Operator Stokesa – Wikipedia, wolna encyklopedia

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vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Wyprowadzenie_alternatywne"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Wyprowadzenie alternatywne</span> </div> </a> <ul id="toc-Wyprowadzenie_alternatywne-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">3</span> <span>Przypisy</span> </div> </a> <ul id="toc-Przypisy-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 Stokesa</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 16 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-16" 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">16 języków</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Materi%C3%A1lov%C3%A1_derivace" title="Materiálová derivace – czeski" lang="cs" hreflang="cs" data-title="Materiálová derivace" 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/Materielt_afledte" title="Materielt afledte – duński" lang="da" hreflang="da" data-title="Materielt afledte" 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/Substantielle_Ableitung" title="Substantielle Ableitung – niemiecki" lang="de" hreflang="de" data-title="Substantielle Ableitung" 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/Materiaalne_tuletis" title="Materiaalne tuletis – estoński" lang="et" hreflang="et" data-title="Materiaalne tuletis" data-language-autonym="Eesti" data-language-local-name="estoński" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Material_derivative" title="Material derivative – angielski" lang="en" hreflang="en" data-title="Material derivative" 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/Derivada_material" title="Derivada material – hiszpański" lang="es" hreflang="es" data-title="Derivada material" data-language-autonym="Español" data-language-local-name="hiszpański" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B4%D8%AA%D9%82_%D9%85%D8%A7%D8%AF%D9%87" title="مشتق ماده – perski" lang="fa" hreflang="fa" data-title="مشتق ماده" data-language-autonym="فارسی" data-language-local-name="perski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Derivata_materiale" title="Derivata materiale – włoski" lang="it" hreflang="it" data-title="Derivata materiale" data-language-autonym="Italiano" data-language-local-name="włoski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA_%D7%97%D7%95%D7%9E%D7%A8%D7%99%D7%AA" title="נגזרת חומרית – hebrajski" lang="he" hreflang="he" data-title="נגזרת חומרית" data-language-autonym="עברית" data-language-local-name="hebrajski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Materi%C3%ABle_afgeleide" title="Materiële afgeleide – niderlandzki" lang="nl" hreflang="nl" data-title="Materiële afgeleide" 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/%E7%89%A9%E8%B3%AA%E5%BE%AE%E5%88%86" title="物質微分 – japoński" lang="ja" hreflang="ja" data-title="物質微分" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Derivada_material" title="Derivada material – portugalski" lang="pt" hreflang="pt" data-title="Derivada material" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D0%B8%D0%B7%D0%B2%D0%BE%D0%B4%D0%BD%D0%B0%D1%8F_%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%B0" title="Производная Лагранжа – rosyjski" lang="ru" hreflang="ru" data-title="Производная Лагранжа" data-language-autonym="Русский" data-language-local-name="rosyjski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Materiaaliderivaatta" title="Materiaaliderivaatta – fiński" lang="fi" hreflang="fi" data-title="Materiaaliderivaatta" data-language-autonym="Suomi" data-language-local-name="fiński" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1o_h%C3%A0m_h%E1%BB%AFu_h%C3%ACnh" title="Đạo hàm hữu hình – wietnamski" lang="vi" hreflang="vi" data-title="Đạo hàm hữu hình" data-language-autonym="Tiếng Việt" data-language-local-name="wietnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%89%A9%E8%B3%AA%E5%B0%8E%E6%95%B8" title="物質導數 – chiński" lang="zh" hreflang="zh" data-title="物質導數" data-language-autonym="中文" data-language-local-name="chiński" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1801562#sitelinks-wikipedia" title="Edytuj linki pomiędzy wersjami językowymi" class="wbc-editpage">Edytuj linki</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Przestrzenie nazw"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Operator_Stokesa" title="Zobacz stronę treści [c]" accesskey="c"><span>Artykuł</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Dyskusja:Operator_Stokesa" rel="discussion" title="Dyskusja o zawartości tej strony [t]" accesskey="t"><span>Dyskusja</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Zmień wariant języka" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">polski</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Widok"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Operator_Stokesa"><span>Czytaj</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Operator_Stokesa&veaction=edit" title="Edytuj tę stronę [v]" accesskey="v"><span>Edytuj</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Operator_Stokesa&action=edit" title="Edycja kodu źródłowego strony [e]" accesskey="e"><span>Edytuj kod źródłowy</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Operator_Stokesa&action=history" title="Starsze wersje tej strony [h]" accesskey="h"><span>Wyświetl historię</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Narzędzia dla stron"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Narzędzia" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Narzędzia</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Narzędzia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">przypnij</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">ukryj</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Więcej opcji" > <div class="vector-menu-heading"> Działania </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Operator_Stokesa"><span>Czytaj</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Operator_Stokesa&veaction=edit" title="Edytuj tę stronę [v]" accesskey="v"><span>Edytuj</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Operator_Stokesa&action=edit" title="Edycja kodu źródłowego strony [e]" accesskey="e"><span>Edytuj kod źródłowy</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Operator_Stokesa&action=history"><span>Wyświetl historię</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Ogólne </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Specjalna:Linkuj%C4%85ce/Operator_Stokesa" title="Pokaż listę wszystkich stron linkujących do tej strony [j]" accesskey="j"><span>Linkujące</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Specjalna:Zmiany_w_linkowanych/Operator_Stokesa" rel="nofollow" title="Ostatnie zmiany w stronach, do których ta strona linkuje [k]" accesskey="k"><span>Zmiany w linkowanych</span></a></li><li id="t-upload" class="mw-list-item"><a href="//pl.wikipedia.org/wiki/Wikipedia:Prześlij_plik" title="Prześlij pliki [u]" accesskey="u"><span>Prześlij plik</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Specjalna:Strony_specjalne" title="Lista wszystkich stron specjalnych [q]" accesskey="q"><span>Strony specjalne</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Operator_Stokesa&oldid=56600914" title="Stały link do tej wersji tej strony"><span>Link do tej wersji</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Operator_Stokesa&action=info" title="Więcej informacji na temat tej strony"><span>Informacje o tej stronie</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Cytuj&page=Operator_Stokesa&id=56600914&wpFormIdentifier=titleform" title="Informacja o tym jak należy cytować tę stronę"><span>Cytowanie tego artykułu</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Skr%C3%B3%C4%87_adres_URL&url=https%3A%2F%2Fpl.wikipedia.org%2Fwiki%2FOperator_Stokesa"><span>Zobacz skrócony adres URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Kod_QR&url=https%3A%2F%2Fpl.wikipedia.org%2Fwiki%2FOperator_Stokesa"><span>Pobierz kod QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Drukuj lub eksportuj </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Specjalna:Ksi%C4%85%C5%BCka&bookcmd=book_creator&referer=Operator+Stokesa"><span>Utwórz książkę</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Specjalna:DownloadAsPdf&page=Operator_Stokesa&action=show-download-screen"><span>Pobierz jako PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Operator_Stokesa&printable=yes" title="Wersja do wydruku [p]" accesskey="p"><span>Wersja do druku</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> W innych projektach </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1801562" title="Link do powiązanego elementu w repozytorium danych [g]" accesskey="g"><span>Element Wikidanych</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Narzędzia dla stron"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Wygląd"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="pl" dir="ltr"><p><b>Operator Stokesa</b> (<i>operator pochodnej materialnej</i>) – <a href="/wiki/Operator_r%C3%B3%C5%BCniczkowy" title="Operator różniczkowy">operator różniczkowy</a> stosowany w <a href="/wiki/Mechanika" title="Mechanika">mechanice</a> do oznaczania <b>różniczkowania wędrownego</b> (inaczej <b>pochodnej substancjalnej</b> lub <b>pochodnej materialnej</b>). Określa tempo zmiany dowolnej własności związanej z elementarną objętością ciała, która może znajdować się w ruchu, w odróżnieniu od różniczkowania lokalnego – związanego z <a href="/wiki/Uk%C5%82ad_odniesienia" title="Układ odniesienia">układem odniesienia</a>, który zwykle uznaje się za nieruchomy<sup id="cite_ref-szantyr_1-0" class="reference"><a href="#cite_note-szantyr-1">[1]</a></sup>. </p><p>Operator używany w <a href="/wiki/Mechanika_p%C5%82yn%C3%B3w" title="Mechanika płynów">mechanice płynów</a>. </p><p>Operator Stokesa zwykle oznaczany jest przez: </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 {D}{Dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5e970b27e8376cbe77ebbf728a6ff71bf46a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.6ex; height:5.176ex;" alt="{\displaystyle {\frac {D}{Dt}}}"></span> lub w sktócie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{t}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ee48f12691290ea520005632a5bf114b6816f3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.397ex; height:2.509ex;" alt="{\displaystyle D_{t}.}"></span></dd></dl> <p>W <a href="/wiki/Analiza_w%C4%99drowna_i_lokalna" title="Analiza wędrowna i lokalna">analizie wędrownej</a> równoważny jest symbolowi: </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 }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/521e1ca46720dbd58a13567163e158e51a6e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.994ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}}"></span></dd></dl> <p>różniczkowania cząstkowego względem czasu <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 t.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e6cc375ac6123d2342be53eba87b92fbbacf07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.486ex; height:2.009ex;" alt="{\displaystyle t.}"></span> </p><p>Natomiast przy użyciu <a href="/wiki/Analiza_w%C4%99drowna_i_lokalna" title="Analiza wędrowna i lokalna">analizy lokalnej</a> symbol równoważny jest operatorowi: </p> <table style="border-collapse: collapse; margin: 1em 1em 1em 1em;"> <tbody><tr style="border: 1px solid #CCCCCC;"> <td style="font-size:0.8em; margin-right:1em;"><a href="/w/index.php?title=Notacja_tensorowa&action=edit&redlink=1" class="new" title="Notacja tensorowa (strona nie istnieje)">Zapis klasyczny</a> </td> <td style="padding: 0.5em;"> <p><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 {D}{Dt}}={\frac {\partial }{\partial t}}+v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>v</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>v</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>v</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb5b7b148fe215276a29d870dd89685bc854737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.856ex; height:6.009ex;" alt="{\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+v_{x}{\frac {\partial }{\partial x}}+v_{y}{\frac {\partial }{\partial y}}+v_{z}{\frac {\partial }{\partial z}}}"></span> </p> </td></tr> <tr style="border: 1px solid #CCCCCC;"> <td style="font-size:0.8em; margin-right:1em;"><a href="/w/index.php?title=Notacja_tensorowa&action=edit&redlink=1" class="new" title="Notacja tensorowa (strona nie istnieje)">Zapis indeksowy</a> </td> <td style="padding: 0.5em;"> <p><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 {D}{Dt}}={\frac {\partial }{\partial t}}+v_{i}\nabla ^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+v_{i}\nabla ^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae466be5084fa664c1dcf04d32fdd068ff5fa5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.196ex; height:5.509ex;" alt="{\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+v_{i}\nabla ^{i}}"></span> </p> </td></tr> <tr style="border: 1px solid #CCCCCC;"> <td style="font-size:0.8em; margin-right:1em;"><a href="/w/index.php?title=Notacja_tensorowa&action=edit&redlink=1" class="new" title="Notacja tensorowa (strona nie istnieje)">Zapis absolutny</a> </td> <td style="padding: 0.5em;"> <p><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 {D}{Dt}}={\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d6e3de72ed3d3ccdd557e538dac3c2e479c11a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.323ex; height:5.509ex;" alt="{\displaystyle {\frac {D}{Dt}}={\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}}"></span> </p> </td></tr></tbody></table> <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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> – <a href="/wiki/Pr%C4%99dko%C5%9B%C4%87" title="Prędkość">prędkość</a> elementu ciała, z którym jest stale związana różniczkowana wielkość. </p><p>Pierwszy składnik po prawej stronie równania nosi nazwę <i>pochodnej lokalnej</i>, drugi (pozostałe w przypadku zapisu klasycznego) <i>pochodnej konwekcyjnej</i> (unoszenia). Pochodna lokalna określa szybkość zmiany wielkości w danym punkcie wynikającą ze zmiany pola w czasie. Pochodna unoszenia określa szybkość zmiany na skutek przemieszczania się płynu<sup id="cite_ref-szantyr_1-1" class="reference"><a href="#cite_note-szantyr-1">[1]</a></sup>. </p><p>Zapisując jawnie różniczkowaną własność jako <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> która w ogólności może być dowolnym polem <a href="/wiki/Tensor" title="Tensor">tensorowym</a>, można wyrazić operator Stokesa przez: </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 {D}{Dt}}\phi =({\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }})\phi ={\frac {\partial \phi }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}\phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}\phi =({\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }})\phi ={\frac {\partial \phi }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}\phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96678ee3ad61c5b697654d11374bbe6bf8c2defd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.204ex; height:5.676ex;" alt="{\displaystyle {\frac {D}{Dt}}\phi =({\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }})\phi ={\frac {\partial \phi }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}\phi .}"></span></dd></dl> <p>Jeżeli funkcja różniczkowana jest prędkością, to pochodna jest przyspieszeniem płynu<sup id="cite_ref-szantyr_1-2" class="reference"><a href="#cite_note-szantyr-1">[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 \phi =v,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mi>v</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi =v,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24fd1ac2c41c945b26da765b550d4d0f1cca17dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.258ex; height:2.509ex;" alt="{\displaystyle \phi =v,}"></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 a={\frac {D}{Dt}}v=({\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }})v={\frac {\partial v}{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>v</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>v</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a={\frac {D}{Dt}}v=({\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }})v={\frac {\partial v}{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}v.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3bf3247385d1c8f24c7573354cf3511a0ee2c18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.501ex; height:5.509ex;" alt="{\displaystyle a={\frac {D}{Dt}}v=({\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }})v={\frac {\partial v}{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}v.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Wyprowadzenie_w_analizie_lokalnej">Wyprowadzenie w analizie lokalnej</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_Stokesa&veaction=edit&section=1" title="Edytuj sekcję: Wyprowadzenie w analizie lokalnej" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_Stokesa&action=edit&section=1" title="Edytuj kod źródłowy sekcji: Wyprowadzenie w analizie lokalnej"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>W układzie współrzędnych Eulera punkt o współrzędnej <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db2dc6ced9cc3bc7e8b9f2707cbec033f6d3759c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {x}}}"></span> w chwili <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 t,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea3ad87830a1055c7b85c04cf940cfd3b847ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.486ex; height:2.343ex;" alt="{\displaystyle t,}"></span> znajdzie się w chwili <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 t+\Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t+\Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e896baa706e07d8100439dc56c5b27771e354e4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.456ex; height:2.343ex;" alt="{\displaystyle t+\Delta t}"></span> w punkcie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {x}}+\Delta {\vec {x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {x}}+\Delta {\vec {x}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b06167b25985a67439f973be8276bfb98a842d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.082ex; height:2.509ex;" alt="{\displaystyle {\vec {x}}+\Delta {\vec {x}}.}"></span> Z definicji pochodnej: </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 {D}{Dt}}\phi =\lim _{\Delta t\to 0}{\frac {\phi (t+\Delta t,{\vec {x}}+\Delta {\vec {x}})-\phi (t,{\vec {x}})}{\Delta t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}\phi =\lim _{\Delta t\to 0}{\frac {\phi (t+\Delta t,{\vec {x}}+\Delta {\vec {x}})-\phi (t,{\vec {x}})}{\Delta t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5cfcb638e628a767a825fddb6d5b686740e9ade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:41.74ex; height:6.009ex;" alt="{\displaystyle {\frac {D}{Dt}}\phi =\lim _{\Delta t\to 0}{\frac {\phi (t+\Delta t,{\vec {x}}+\Delta {\vec {x}})-\phi (t,{\vec {x}})}{\Delta t}}.}"></span></dd></dl> <p>Oznaczając: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}'={\frac {\Delta {\vec {x}}}{\Delta t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}'={\frac {\Delta {\vec {x}}}{\Delta t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d475ad2fd66ef33102026963683615fcd9a3dca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.707ex; height:5.509ex;" alt="{\displaystyle {\vec {v}}'={\frac {\Delta {\vec {x}}}{\Delta t}},}"></span></dd></dl> <p>można zauważyć, że: </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 \lim _{\Delta t\to 0}{\vec {v}}'={\vec {v}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{\Delta t\to 0}{\vec {v}}'={\vec {v}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/725a0ab0e8d15f3fd32965b3f7b5cf50224ea323" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.596ex; height:4.509ex;" alt="{\displaystyle \lim _{\Delta t\to 0}{\vec {v}}'={\vec {v}}.}"></span></dd></dl> <p>Rozwijając różniczkowaną funkcję wokół punktu <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 (t,x,y,z),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</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 (t,x,y,z),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d600e2b6cbe69e515f5179a2e5e718b6aede2652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.971ex; height:2.843ex;" alt="{\displaystyle (t,x,y,z),}"></span> otrzymuje się: </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}\phi (t+\Delta t,{\vec {x}}+\Delta {\vec {x}})&=\phi (t,{\vec {x}})+{\frac {\partial \phi }{\partial {\vec {x}}}}\cdot \Delta {\vec {x}}+{\frac {\partial \phi }{\partial t}}\Delta t+{\mathcal {O}}(\Delta {\vec {x}}\,\Delta {\vec {x}})+{\mathcal {O}}(\Delta t^{2})\\&=\phi (t,{\vec {x}})+\left({\frac {\partial \phi }{\partial {\vec {x}}}}\cdot {\vec {v}}'+{\frac {\partial \phi }{\partial t}}\right)\Delta t+{\mathcal {O}}(\Delta t^{2}).\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> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\phi (t+\Delta t,{\vec {x}}+\Delta {\vec {x}})&=\phi (t,{\vec {x}})+{\frac {\partial \phi }{\partial {\vec {x}}}}\cdot \Delta {\vec {x}}+{\frac {\partial \phi }{\partial t}}\Delta t+{\mathcal {O}}(\Delta {\vec {x}}\,\Delta {\vec {x}})+{\mathcal {O}}(\Delta t^{2})\\&=\phi (t,{\vec {x}})+\left({\frac {\partial \phi }{\partial {\vec {x}}}}\cdot {\vec {v}}'+{\frac {\partial \phi }{\partial t}}\right)\Delta t+{\mathcal {O}}(\Delta t^{2}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b2d6088276a9a817f4e3ff7e07b13674434e2e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:72.595ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\phi (t+\Delta t,{\vec {x}}+\Delta {\vec {x}})&=\phi (t,{\vec {x}})+{\frac {\partial \phi }{\partial {\vec {x}}}}\cdot \Delta {\vec {x}}+{\frac {\partial \phi }{\partial t}}\Delta t+{\mathcal {O}}(\Delta {\vec {x}}\,\Delta {\vec {x}})+{\mathcal {O}}(\Delta t^{2})\\&=\phi (t,{\vec {x}})+\left({\frac {\partial \phi }{\partial {\vec {x}}}}\cdot {\vec {v}}'+{\frac {\partial \phi }{\partial t}}\right)\Delta t+{\mathcal {O}}(\Delta t^{2}).\end{aligned}}}"></span></dd></dl> <p>Stąd: </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 {D}{Dt}}\phi =\lim _{\Delta t\to 0}\left({\frac {\partial \phi }{\partial {\vec {x}}}}\cdot {\vec {v}}'+{\frac {\partial \phi }{\partial t}}\right)=\lim _{\Delta t\to 0}\left({\vec {v}}'\cdot {\vec {\nabla }}+{\frac {\partial }{\partial t}}\right)\phi =\left({\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}\right)\phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>ϕ</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <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"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>′</mo> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mi>ϕ</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="normal">∇</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mrow> <mo>)</mo> </mrow> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}\phi =\lim _{\Delta t\to 0}\left({\frac {\partial \phi }{\partial {\vec {x}}}}\cdot {\vec {v}}'+{\frac {\partial \phi }{\partial t}}\right)=\lim _{\Delta t\to 0}\left({\vec {v}}'\cdot {\vec {\nabla }}+{\frac {\partial }{\partial t}}\right)\phi =\left({\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}\right)\phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63857b9df23f0f809a8108b140c01cf925d9717e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:73.758ex; height:6.176ex;" alt="{\displaystyle {\frac {D}{Dt}}\phi =\lim _{\Delta t\to 0}\left({\frac {\partial \phi }{\partial {\vec {x}}}}\cdot {\vec {v}}'+{\frac {\partial \phi }{\partial t}}\right)=\lim _{\Delta t\to 0}\left({\vec {v}}'\cdot {\vec {\nabla }}+{\frac {\partial }{\partial t}}\right)\phi =\left({\frac {\partial }{\partial t}}+{\vec {v}}\cdot {\vec {\nabla }}\right)\phi .}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Wyprowadzenie_alternatywne">Wyprowadzenie alternatywne</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Operator_Stokesa&veaction=edit&section=2" title="Edytuj sekcję: Wyprowadzenie alternatywne" class="mw-editsection-visualeditor"><span>edytuj</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Operator_Stokesa&action=edit&section=2" title="Edytuj kod źródłowy sekcji: Wyprowadzenie alternatywne"><span>edytuj kod</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/R%C3%B3%C5%BCniczka_zupe%C5%82na" title="Różniczka zupełna">Różniczka zupełna</a> funkcji <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 \phi (t,x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</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 \phi (t,x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e480cb12f6e0b9a743ae2fb8a218fdea07720a1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.709ex; height:2.843ex;" alt="{\displaystyle \phi (t,x,y,z)}"></span> ma postać: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\phi ={\frac {\partial \phi }{\partial t}}dt+{\frac {\partial \phi }{\partial x}}dx+{\frac {\partial \phi }{\partial y}}dy+{\frac {\partial \phi }{\partial z}}dz,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ϕ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> <mo>+</mo> <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> <mi>d</mi> <mi>x</mi> <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> <mi>d</mi> <mi>y</mi> <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> <mi>d</mi> <mi>z</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\phi ={\frac {\partial \phi }{\partial t}}dt+{\frac {\partial \phi }{\partial x}}dx+{\frac {\partial \phi }{\partial y}}dy+{\frac {\partial \phi }{\partial z}}dz,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a92b20157d40eb161639c1eefcb7c72049c107" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.303ex; height:6.176ex;" alt="{\displaystyle d\phi ={\frac {\partial \phi }{\partial t}}dt+{\frac {\partial \phi }{\partial x}}dx+{\frac {\partial \phi }{\partial y}}dy+{\frac {\partial \phi }{\partial z}}dz,}"></span></dd></dl> <p>dzieląc 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 dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993271528b71660c0c4282c0d0b0f4cbf88e5e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.702ex; height:2.509ex;" alt="{\displaystyle dt,}"></span> możemy zapisać: </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 {d\phi }{dt}}={\frac {\partial \phi }{\partial t}}+{\frac {\partial \phi }{\partial x}}{\frac {dx}{dt}}+{\frac {\partial \phi }{\partial y}}{\frac {dy}{dt}}+{\frac {\partial \phi }{\partial z}}{\frac {dz}{dt}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</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>t</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>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</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 class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\phi }{dt}}={\frac {\partial \phi }{\partial t}}+{\frac {\partial \phi }{\partial x}}{\frac {dx}{dt}}+{\frac {\partial \phi }{\partial y}}{\frac {dy}{dt}}+{\frac {\partial \phi }{\partial z}}{\frac {dz}{dt}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cf4cc767d8717761983415c55d26ad7ac028e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.592ex; height:6.176ex;" alt="{\displaystyle {\frac {d\phi }{dt}}={\frac {\partial \phi }{\partial t}}+{\frac {\partial \phi }{\partial x}}{\frac {dx}{dt}}+{\frac {\partial \phi }{\partial y}}{\frac {dy}{dt}}+{\frac {\partial \phi }{\partial z}}{\frac {dz}{dt}},}"></span></dd></dl> <p>uwzględniając, że <a href="/wiki/Pr%C4%99dko%C5%9B%C4%87" title="Prędkość">prędkość</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}=[u,v,w]=\left[{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>u</mi> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>z</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}=[u,v,w]=\left[{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb81fedb364893c4622a56f586f4304f3cd53f4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.107ex; height:6.176ex;" alt="{\displaystyle {\vec {v}}=[u,v,w]=\left[{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right]}"></span> otrzymujemy: </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 {d\phi }{dt}}={\frac {\partial \phi }{\partial t}}+u{\frac {\partial \phi }{\partial x}}+v{\frac {\partial \phi }{\partial y}}+w{\frac {\partial \phi }{\partial z}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</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>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>u</mi> <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> <mi>v</mi> <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> <mi>w</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\phi }{dt}}={\frac {\partial \phi }{\partial t}}+u{\frac {\partial \phi }{\partial x}}+v{\frac {\partial \phi }{\partial y}}+w{\frac {\partial \phi }{\partial z}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a688d6d4593e39883b14defcd961673d93a14a6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.984ex; height:6.176ex;" alt="{\displaystyle {\frac {d\phi }{dt}}={\frac {\partial \phi }{\partial t}}+u{\frac {\partial \phi }{\partial x}}+v{\frac {\partial \phi }{\partial y}}+w{\frac {\partial \phi }{\partial z}}.}"></span></dd></dl> <p>Co można zapisać, używając operatorów: </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 {d\phi }{dt}}={\frac {\partial \phi }{\partial t}}+{\vec {v}}\cdot \nabla \phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>ϕ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</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>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\phi }{dt}}={\frac {\partial \phi }{\partial t}}+{\vec {v}}\cdot \nabla \phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf6e8402996753b184b73299357c6441d5e6d96e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.739ex; height:5.676ex;" alt="{\displaystyle {\frac {d\phi }{dt}}={\frac {\partial \phi }{\partial t}}+{\vec {v}}\cdot \nabla \phi .}"></span></dd></dl> <p>W literaturze oznaczenia <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 {d}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e262410b19a3eeab08b01224e89815c110c8ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.892ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}}"></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 {\frac {D}{Dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>D</mi> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D}{Dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5e970b27e8376cbe77ebbf728a6ff71bf46a84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.6ex; height:5.176ex;" alt="{\displaystyle {\frac {D}{Dt}}}"></span> używane są zamiennie. </p> <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_Stokesa&veaction=edit&section=3" 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_Stokesa&action=edit&section=3" 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-szantyr-1"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-szantyr_1-0">a</a></sup> <sup><a href="#cite_ref-szantyr_1-1">b</a></sup> <sup><a href="#cite_ref-szantyr_1-2">c</a></sup></span> <span class="reference-text"><cite class="citation web">J. Szantyr: <a rel="nofollow" class="external text" href="http://www.pg.gda.pl/~krzyte/students/MPW6.pdf">Kinetyka płynów</a>. [dostęp 2018-11-16].</cite></span> </li> </ol></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_Stokesa&oldid=56600914">https://pl.wikipedia.org/w/index.php?title=Operator_Stokesa&oldid=56600914</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">Kategorie</a>: <ul><li><a href="/wiki/Kategoria:Rachunek_r%C3%B3%C5%BCniczkowy_i_ca%C5%82kowy" title="Kategoria:Rachunek różniczkowy i całkowy">Rachunek różniczkowy i całkowy</a></li><li><a href="/wiki/Kategoria:Mechanika_p%C5%82yn%C3%B3w" title="Kategoria:Mechanika płynów">Mechanika płynów</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 4 maj 2019, 23:07.</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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