Divergență - Wikipedia

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Disponibil în 50 limbi" > <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-50" 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">50 limbi</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/%D8%AA%D8%A8%D8%A7%D8%B9%D8%AF_(%D9%85%D8%AA%D8%AC%D9%87%D8%A7%D8%AA)" title="تباعد (متجهات) – arabă" lang="ar" hreflang="ar" data-title="تباعد (متجهات)" data-language-autonym="العربية" data-language-local-name="arabă" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D0%B3%D0%B5%D0%BD%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Дивергенция (математика) – bulgară" lang="bg" hreflang="bg" data-title="Дивергенция (математика)" data-language-autonym="Български" data-language-local-name="bulgară" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A1%E0%A6%BE%E0%A6%87%E0%A6%AD%E0%A6%BE%E0%A6%B0%E0%A6%9C%E0%A7%87%E0%A6%A8%E0%A7%8D%E0%A6%B8" title="ডাইভারজেন্স – bengaleză" lang="bn" hreflang="bn" data-title="ডাইভারজেন্স" data-language-autonym="বাংলা" data-language-local-name="bengaleză" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Divergencija" title="Divergencija – bosniacă" lang="bs" hreflang="bs" data-title="Divergencija" data-language-autonym="Bosanski" data-language-local-name="bosniacă" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Diverg%C3%A8ncia" title="Divergència – catalană" lang="ca" hreflang="ca" data-title="Divergència" data-language-autonym="Català" data-language-local-name="catalană" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Divergence_(oper%C3%A1tor)" title="Divergence (operátor) – cehă" lang="cs" hreflang="cs" data-title="Divergence (operátor)" data-language-autonym="Čeština" data-language-local-name="cehă" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D0%B3%D0%B5%D0%BD%D1%86%D0%B8" title="Дивергенци – ciuvașă" lang="cv" hreflang="cv" data-title="Дивергенци" data-language-autonym="Чӑвашла" data-language-local-name="ciuvașă" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Divergenz_eines_Vektorfeldes" title="Divergenz eines Vektorfeldes – germană" lang="de" hreflang="de" data-title="Divergenz eines Vektorfeldes" data-language-autonym="Deutsch" data-language-local-name="germană" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Divergence" title="Divergence – engleză" lang="en" hreflang="en" data-title="Divergence" data-language-autonym="English" data-language-local-name="engleză" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diver%C4%9Denco" title="Diverĝenco – esperanto" lang="eo" hreflang="eo" data-title="Diverĝenco" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Divergencia_(matem%C3%A1tica)" title="Divergencia (matemática) – spaniolă" lang="es" hreflang="es" data-title="Divergencia (matemática)" data-language-autonym="Español" data-language-local-name="spaniolă" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Divergents_(matemaatika)" title="Divergents (matemaatika) – estonă" lang="et" hreflang="et" data-title="Divergents (matemaatika)" data-language-autonym="Eesti" data-language-local-name="estonă" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Dibergentzia_(matematika)" title="Dibergentzia (matematika) – bască" lang="eu" hreflang="eu" data-title="Dibergentzia (matematika)" data-language-autonym="Euskara" data-language-local-name="bască" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%DB%8C%D9%88%D8%B1%DA%98%D8%A7%D9%86%D8%B3" title="دیورژانس – persană" lang="fa" hreflang="fa" data-title="دیورژانس" data-language-autonym="فارسی" data-language-local-name="persană" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Divergenssi" title="Divergenssi – finlandeză" lang="fi" hreflang="fi" data-title="Divergenssi" data-language-autonym="Suomi" data-language-local-name="finlandeză" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Divergence_(analyse_vectorielle)" title="Divergence (analyse vectorielle) – franceză" lang="fr" hreflang="fr" data-title="Divergence (analyse vectorielle)" data-language-autonym="Français" data-language-local-name="franceză" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%99%D7%91%D7%A8%D7%92%D7%A0%D7%A5" title="דיברגנץ – ebraică" lang="he" hreflang="he" data-title="דיברגנץ" data-language-autonym="עברית" data-language-local-name="ebraică" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%9A%E0%A4%B2%E0%A4%A8" title="विचलन – hindi" lang="hi" hreflang="hi" data-title="विचलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Divergencija_polja" title="Divergencija polja – croată" lang="hr" hreflang="hr" data-title="Divergencija polja" data-language-autonym="Hrvatski" data-language-local-name="croată" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Divergencia_(vektoranal%C3%ADzis)" title="Divergencia (vektoranalízis) – maghiară" lang="hu" hreflang="hu" data-title="Divergencia (vektoranalízis)" data-language-autonym="Magyar" data-language-local-name="maghiară" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A1%D6%80%D5%A1%D5%B4%D5%AB%D5%BF%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Տարամիտություն – armeană" lang="hy" hreflang="hy" data-title="Տարամիտություն" data-language-autonym="Հայերեն" data-language-local-name="armeană" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sundurleitni" title="Sundurleitni – islandeză" lang="is" hreflang="is" data-title="Sundurleitni" data-language-autonym="Íslenska" data-language-local-name="islandeză" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Divergenza" title="Divergenza – italiană" lang="it" hreflang="it" data-title="Divergenza" data-language-autonym="Italiano" data-language-local-name="italiană" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%99%BA%E6%95%A3_(%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E8%A7%A3%E6%9E%90)" title="発散 (ベクトル解析) – japoneză" lang="ja" hreflang="ja" data-title="発散 (ベクトル解析)" data-language-autonym="日本語" data-language-local-name="japoneză" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%98%E1%83%95%E1%83%94%E1%83%A0%E1%83%92%E1%83%94%E1%83%9C%E1%83%AA%E1%83%98%E1%83%90" title="დივერგენცია – georgiană" lang="ka" hreflang="ka" data-title="დივერგენცია" data-language-autonym="ქართული" data-language-local-name="georgiană" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D0%B3%D0%B5%D0%BD%D1%86%D0%B8%D1%8F" title="Дивергенция – kazahă" lang="kk" hreflang="kk" data-title="Дивергенция" data-language-autonym="Қазақша" data-language-local-name="kazahă" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%9C%EC%82%B0_(%EB%B2%A1%ED%84%B0)" title="발산 (벡터) – coreeană" lang="ko" hreflang="ko" data-title="발산 (벡터)" data-language-autonym="한국어" data-language-local-name="coreeană" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Divergenza" title="Divergenza – Lombard" lang="lmo" hreflang="lmo" data-title="Divergenza" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Divergencija" title="Divergencija – lituaniană" lang="lt" hreflang="lt" data-title="Divergencija" data-language-autonym="Lietuvių" data-language-local-name="lituaniană" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Divergentie_(vectorveld)" title="Divergentie (vectorveld) – neerlandeză" lang="nl" hreflang="nl" data-title="Divergentie (vectorveld)" data-language-autonym="Nederlands" data-language-local-name="neerlandeză" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Divergens" title="Divergens – norvegiană nynorsk" lang="nn" hreflang="nn" data-title="Divergens" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegiană nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Divergens" title="Divergens – norvegiană bokmål" lang="nb" hreflang="nb" data-title="Divergens" data-language-autonym="Norsk bokmål" data-language-local-name="norvegiană bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dywergencja" title="Dywergencja – poloneză" lang="pl" hreflang="pl" data-title="Dywergencja" data-language-autonym="Polski" data-language-local-name="poloneză" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Diverg%C3%AAncia" title="Divergência – portugheză" lang="pt" hreflang="pt" data-title="Divergência" data-language-autonym="Português" data-language-local-name="portugheză" 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%94%D0%B8%D0%B2%D0%B5%D1%80%D0%B3%D0%B5%D0%BD%D1%86%D0%B8%D1%8F" title="Дивергенция – rusă" lang="ru" hreflang="ru" data-title="Дивергенция" data-language-autonym="Русский" data-language-local-name="rusă" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Divergencija" title="Divergencija – sârbo-croată" lang="sh" hreflang="sh" data-title="Divergencija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="sârbo-croată" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Divergence" title="Divergence – Simple English" lang="en-simple" hreflang="en-simple" data-title="Divergence" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Divergencia_(vektorov%C3%A9_pole)" title="Divergencia (vektorové pole) – slovacă" lang="sk" hreflang="sk" data-title="Divergencia (vektorové pole)" data-language-autonym="Slovenčina" data-language-local-name="slovacă" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Divergenca" title="Divergenca – slovenă" lang="sl" hreflang="sl" data-title="Divergenca" data-language-autonym="Slovenščina" data-language-local-name="slovenă" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Divergjenca" title="Divergjenca – albaneză" lang="sq" hreflang="sq" data-title="Divergjenca" data-language-autonym="Shqip" data-language-local-name="albaneză" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D0%B3%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0" title="Дивергенција – sârbă" lang="sr" hreflang="sr" data-title="Дивергенција" data-language-autonym="Српски / srpski" data-language-local-name="sârbă" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Divergens_(vektoranalys)" title="Divergens (vektoranalys) – suedeză" lang="sv" hreflang="sv" data-title="Divergens (vektoranalys)" data-language-autonym="Svenska" data-language-local-name="suedeză" 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%B5%E0%AE%BF%E0%AE%B0%E0%AE%BF%E0%AE%A4%E0%AE%B2%E0%AF%8D_(%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AF%E0%AE%A9%E0%AF%8D_%E0%AE%A8%E0%AF%81%E0%AE%A3%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="விரிதல் (திசையன் நுண்கணிதம்) – tamilă" lang="ta" hreflang="ta" data-title="விரிதல் (திசையன் நுண்கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tamilă" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Diberhensiya_(matematika)" title="Diberhensiya (matematika) – tagalog" lang="tl" hreflang="tl" data-title="Diberhensiya (matematika)" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Diverjans" title="Diverjans – turcă" lang="tr" hreflang="tr" data-title="Diverjans" data-language-autonym="Türkçe" data-language-local-name="turcă" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D0%B3%D0%B5%D0%BD%D1%86%D0%B8%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Дивергенция (математика) – tătară" lang="tt" hreflang="tt" data-title="Дивергенция (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="tătară" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D0%B2%D0%B5%D1%80%D0%B3%D0%B5%D0%BD%D1%86%D1%96%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Дивергенція (математика) – ucraineană" lang="uk" hreflang="uk" data-title="Дивергенція (математика)" data-language-autonym="Українська" data-language-local-name="ucraineană" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/To%C3%A1n_t%E1%BB%AD_div" title="Toán tử div – vietnameză" lang="vi" hreflang="vi" data-title="Toán tử div" data-language-autonym="Tiếng Việt" data-language-local-name="vietnameză" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%A3%E5%BA%A6" title="散度 – chineză" lang="zh" hreflang="zh" data-title="散度" data-language-autonym="中文" data-language-local-name="chineză" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%95%A3%E5%BA%A6" title="散度 – cantoneză" lang="yue" hreflang="yue" data-title="散度" data-language-autonym="粵語" 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mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Tipărire/exportare </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=Special:Carte&bookcmd=book_creator&referer=Divergen%C8%9B%C4%83"><span>Creare carte</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Divergen%C8%9B%C4%83&action=show-download-screen"><span>Descărcare ca PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Divergen%C8%9B%C4%83&printable=yes" title="Versiunea de tipărit a acestei pagini [p]" accesskey="p"><span>Versiune de tipărit</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"> În alte proiecte </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Divergence" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q189000" title="Legătură către elementul asociat din depozitul de date [g]" accesskey="g"><span>Element Wikidata</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="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Aspect"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspect</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ascunde</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">De la Wikipedia, enciclopedia liberă</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="ro" dir="ltr"><p><b>Divergența</b> este o noțiune din <a href="/w/index.php?title=Teoria_c%C3%A2mpurilor&action=edit&redlink=1" class="new" title="Teoria câmpurilor — pagină inexistentă">teoria câmpurilor</a>, un operator vectorial care exprimă <a href="/w/index.php?title=Fluxul_unui_c%C3%A2mp_vectorial&action=edit&redlink=1" class="new" title="Fluxul unui câmp vectorial — pagină inexistentă">fluxul unui câmp vectorial</a> care trece (latina: divergere) printr-o suprafață închisă. Divergența într-un punct (P) al spațiului este fluxul câmpului care trece prin suprafața închisă care înconjoară un volum infinitezimal, care cuprinde punctul (P) al spațiului. Divergența fluxului care străbate suprafața închisă din interior spre exterior se consideră pozitivă, iar divergența fluxul care străbate suprafața închisă dinspre exterior spre interior se consideră negativă. Astfel divergența unui câmp care străbate suprafața închisă intrând pe o parte si ieșind pe altă parte este egală cu zero. Divergența este diferită de zero când într-un punct din interiorul suprafeței închise se află o sursă a câmpului, exprimat in [legea lui Gauss]] din electrostatică. Intuitiv se spune câmpul are în acel punct un 'izvor'. Tratarea matematică se face in <a href="/wiki/Analiza_matematic%C4%83" title="Analiza matematică">analiza matematică</a> cât si in <a href="/wiki/Analiza_vectorial%C4%83" class="mw-redirect" title="Analiza vectorială">analiza vectorială</a> abordând tematica cu mijloacele lor specifice. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definiția_matematică"><span id="Defini.C8.9Bia_matematic.C4.83"></span>Definiția matematică</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divergen%C8%9B%C4%83&veaction=edit&section=1" title="Modifică secțiunea: Definiția matematică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divergen%C8%9B%C4%83&action=edit&section=1" title="Edit section's source code: Definiția matematică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Divergența unui câmp vectorial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef40edff397a115ecdce7d3518001dfcc7f37d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.771ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}}"></span> este definită ca:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} \,{\vec {F}}=\nabla \cdot {\vec {F}}=\lim _{|\Delta V|\to 0}\left({\frac {1}{|\Delta V|}}\oint _{\partial (\Delta V)}{\vec {F}}\;\cdot {\vec {n}}\,dS\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>div</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thickmathspace" /> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>n</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>S</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {div} \,{\vec {F}}=\nabla \cdot {\vec {F}}=\lim _{|\Delta V|\to 0}\left({\frac {1}{|\Delta V|}}\oint _{\partial (\Delta V)}{\vec {F}}\;\cdot {\vec {n}}\,dS\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d53e3da36d52cdd48521c46a64b064ed48d5655" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.091ex; height:6.343ex;" alt="{\displaystyle \operatorname {div} \,{\vec {F}}=\nabla \cdot {\vec {F}}=\lim _{|\Delta V|\to 0}\left({\frac {1}{|\Delta V|}}\oint _{\partial (\Delta V)}{\vec {F}}\;\cdot {\vec {n}}\,dS\right)\,}"></span></dd></dl> <p>unde <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> </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/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \nabla }"></span> (nabla) este operatorul diferențial: </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 ={\frac {\partial }{\partial \ x}}\cdot {\vec {i}}+{\frac {\partial }{\partial \ y}}\cdot {\vec {j}}+{\frac {\partial }{\partial \ z}}\cdot {\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mtext> </mtext> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</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> <mtext> </mtext> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mtext> </mtext> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ={\frac {\partial }{\partial \ x}}\cdot {\vec {i}}+{\frac {\partial }{\partial \ y}}\cdot {\vec {j}}+{\frac {\partial }{\partial \ z}}\cdot {\vec {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8efc7f9d07da2631c638bda695dc78f0e686a419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.929ex; height:6.009ex;" alt="{\displaystyle \nabla ={\frac {\partial }{\partial \ x}}\cdot {\vec {i}}+{\frac {\partial }{\partial \ y}}\cdot {\vec {j}}+{\frac {\partial }{\partial \ z}}\cdot {\vec {k}}}"></span></dd></dl> <p>Astfel divergența este un scalar, fiind produsul scalar a doi vectori. </p> <div class="mw-heading mw-heading2"><h2 id="Expresia_matematică_[2]"><span id="Expresia_matematic.C4.83_.5B2.5D"></span>Expresia matematică <sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divergen%C8%9B%C4%83&veaction=edit&section=2" title="Modifică secțiunea: Expresia matematică [2]" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divergen%C8%9B%C4%83&action=edit&section=2" title="Edit section's source code: Expresia matematică [2]"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>În <a href="/wiki/Coordonate_carteziene" title="Coordonate carteziene">coordonate carteziene</a>, expresia matematicä a divergenței unui câmp vectorial, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}=F_{x}\cdot {\vec {i}}+F_{y}\cdot {\vec {j}}+F_{z}\cdot k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>j</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>⋅</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}=F_{x}\cdot {\vec {i}}+F_{y}\cdot {\vec {j}}+F_{z}\cdot k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1e8d7a3742dea749e24d8239811f8742e38084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.694ex; height:3.509ex;" alt="{\displaystyle {\vec {F}}=F_{x}\cdot {\vec {i}}+F_{y}\cdot {\vec {j}}+F_{z}\cdot k}"></span>, rezultă folosind operatorul <a href="/wiki/Nabla" title="Nabla">nabla</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 \operatorname {div} {\vec {F}}=\nabla \cdot {\vec {F}}\,=\,{\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>div</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {div} {\vec {F}}=\nabla \cdot {\vec {F}}\,=\,{\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c4a6d6edbcc96de8cf5d8401b3e2a7360d243e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.533ex; height:6.343ex;" alt="{\displaystyle \operatorname {div} {\vec {F}}=\nabla \cdot {\vec {F}}\,=\,{\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}}"></span></dd></dl> <p>În <a href="/wiki/Coordonate_polare" title="Coordonate polare">coordonate polare</a> expresia matematicä a divergenței unui câmp vectorial, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}=F_{\rho }\cdot {\vec {e}}_{\rho }+F_{\varphi }\cdot {\vec {e}}_{\varphi }+F_{z}\cdot {\vec {e}}_{z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}=F_{\rho }\cdot {\vec {e}}_{\rho }+F_{\varphi }\cdot {\vec {e}}_{\varphi }+F_{z}\cdot {\vec {e}}_{z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b72cd76f024452df1b3ac4ab3bb13b820e074f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.522ex; height:3.509ex;" alt="{\displaystyle {\vec {F}}=F_{\rho }\cdot {\vec {e}}_{\rho }+F_{\varphi }\cdot {\vec {e}}_{\varphi }+F_{z}\cdot {\vec {e}}_{z}}"></span> este: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} {\vec {F}}=\nabla \cdot {\vec {F}}\,={\frac {1}{\rho }}\cdot {\frac {\partial {(\rho \cdot F_{\rho })}}{\partial \rho }}+{\frac {1}{\rho }}\cdot {\frac {\partial {F_{\varphi }}}{\partial \varphi }}+{\frac {\partial {F_{z}}}{\partial z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>div</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {div} {\vec {F}}=\nabla \cdot {\vec {F}}\,={\frac {1}{\rho }}\cdot {\frac {\partial {(\rho \cdot F_{\rho })}}{\partial \rho }}+{\frac {1}{\rho }}\cdot {\frac {\partial {F_{\varphi }}}{\partial \varphi }}+{\frac {\partial {F_{z}}}{\partial z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9b5918c4793a5a4a500f166c338fc40bae8f77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.438ex; height:6.343ex;" alt="{\displaystyle \operatorname {div} {\vec {F}}=\nabla \cdot {\vec {F}}\,={\frac {1}{\rho }}\cdot {\frac {\partial {(\rho \cdot F_{\rho })}}{\partial \rho }}+{\frac {1}{\rho }}\cdot {\frac {\partial {F_{\varphi }}}{\partial \varphi }}+{\frac {\partial {F_{z}}}{\partial z}}}"></span></dd></dl> <p>În <a href="/wiki/Coordonate_sferice" title="Coordonate sferice">coordonate sferice</a> expresia matematică a divergenței unui câmp vectorial, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {F}}=F_{\rho }\cdot {\vec {e}}_{\rho }+F_{\eta }\cdot {\vec {e}}_{\eta }+F_{\varphi }\cdot {\vec {e}}_{\varphi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>e</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}=F_{\rho }\cdot {\vec {e}}_{\rho }+F_{\eta }\cdot {\vec {e}}_{\eta }+F_{\varphi }\cdot {\vec {e}}_{\varphi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23c0ac76b44bd83c26ed65c3a7a5b278c034086b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.637ex; height:3.509ex;" alt="{\displaystyle {\vec {F}}=F_{\rho }\cdot {\vec {e}}_{\rho }+F_{\eta }\cdot {\vec {e}}_{\eta }+F_{\varphi }\cdot {\vec {e}}_{\varphi }}"></span> este: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} {\vec {F}}=\nabla \cdot {\vec {F}}\,={\frac {1}{\rho ^{2}}}\cdot {\frac {\partial {(\rho ^{2}\cdot F_{\rho })}}{\partial \rho }}+{\frac {1}{\rho \cdot \sin(\eta )}}\cdot {\frac {\partial {(\sin(\eta )\cdot F_{\eta })}}{\partial \eta }}+{\frac {1}{\rho \cdot \sin(\eta )}}\cdot {\frac {\partial F_{\varphi }}{\partial \varphi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>div</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ρ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ρ</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>η</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>η</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ρ</mi> <mo>⋅</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>η</mi> <mo stretchy="false">)</mo> </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>φ</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>φ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {div} {\vec {F}}=\nabla \cdot {\vec {F}}\,={\frac {1}{\rho ^{2}}}\cdot {\frac {\partial {(\rho ^{2}\cdot F_{\rho })}}{\partial \rho }}+{\frac {1}{\rho \cdot \sin(\eta )}}\cdot {\frac {\partial {(\sin(\eta )\cdot F_{\eta })}}{\partial \eta }}+{\frac {1}{\rho \cdot \sin(\eta )}}\cdot {\frac {\partial F_{\varphi }}{\partial \varphi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4f6c098d38366cacbe3406baf2e4933745ad181" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:79.67ex; height:6.843ex;" alt="{\displaystyle \operatorname {div} {\vec {F}}=\nabla \cdot {\vec {F}}\,={\frac {1}{\rho ^{2}}}\cdot {\frac {\partial {(\rho ^{2}\cdot F_{\rho })}}{\partial \rho }}+{\frac {1}{\rho \cdot \sin(\eta )}}\cdot {\frac {\partial {(\sin(\eta )\cdot F_{\eta })}}{\partial \eta }}+{\frac {1}{\rho \cdot \sin(\eta )}}\cdot {\frac {\partial F_{\varphi }}{\partial \varphi }}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divergen%C8%9B%C4%83&veaction=edit&section=3" title="Modifică secțiunea: Note" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divergen%C8%9B%C4%83&action=edit&section=3" title="Edit section's source code: Note"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text"><cite class="citation web">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Divergence.html">„Divergență”</a>. <i>MathWorld--A Wolfram Web Resource</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld--A+Wolfram+Web+Resource&rft.atitle=Divergen%C8%9B%C4%83&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FDivergence.html&rfr_id=info%3Asid%2Fro.wikipedia.org%3ADivergen%C8%9B%C4%83" class="Z3988"><span style="display:none;"> </span></span><style data-mw-deduplicate="TemplateStyles:r16236537">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"„""”""«""»"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}</style></span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text"> Bronstein, Semendiajew, "Lexikon der Mathematik" </span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Vezi_și"><span id="Vezi_.C8.99i"></span>Vezi și</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Divergen%C8%9B%C4%83&veaction=edit&section=4" title="Modifică secțiunea: Vezi și" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Divergen%C8%9B%C4%83&action=edit&section=4" title="Edit section's source code: Vezi și"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Laplacian" title="Laplacian">Laplacian</a></li> <li><a href="/wiki/Ecua%C8%9Bie_cu_derivate_par%C8%9Biale" title="Ecuație cu derivate parțiale">Ecuație cu derivate parțiale</a></li></ul> <div class="noprint tright portal" style="border:solid #aaa 1px; 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Divergență - Wikipedia