiki/Iloczyn_diadyczny" title="Iloczyn diadyczny">iloczynu diadycznego</a>, w miejsce iloczynu skalarnego dla dywergencji i iloczynu wektorowego dla rotacji opisuje <i>pochodną kowariantną</i>; dokładniej: jeśli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6194e680a4e7c521f2178c50eea302843a852d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.053ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} }"></span> jest trójwymiarowym polem wektorowym, to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \otimes \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \otimes \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b43addd07ae0c340538af97f0ffb15892914b9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.83ex; height:2.343ex;" alt="{\displaystyle \nabla \otimes \mathbf {f} }"></span> jest <a href="/wiki/Tensor" title="Tensor">tensorem</a> drugiego rzędu odpowiadającym pochodnej kowariantnej <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 \mathrm {D} \mathbf {f} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} \mathbf {f} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25654a35ef7f187284d1e2916e648608a8cc701" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.476ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} \mathbf {f} ,}"></span> którą można przedstawić za pomocą <a href="/wiki/Macierz" title="Macierz">macierzy</a> równoważnej <a href="/wiki/Macierz_Jacobiego" title="Macierz Jacobiego">macierzy Jacobiego</a> pola wektorowego <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb9155bf932307f4b9b8d78b76f28413aa119f62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.7ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} .}"></span> Notację tę stosuje się również do opisu zmiany pola wektorowego <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {f} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {f} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c6269253a8421a54a5846ca1dac81e68bb673a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.102ex; height:2.343ex;" alt="{\displaystyle \delta \mathbf {f} }"></span> przy małym <a href="/wiki/Przemieszczenie_(fizyka)" title="Przemieszczenie (fizyka)">przemieszczeniu</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 \delta \mathbf {r} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {r} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68aee3109a8eaaa4a726566fd05076d5e0f59b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.798ex; height:2.676ex;" alt="{\displaystyle \delta \mathbf {r} ,}"></span> mianowicie </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta \mathbf {f} =(\nabla \otimes \mathbf {f} )\cdot \delta \mathbf {r} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta \mathbf {f} =(\nabla \otimes \mathbf {f} )\cdot \delta \mathbf {r} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d418260e4b541045a35c771787a5deed84d4a44c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.316ex; height:2.843ex;" alt="{\displaystyle \delta \mathbf {f} =(\nabla \otimes \mathbf {f} )\cdot \delta \mathbf {r} .}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Złożenia"><span id="Z.C5.82o.C5.BCenia"></span>Złożenia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=8" title="Edytuj sekcję: Złożenia" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=8" title="Edytuj kod źródłowy sekcji: Złożenia"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Plik:DCG_chart.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/DCG_chart.svg/220px-DCG_chart.svg.png" decoding="async" width="220" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/DCG_chart.svg/330px-DCG_chart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/DCG_chart.svg/440px-DCG_chart.svg.png 2x" data-file-width="482" data-file-height="323" /></a><figcaption>Następujący diagram demonstruje wszystkie zasady dotyczące złożeń różnych operatorów: symbole <i>D</i>, <i>C</i>, <i>G</i>, <i>L<sub>scalar</sub></i>, <i>L<sub>vect</sub></i> oraz <i>CC</i> oznaczają kolejno dywergencję, rotację, gradient, laplasjan skalarny i wektorowy oraz rotację rotacji; niebieskie strzałki przedstawiają istnienie złożenia wskazywanego za pomocą strzałki, niebieski okrąg obrazuje możliwość dwukrotnego złożenia rotacji, czerwone okręgi (przerywane) oddają niemożność złożenia dywergencji i gradientu samych ze sobą.</figcaption></figure> <p>Rozpatrując możliwość „brania różnych iloczynów” nabli przez pola skalarne i wektorowe, które dają inne pola skalarne bądź wektorowe, można wyróżnić wiele możliwości złożeń uzyskanych operatorów; zgodność poszczególnych operatorów umożliwia wykonanie następujących złożeń: </p> <ul><li>trzech operacji na polu wektorowym uzyskanym jako gradient pola skalarnego,</li></ul> <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 \mathrm {div} \;(\mathrm {grad} \;\varphi )=\nabla \cdot (\nabla \varphi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thickmathspace" /> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {div} \;(\mathrm {grad} \;\varphi )=\nabla \cdot (\nabla \varphi ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15ab6b7c898ca62a20e603d62d73b2e67cb97ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.941ex; height:2.843ex;" alt="{\displaystyle \mathrm {div} \;(\mathrm {grad} \;\varphi )=\nabla \cdot (\nabla \varphi ),}"></span></dd> <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 \mathrm {rot} \;(\mathrm {grad} \;\varphi )=\nabla \times (\nabla \varphi ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thickmathspace" /> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {rot} \;(\mathrm {grad} \;\varphi )=\nabla \times (\nabla \varphi ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f239e0820377c8299b34ff1a6498e3a43024a7eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.914ex; height:2.843ex;" alt="{\displaystyle \mathrm {rot} \;(\mathrm {grad} \;\varphi )=\nabla \times (\nabla \varphi ),}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \varphi =\nabla ^{2}\varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>φ</mi> <mo>=</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \varphi =\nabla ^{2}\varphi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa840ca9e171d9e474e200dc40e621a4ede1b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.712ex; height:3.176ex;" alt="{\displaystyle \Delta \varphi =\nabla ^{2}\varphi ,}"></span></dd></dl> <ul><li>operacji na polu skalarnym uzyskanym jako dywergencja pola wektorowego,</li></ul> <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 \mathrm {grad} \;(\mathrm {div} \;\mathbf {f} )=\nabla (\nabla \cdot \mathbf {f} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {grad} \;(\mathrm {div} \;\mathbf {f} )=\nabla (\nabla \cdot \mathbf {f} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6917dfea822d57191e65345f2eefcb606da83ecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.008ex; height:2.843ex;" alt="{\displaystyle \mathrm {grad} \;(\mathrm {div} \;\mathbf {f} )=\nabla (\nabla \cdot \mathbf {f} ),}"></span></dd></dl> <ul><li>dwóch operacji na polu wektorowym uzyskanym jako rotacja pola wektorowego,</li></ul> <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 \mathrm {div} \;(\mathrm {rot} \;\mathbf {f} )=\nabla \cdot (\nabla \times \mathbf {f} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {div} \;(\mathrm {rot} \;\mathbf {f} )=\nabla \cdot (\nabla \times \mathbf {f} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9ce379455b7569fe85dda45732ad86d0637fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.298ex; height:2.843ex;" alt="{\displaystyle \mathrm {div} \;(\mathrm {rot} \;\mathbf {f} )=\nabla \cdot (\nabla \times \mathbf {f} ),}"></span></dd> <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 \mathrm {rot} \;(\mathrm {rot} \;\mathbf {f} )=\nabla \times (\nabla \times \mathbf {f} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {rot} \;(\mathrm {rot} \;\mathbf {f} )=\nabla \times (\nabla \times \mathbf {f} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f003a1934649ddc27750c1bb3cf520e42f8522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.271ex; height:2.843ex;" alt="{\displaystyle \mathrm {rot} \;(\mathrm {rot} \;\mathbf {f} )=\nabla \times (\nabla \times \mathbf {f} ),}"></span></dd></dl> <ul><li>operacji laplasjanu wektorowego,</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \mathbf {f} =\nabla ^{2}\mathbf {f} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {f} =\nabla ^{2}\mathbf {f} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/423e161fe65f61ea112004a2b11a200bbb85c78d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.778ex; height:3.009ex;" alt="{\displaystyle \Delta \mathbf {f} =\nabla ^{2}\mathbf {f} ,}"></span></dd></dl> <p>przy czym dwa z nich są zawsze równe, </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 \mathrm {div} \;(\mathrm {grad} \;\varphi )=\nabla \cdot (\nabla \varphi )=\nabla ^{2}\varphi =\Delta \varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thickmathspace" /> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>φ</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {div} \;(\mathrm {grad} \;\varphi )=\nabla \cdot (\nabla \varphi )=\nabla ^{2}\varphi =\Delta \varphi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ba5a4cb91bd1122c0f0ed05f54a66d7c6ee6b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.104ex; height:3.176ex;" alt="{\displaystyle \mathrm {div} \;(\mathrm {grad} \;\varphi )=\nabla \cdot (\nabla \varphi )=\nabla ^{2}\varphi =\Delta \varphi ,}"></span></dd></dl> <p>zaś następujące dwa zawsze znikają, o ile pola są <i>wystarczająco regularne</i>: </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 \mathrm {rot} \;(\mathrm {grad} \;\varphi )=\nabla \times (\nabla \varphi )=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thickmathspace" /> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {rot} \;(\mathrm {grad} \;\varphi )=\nabla \times (\nabla \varphi )=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed8eb0a9931fd72ac2978370dbc8b5e796994f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.175ex; height:2.843ex;" alt="{\displaystyle \mathrm {rot} \;(\mathrm {grad} \;\varphi )=\nabla \times (\nabla \varphi )=0,}"></span></dd> <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 \mathrm {div} \;(\mathrm {rot} \;\mathbf {f} )=\nabla \cdot (\nabla \times \mathbf {f} )=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {div} \;(\mathrm {rot} \;\mathbf {f} )=\nabla \cdot (\nabla \times \mathbf {f} )=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b66c8943dd50cfcede2177d5a47cd53cd34656e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.559ex; height:2.843ex;" alt="{\displaystyle \mathrm {div} \;(\mathrm {rot} \;\mathbf {f} )=\nabla \cdot (\nabla \times \mathbf {f} )=0.}"></span></dd></dl> <p>Zachodzi również tożsamość przypominająca <a href="/wiki/Iloczyn_wektorowy#Wzór_Lagrange'a" title="Iloczyn wektorowy">wzór Lagrange’a</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 \nabla \times (\nabla \times \mathbf {f} )=\nabla (\nabla \cdot \mathbf {f} )-\nabla ^{2}\mathbf {f} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times (\nabla \times \mathbf {f} )=\nabla (\nabla \cdot \mathbf {f} )-\nabla ^{2}\mathbf {f} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2888e18fe37d6ce9f4800d41ae8eeeffd3b012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.458ex; height:3.176ex;" alt="{\displaystyle \nabla \times (\nabla \times \mathbf {f} )=\nabla (\nabla \cdot \mathbf {f} )-\nabla ^{2}\mathbf {f} ,}"></span></dd></dl> <p>gdyż </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 \mathrm {rot} \;(\mathrm {rot} \;\mathbf {f} )=\mathrm {grad} \;(\mathrm {div} \;\mathbf {f} )-\Delta \mathbf {f} ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {rot} \;(\mathrm {rot} \;\mathbf {f} )=\mathrm {grad} \;(\mathrm {div} \;\mathbf {f} )-\Delta \mathbf {f} ;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72b68ceda8cac71166137b4aca908764f4a6611" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.534ex; height:2.843ex;" alt="{\displaystyle \mathrm {rot} \;(\mathrm {rot} \;\mathbf {f} )=\mathrm {grad} \;(\mathrm {div} \;\mathbf {f} )-\Delta \mathbf {f} ;}"></span></dd></dl> <p>jeśli pola są <i>wystarczająco regularne</i>, to jeden z operatorów można wyrazić za pomocą iloczynu tensorowego: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla (\nabla \cdot \mathbf {f} )=\nabla \cdot (\nabla \otimes \mathbf {f} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla (\nabla \cdot \mathbf {f} )=\nabla \cdot (\nabla \otimes \mathbf {f} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beecf3cb3e82c30e85b033cb263a5e9c03bdd7e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.413ex; height:2.843ex;" alt="{\displaystyle \nabla (\nabla \cdot \mathbf {f} )=\nabla \cdot (\nabla \otimes \mathbf {f} ),}"></span></dd></dl> <p>ponieważ </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 \mathrm {grad} \;(\mathrm {div} \;\mathbf {f} )=\mathrm {div} (\mathrm {D} \mathbf {f} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">d</mi> </mrow> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {grad} \;(\mathrm {div} \;\mathbf {f} )=\mathrm {div} (\mathrm {D} \mathbf {f} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6117df6d561b3e8d5e028e3fbaca51a04b7936d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.399ex; height:2.843ex;" alt="{\displaystyle \mathrm {grad} \;(\mathrm {div} \;\mathbf {f} )=\mathrm {div} (\mathrm {D} \mathbf {f} ).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Zobacz_też"><span id="Zobacz_te.C5.BC"></span>Zobacz też</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=9" title="Edytuj sekcję: Zobacz też" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=9" title="Edytuj kod źródłowy sekcji: Zobacz też"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="infobox noprint plainlinks" cellpadding="4" role="presentation"> <tbody><tr> <td style="vertical-align:middle; text-align:center; width:30px;"><span class="notpageimage" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/28px-WiktionaryPl_nodesc.svg.png" decoding="async" width="28" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/42px-WiktionaryPl_nodesc.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/WiktionaryPl_nodesc.svg/56px-WiktionaryPl_nodesc.svg.png 2x" data-file-width="122" data-file-height="117" /></span></span> </td> <td style="line-height:normal; vertical-align:middle; text-align:center; flex:unset;"><a href="https://pl.wiktionary.org/wiki/nabla" class="extiw" title="wikt:nabla"><strong>Zobacz hasło</strong> <em>nabla</em> w Wikisłowniku</a> </td></tr></tbody></table> <ul><li><a href="/wiki/Operator_nabla_w_r%C3%B3%C5%BCnych_uk%C5%82adach_wsp%C3%B3%C5%82rz%C4%99dnych" title="Operator nabla w różnych układach współrzędnych">operator nabla w różnych układach współrzędnych</a></li> <li><a href="/wiki/Pochodne_Wirtingera" title="Pochodne Wirtingera">pochodne Wirtingera</a></li> <li><a href="/wiki/R%C3%B3wnania_Maxwella" title="Równania Maxwella">równania Maxwella</a></li> <li><a href="/wiki/R%C3%B3wnania_Naviera-Stokesa" title="Równania Naviera-Stokesa">równania Naviera-Stokesa</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Zastrzeżenia"><span id="Zastrze.C5.BCenia"></span>Zastrzeżenia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=10" title="Edytuj sekcję: Zastrzeżenia" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=10" title="Edytuj kod źródłowy sekcji: Zastrzeżenia"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Większość z powyższych własności zdaje się być zwykłymi tożsamościami dotyczącymi wektorów – w szczególności podstawienie zamiast nabli wektora zawsze da prawdziwą tożsamość wektorową (poza tymi, które dotyczą własności różniczkowych, np. <a href="/w/index.php?title=Regu%C5%82a_Leibniza&action=edit&redlink=1" class="new" title="Reguła Leibniza (strona nie istnieje)">reguła iloczynu</a>). Jest to istotne ułatwienie, które niekiedy może być zdradliwe, gdyż stosowanie nabli wymaga zachowania kolejności czynników poszczególnych mnożeń. Wynika to z faktu, iż wektor jest obiektem mającym jednoznacznie określone liczbowo współrzędne, zaś nabla nie przedstawia żadnej wartości dopóki nie zadziała na pewnym polu. </p><p>Przykładowo tożsamość wektorowa </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 (\mathbf {u} \cdot \mathbf {v} )\varphi =(\mathbf {v} \cdot \mathbf {u} )\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mi>φ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {u} \cdot \mathbf {v} )\varphi =(\mathbf {v} \cdot \mathbf {u} )\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b76a1712c7d51b8fd5773088c8f04be8ebb5ebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.908ex; height:2.843ex;" alt="{\displaystyle (\mathbf {u} \cdot \mathbf {v} )\varphi =(\mathbf {v} \cdot \mathbf {u} )\varphi }"></span></dd></dl> <p>zastosowana dla dywergencji pola wektorowego przestaje być prawdziwa: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\nabla \cdot \mathbf {f} )\varphi \neq (\mathbf {f} \cdot \nabla )\varphi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mi>φ</mi> <mo>≠</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mi>φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\nabla \cdot \mathbf {f} )\varphi \neq (\mathbf {f} \cdot \nabla )\varphi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5445feb03664a3f0341df9bbdb9ac8dcef9b2173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.741ex; height:2.843ex;" alt="{\displaystyle (\nabla \cdot \mathbf {f} )\varphi \neq (\mathbf {f} \cdot \nabla )\varphi .}"></span></dd></dl> <p>Otóż </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 {\begin{aligned}(\nabla \cdot \mathbf {f} )\varphi &=\left({\frac {\partial f_{x}}{\partial x}}+{\frac {\partial f_{y}}{\partial y}}+{\frac {\partial f_{z}}{\partial z}}\right)\varphi \\&={\frac {\partial f_{x}}{\partial x}}\varphi +{\frac {\partial f_{y}}{\partial y}}\varphi +{\frac {\partial f_{z}}{\partial z}}\varphi ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo stretchy="false">)</mo> <mi>φ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </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> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mi>φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mi>φ</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\nabla \cdot \mathbf {f} )\varphi &=\left({\frac {\partial f_{x}}{\partial x}}+{\frac {\partial f_{y}}{\partial y}}+{\frac {\partial f_{z}}{\partial z}}\right)\varphi \\&={\frac {\partial f_{x}}{\partial x}}\varphi +{\frac {\partial f_{y}}{\partial y}}\varphi +{\frac {\partial f_{z}}{\partial z}}\varphi ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1afd55b8c5b4a2f63728db49f06c89a79fedfd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:35.961ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}(\nabla \cdot \mathbf {f} )\varphi &=\left({\frac {\partial f_{x}}{\partial x}}+{\frac {\partial f_{y}}{\partial y}}+{\frac {\partial f_{z}}{\partial z}}\right)\varphi \\&={\frac {\partial f_{x}}{\partial x}}\varphi +{\frac {\partial f_{y}}{\partial y}}\varphi +{\frac {\partial f_{z}}{\partial z}}\varphi ,\end{aligned}}}"></span></dd></dl> <p>zaś </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 {\begin{aligned}(\mathbf {f} \cdot \nabla )\varphi &=\left(f_{x}{\frac {\partial }{\partial x}}+f_{y}{\frac {\partial }{\partial y}}+f_{z}{\frac {\partial }{\partial z}}\right)\varphi \\&=f_{x}{\frac {\partial \varphi }{\partial x}}+f_{y}{\frac {\partial \varphi }{\partial y}}+f_{z}{\frac {\partial \varphi }{\partial z}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mi>φ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\mathbf {f} \cdot \nabla )\varphi &=\left(f_{x}{\frac {\partial }{\partial x}}+f_{y}{\frac {\partial }{\partial y}}+f_{z}{\frac {\partial }{\partial z}}\right)\varphi \\&=f_{x}{\frac {\partial \varphi }{\partial x}}+f_{y}{\frac {\partial \varphi }{\partial y}}+f_{z}{\frac {\partial \varphi }{\partial z}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128441ac697c19eda6850df03290bf3650d3af04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:39.534ex; height:12.343ex;" alt="{\displaystyle {\begin{aligned}(\mathbf {f} \cdot \nabla )\varphi &=\left(f_{x}{\frac {\partial }{\partial x}}+f_{y}{\frac {\partial }{\partial y}}+f_{z}{\frac {\partial }{\partial z}}\right)\varphi \\&=f_{x}{\frac {\partial \varphi }{\partial x}}+f_{y}{\frac {\partial \varphi }{\partial y}}+f_{z}{\frac {\partial \varphi }{\partial z}},\end{aligned}}}"></span></dd></dl> <p>gdzie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} =(f_{x},f_{y},f_{z}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} =(f_{x},f_{y},f_{z}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc1206cf367623c647151e7f07af93493440c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.317ex; height:3.009ex;" alt="{\displaystyle \mathbf {f} =(f_{x},f_{y},f_{z}).}"></span> </p><p>Przy korzystaniu z własności różniczkowych nabli również wymagana jest ostrożność: niech <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d135f308e43463a63104ad85008b3b072c3e938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.456ex; height:2.676ex;" alt="{\displaystyle \nabla \varphi }"></span> oznacza gradient pola skalarnego <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 \varphi ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb4baf1e617abd3f5384bab1851bf109ea0b614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.167ex; height:2.176ex;" alt="{\displaystyle \varphi ,}"></span> podczas gdy napis <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 \varphi \nabla }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mi mathvariant="normal">∇</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \nabla }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43cb8f84da76327f2486250b257cf6e6b9ab05fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.456ex; height:2.676ex;" alt="{\displaystyle \varphi \nabla }"></span> reprezentuje iloczyn pola <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 \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span> oraz gradientu jeszcze niewskazanego pola skalarnego, czyli jako taki przedstawia funkcję pochodnej, będąc tym samym kolejnym operatorem różniczkowym. Podobnie jeżeli <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 \varphi (x,y,z)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (x,y,z)=x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed8c6509637ff7c820604832775d768971ed4b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.399ex; height:2.843ex;" alt="{\displaystyle \varphi (x,y,z)=x}"></span> oraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi (x,y,z)=y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi (x,y,z)=y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f46f2f0493dc91d915ab667b7520eadb178f7d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.864ex; height:2.843ex;" alt="{\displaystyle \psi (x,y,z)=y,}"></span> to </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 {\begin{aligned}(\nabla \varphi )\times (\nabla \psi )&=\left(\mathbf {i} {\frac {\partial \varphi }{\partial x}}+\mathbf {j} {\frac {\partial \varphi }{\partial y}}+\mathbf {k} {\frac {\partial \varphi }{\partial z}}\right)\times \left(\mathbf {i} {\frac {\partial \psi }{\partial x}}+\mathbf {j} {\frac {\partial \psi }{\partial y}}+\mathbf {k} {\frac {\partial \psi }{\partial z}}\right)\\[2pt]&=(\mathbf {i} \cdot 1+\mathbf {j} \cdot 0+\mathbf {k} \cdot 0)\times (\mathbf {i} \cdot 0+\mathbf {j} \cdot 1+\mathbf {k} \cdot 0)\\[2pt]&=\mathbf {i} \times \mathbf {j} =\mathbf {k} ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>⋅</mo> <mn>0</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">i</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(\nabla \varphi )\times (\nabla \psi )&=\left(\mathbf {i} {\frac {\partial \varphi }{\partial x}}+\mathbf {j} {\frac {\partial \varphi }{\partial y}}+\mathbf {k} {\frac {\partial \varphi }{\partial z}}\right)\times \left(\mathbf {i} {\frac {\partial \psi }{\partial x}}+\mathbf {j} {\frac {\partial \psi }{\partial y}}+\mathbf {k} {\frac {\partial \psi }{\partial z}}\right)\\[2pt]&=(\mathbf {i} \cdot 1+\mathbf {j} \cdot 0+\mathbf {k} \cdot 0)\times (\mathbf {i} \cdot 0+\mathbf {j} \cdot 1+\mathbf {k} \cdot 0)\\[2pt]&=\mathbf {i} \times \mathbf {j} =\mathbf {k} ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d843344a5d7e9d3a4ff80301818dffc8175313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:66.222ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}(\nabla \varphi )\times (\nabla \psi )&=\left(\mathbf {i} {\frac {\partial \varphi }{\partial x}}+\mathbf {j} {\frac {\partial \varphi }{\partial y}}+\mathbf {k} {\frac {\partial \varphi }{\partial z}}\right)\times \left(\mathbf {i} {\frac {\partial \psi }{\partial x}}+\mathbf {j} {\frac {\partial \psi }{\partial y}}+\mathbf {k} {\frac {\partial \psi }{\partial z}}\right)\\[2pt]&=(\mathbf {i} \cdot 1+\mathbf {j} \cdot 0+\mathbf {k} \cdot 0)\times (\mathbf {i} \cdot 0+\mathbf {j} \cdot 1+\mathbf {k} \cdot 0)\\[2pt]&=\mathbf {i} \times \mathbf {j} =\mathbf {k} ,\end{aligned}}}"></span></dd></dl> <p>podczas gdy </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 (\mathbf {u} \varphi )\times (\mathbf {u} \psi )=\varphi \psi (\mathbf {u} \times \mathbf {u} )=\varphi \psi \mathbf {0} =\mathbf {0} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mi>φ</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mi>ψ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mi>ψ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {u} \varphi )\times (\mathbf {u} \psi )=\varphi \psi (\mathbf {u} \times \mathbf {u} )=\varphi \psi \mathbf {0} =\mathbf {0} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9344d1cf5505f55ea6da1fd4f3a9b18ea721f69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.765ex; height:2.843ex;" alt="{\displaystyle (\mathbf {u} \varphi )\times (\mathbf {u} \psi )=\varphi \psi (\mathbf {u} \times \mathbf {u} )=\varphi \psi \mathbf {0} =\mathbf {0} .}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Uwagi">Uwagi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=11" title="Edytuj sekcję: Uwagi" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=11" title="Edytuj kod źródłowy sekcji: Uwagi"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="do-not-make-smaller refsection refsection-uwagi ll-script ll-script-uwagi"><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">Etymologia w artykule dot. <a href="/wiki/Symbol_nabla" title="Symbol nabla">symbolu nabla</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Przypisy">Przypisy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=12" title="Edytuj sekcję: Przypisy" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=12" title="Edytuj kod źródłowy sekcji: Przypisy"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="do-not-make-smaller refsection"><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-epwn-2"><span class="mw-cite-backlink"><a href="#cite_ref-epwn_2-0">↑</a></span> <span class="reference-text"><cite class="citation web open-access"><a rel="nofollow" class="external text" href="https://encyklopedia.pwn.pl/haslo/;3945039"><i>nabla</i></a>, [w:] <i><a href="/wiki/Encyklopedia_PWN_(internetowa)" title="Encyklopedia PWN (internetowa)">Encyklopedia PWN</a></i> [online], <a href="/wiki/Wydawnictwo_Naukowe_PWN" title="Wydawnictwo Naukowe PWN">Wydawnictwo Naukowe PWN</a><span class="accessdate"> [dostęp 2021-10-02]</span><span class="Z3988" title="ctx_ver=Z39.88-2004&rft.gengre=unknown&rft.atitle=nabla&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.jtitle=%5B%5BWydawnictwo+Naukowe+PWN%5D%5D&rft_id=https%3A%2F%2Fencyklopedia.pwn.pl%2Fhaslo%2F%3B3945039" style="display:none"> </span>.</cite></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Linki_zewnętrzne"><span id="Linki_zewn.C4.99trzne"></span>Linki zewnętrzne</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_nabla&veaction=edit&section=13" title="Edytuj sekcję: Linki zewnętrzne" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_nabla&action=edit&section=13" title="Edytuj kod źródłowy sekcji: Linki zewnętrzne"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20230608162941/https://www.mat.univie.ac.at/~neum/contrib/nabla.txt">Historia nabli</a> – zapis wątku z 1998 z uniwersyteckiej <a href="/wiki/Lista_dyskusyjna" title="Lista dyskusyjna">listy dyskusyjnej</a>, kopia w <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>; <a rel="nofollow" class="external text" href="http://www.mat.univie.ac.at/~neum/contrib/nabla.txt">Link oryginalny</a> – nieaktywny <span class="lang-list">(<abbr title="Treść w języku angielskim (English)">ang.</abbr>)</span></li> <li><a rel="nofollow" class="external text" href="http://hdl.handle.net/2027.42/7869">Badanie nieprawidłowego użycia ∇ w analizie wektorowej</a> <span class="lang-list">(<abbr title="Treść w języku angielskim (English)">ang.</abbr>)</span> (1994) Tai, Chen</li> <li><span typeof="mw:File"><a href="/wiki/Otwarty_dost%C4%99p" title="publikacja w otwartym dostępie – możesz ją przeczytać"><img alt="publikacja w otwartym dostępie – możesz ją przeczytać" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Open_Access_logo_green_alt2.svg/8px-Open_Access_logo_green_alt2.svg.png" decoding="async" width="8" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Open_Access_logo_green_alt2.svg/12px-Open_Access_logo_green_alt2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Open_Access_logo_green_alt2.svg/16px-Open_Access_logo_green_alt2.svg.png 2x" data-file-width="640" data-file-height="1000" /></a></span> <i><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Hamilton_operator">Hamilton operator</a></i> <span class="lang-list">(<abbr title="Treść w języku angielskim (English)">ang.</abbr>)</span>, Encyclopedia of Mathematics, encyclopediaofmath.org [dostęp 2024-04-05].</li></ul> <div class="navbox do-not-make-smaller mw-collapsible mw-collapsed" data-expandtext="pokaż" data-collapsetext="ukryj"><style data-mw-deduplicate="TemplateStyles:r74983602">.mw-parser-output .navbox{border:1px solid var(--border-color-base,#a2a9b1);margin:auto;text-align:center;padding:3px;margin-top:1em;clear:both}.mw-parser-output table.navbox:not(.pionowy){width:100%}.mw-parser-output .navbox+.navbox{border-top:0;margin-top:0}.mw-parser-output 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Użyj przycisku podglądu przed zapisaniem zmian."><a class="external text" href="https://pl.wikipedia.org/w/index.php?title=Szablon:Operatory_r%C3%B3%C5%BCniczkowe&action=edit">e</a></li></ul><div class="navbox-title caption"><a href="/wiki/Operator_r%C3%B3%C5%BCniczkowy" title="Operator różniczkowy">Operatory różniczkowe</a></div><div class="mw-collapsible-content"><table class="navbox-main-content inner-standard"><tbody><tr class="a1"><th class="navbox-group opis" scope="row">na <a href="/wiki/Pole_skalarne" title="Pole skalarne">polach skalarnych</a></th><td class="navbox-list spis hlist navbox-odd"> <ul><li><a class="mw-selflink selflink">nabla</a></li> <li><a href="/wiki/Operator_d%E2%80%99Alemberta" title="Operator d’Alemberta">dalambercjan</a></li> <li><a href="/wiki/Operator_Laplace%E2%80%99a" title="Operator Laplace’a">laplasjan</a></li></ul> </td></tr><tr class="a2"><th class="navbox-group opis" scope="row">na <a href="/wiki/Pole_wektorowe" title="Pole wektorowe">polach wektorowych</a></th><td class="navbox-list spis hlist navbox-even"> <ul><li><a href="/wiki/Dywergencja" title="Dywergencja">dywergencja</a></li> <li><a href="/wiki/Rotacja" title="Rotacja">rotacja</a></li></ul> </td></tr></tbody></table></div></div> <style data-mw-deduplicate="TemplateStyles:r74016753">.mw-parser-output #normdaten>div+div{margin-top:0.5em}.mw-parser-output #normdaten>div>div{background:var(--background-color-neutral,#eaecf0);padding:.2em .5em}.mw-parser-output #normdaten ul{margin:0;padding:0}.mw-parser-output #normdaten ul li:first-child{padding-left:.5em;border-left:1px solid var(--border-color-base,#a2a9b1)}</style> <div id="normdaten" class="catlinks"><div class="normdaten-typ-fehlt"><div><a href="/wiki/Encyklopedia_internetowa" title="Encyklopedia internetowa">Encyklopedie internetowe</a> (<span class="description">operator</span>):</div> <ul><li><a href="/wiki/Den_Store_Danske_Encyklop%C3%A6di" title="Den Store Danske Encyklopædi">DSDE</a>: <span class="uid"><a class="external text" href="https://wikidata-externalid-url.toolforge.org/?p=8313&url_prefix=https://lex.dk/&id=nabla">nabla</a></span></li></ul> </div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Źródło: „<a dir="ltr" href="https://pl.wikipedia.org/w/index.php?title=Operator_nabla&oldid=73472554">https://pl.wikipedia.org/w/index.php?title=Operator_nabla&oldid=73472554</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Specjalna:Kategorie" title="Specjalna:Kategorie">Kategoria</a>: <ul><li><a href="/wiki/Kategoria:Rachunek_wektorowy" title="Kategoria:Rachunek wektorowy">Rachunek wektorowy</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Tę stronę ostatnio edytowano 10 kwi 2024, 07:19.</li> <li id="footer-info-copyright">Tekst udostępniany na licencji <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pl">Creative Commons: uznanie autorstwa, na tych samych warunkach</a>, z możliwością obowiązywania dodatkowych ograniczeń. 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