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id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EA%B8%B0%EC%9A%B8%EA%B8%B0+%28%EB%B2%A1%ED%84%B0%29" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%EA%B8%B0%EC%9A%B8%EA%B8%B0+%28%EB%B2%A1%ED%84%B0%29" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%EA%B8%B0%EC%9A%B8%EA%B8%B0+%28%EB%B2%A1%ED%84%B0%29" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="사이트"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="목차" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">처음 위치</div> </a> </li> <li id="toc-기울기의_의미" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#기울기의_의미"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>기울기의 의미</span> </div> </a> <ul id="toc-기울기의_의미-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-기울기의_수학적_정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#기울기의_수학적_정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>기울기의 수학적 정의</span> </div> </a> <ul id="toc-기울기의_수학적_정의-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-성질" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#성질"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>성질</span> </div> </a> <ul id="toc-성질-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-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="목차" 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="목차 토글" > <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">목차 토글</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">기울기 (벡터)</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="다른 언어로 문서를 방문합니다. 63개 언어로 읽을 수 있습니다" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-63" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">63개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%89%80%E1%89%A0%E1%89%B5" title="አቀበት – 암하라어" lang="am" hreflang="am" data-title="አቀበት" data-language-autonym="አማርኛ" data-language-local-name="암하라어" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AF%D8%B1%D8%AC_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="تدرج (رياضيات) – 아랍어" lang="ar" hreflang="ar" data-title="تدرج (رياضيات)" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Qradiyent" title="Qradiyent – 아제르바이잔어" lang="az" hreflang="az" data-title="Qradiyent" data-language-autonym="Azərbaycanca" data-language-local-name="아제르바이잔어" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D1%8B%D0%B5%D0%BD%D1%82" title="Градыент – 벨라루스어" lang="be" hreflang="be" data-title="Градыент" data-language-autonym="Беларуская" data-language-local-name="벨라루스어" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D1%8B%D0%B5%D0%BD%D1%82" title="Градыент – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Градыент" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – 불가리아어" lang="bg" hreflang="bg" data-title="Градиент" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8D%E0%A6%B0%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%A1%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A7%87%E0%A6%A8%E0%A7%8D%E0%A6%9F" title="গ্র্যাডিয়েন্ট – 벵골어" lang="bn" hreflang="bn" data-title="গ্র্যাডিয়েন্ট" data-language-autonym="বাংলা" data-language-local-name="벵골어" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Gradijent" title="Gradijent – 보스니아어" lang="bs" hreflang="bs" data-title="Gradijent" data-language-autonym="Bosanski" data-language-local-name="보스니아어" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Gradient_(matem%C3%A0tiques)" title="Gradient (matemàtiques) – 카탈로니아어" lang="ca" hreflang="ca" data-title="Gradient (matemàtiques)" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Gradient_(matematika)" title="Gradient (matematika) – 체코어" lang="cs" hreflang="cs" data-title="Gradient (matematika)" data-language-autonym="Čeština" data-language-local-name="체코어" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – 추바시어" lang="cv" hreflang="cv" data-title="Градиент" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Gradient" title="Gradient – 덴마크어" lang="da" hreflang="da" data-title="Gradient" data-language-autonym="Dansk" data-language-local-name="덴마크어" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gradient_(Mathematik)" title="Gradient (Mathematik) – 독일어" lang="de" hreflang="de" data-title="Gradient (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="독일어" 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/Gradient" title="Gradient – 영어" lang="en" hreflang="en" data-title="Gradient" data-language-autonym="English" data-language-local-name="영어" 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/Gradiento_(matematiko)" title="Gradiento (matematiko) – 에스페란토어" lang="eo" hreflang="eo" data-title="Gradiento (matematiko)" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Gradiente" title="Gradiente – 스페인어" lang="es" hreflang="es" data-title="Gradiente" data-language-autonym="Español" data-language-local-name="스페인어" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Gradient" title="Gradient – 에스토니아어" lang="et" hreflang="et" data-title="Gradient" data-language-autonym="Eesti" data-language-local-name="에스토니아어" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gradiente" title="Gradiente – 바스크어" lang="eu" hreflang="eu" data-title="Gradiente" data-language-autonym="Euskara" data-language-local-name="바스크어" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D8%A7%D8%AF%DB%8C%D8%A7%D9%86" title="گرادیان – 페르시아어" lang="fa" hreflang="fa" data-title="گرادیان" data-language-autonym="فارسی" data-language-local-name="페르시아어" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Gradientti" title="Gradientti – 핀란드어" lang="fi" hreflang="fi" data-title="Gradientti" data-language-autonym="Suomi" data-language-local-name="핀란드어" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Gradient" title="Gradient – 프랑스어" lang="fr" hreflang="fr" data-title="Gradient" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Gr%C3%A1d%C3%A1n" title="Grádán – 아일랜드어" lang="ga" hreflang="ga" data-title="Grádán" data-language-autonym="Gaeilge" data-language-local-name="아일랜드어" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Gradiente" title="Gradiente – 갈리시아어" lang="gl" hreflang="gl" data-title="Gradiente" data-language-autonym="Galego" data-language-local-name="갈리시아어" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%A8%D7%93%D7%99%D7%90%D7%A0%D7%98" title="גרדיאנט – 히브리어" lang="he" hreflang="he" data-title="גרדיאנט" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gradiens" title="Gradiens – 헝가리어" lang="hu" hreflang="hu" data-title="Gradiens" data-language-autonym="Magyar" data-language-local-name="헝가리어" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B3%D6%80%D5%A1%D5%A4%D5%AB%D5%A5%D5%B6%D5%BF" title="Գրադիենտ – 아르메니아어" lang="hy" hreflang="hy" data-title="Գրադիենտ" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gradien" title="Gradien – 인도네시아어" lang="id" hreflang="id" data-title="Gradien" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Gradiento" title="Gradiento – 이도어" lang="io" hreflang="io" data-title="Gradiento" data-language-autonym="Ido" data-language-local-name="이도어" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Stigull" title="Stigull – 아이슬란드어" lang="is" hreflang="is" data-title="Stigull" data-language-autonym="Íslenska" data-language-local-name="아이슬란드어" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gradiente" title="Gradiente – 이탈리아어" lang="it" hreflang="it" data-title="Gradiente" data-language-autonym="Italiano" data-language-local-name="이탈리아어" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8B%BE%E9%85%8D_(%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E8%A7%A3%E6%9E%90)" title="勾配 (ベクトル解析) – 일본어" lang="ja" hreflang="ja" data-title="勾配 (ベクトル解析)" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%92%E1%83%A0%E1%83%90%E1%83%93%E1%83%98%E1%83%94%E1%83%9C%E1%83%A2%E1%83%98" title="გრადიენტი – 조지아어" lang="ka" hreflang="ka" data-title="გრადიენტი" data-language-autonym="ქართული" data-language-local-name="조지아어" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – 카자흐어" lang="kk" hreflang="kk" data-title="Градиент" data-language-autonym="Қазақша" data-language-local-name="카자흐어" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – 키르기스어" lang="ky" hreflang="ky" data-title="Градиент" data-language-autonym="Кыргызча" data-language-local-name="키르기스어" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Gradientas" title="Gradientas – 리투아니아어" lang="lt" hreflang="lt" data-title="Gradientas" data-language-autonym="Lietuvių" data-language-local-name="리투아니아어" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Gradients" title="Gradients – 라트비아어" lang="lv" hreflang="lv" data-title="Gradients" data-language-autonym="Latviešu" data-language-local-name="라트비아어" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B5%87%E0%B4%A1%E0%B4%BF%E0%B4%AF%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%8D" title="ഗ്രേഡിയന്റ് – 말라얄람어" lang="ml" hreflang="ml" data-title="ഗ്രേഡിയന്റ്" data-language-autonym="മലയാളം" data-language-local-name="말라얄람어" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Gradi%C3%ABnt_(wiskunde)" title="Gradiënt (wiskunde) – 네덜란드어" lang="nl" hreflang="nl" data-title="Gradiënt (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Gradient" title="Gradient – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Gradient" data-language-autonym="Norsk nynorsk" data-language-local-name="노르웨이어(니노르스크)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gradient" title="Gradient – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Gradient" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%8D%E0%A8%B0%E0%A9%87%E0%A8%A1%E0%A9%80%E0%A8%85%E0%A9%B0%E0%A8%9F" title="ਗ੍ਰੇਡੀਅੰਟ – 펀잡어" lang="pa" hreflang="pa" data-title="ਗ੍ਰੇਡੀਅੰਟ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="펀잡어" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Gradient_(matematyka)" title="Gradient (matematyka) – 폴란드어" lang="pl" hreflang="pl" data-title="Gradient (matematyka)" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Gradiente" title="Gradiente – 포르투갈어" lang="pt" hreflang="pt" data-title="Gradiente" data-language-autonym="Português" data-language-local-name="포르투갈어" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Gradient" title="Gradient – 루마니아어" lang="ro" hreflang="ro" data-title="Gradient" data-language-autonym="Română" data-language-local-name="루마니아어" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – 러시아어" lang="ru" hreflang="ru" data-title="Градиент" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Gradijent" title="Gradijent – 세르비아-크로아티아어" lang="sh" hreflang="sh" data-title="Gradijent" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="세르비아-크로아티아어" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Gradient" title="Gradient – Simple English" lang="en-simple" hreflang="en-simple" data-title="Gradient" 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/Gradient_(matematika)" title="Gradient (matematika) – 슬로바키아어" lang="sk" hreflang="sk" data-title="Gradient (matematika)" data-language-autonym="Slovenčina" data-language-local-name="슬로바키아어" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Gradient" title="Gradient – 슬로베니아어" lang="sl" hreflang="sl" data-title="Gradient" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Gradient" title="Gradient – 알바니아어" lang="sq" hreflang="sq" data-title="Gradient" data-language-autonym="Shqip" data-language-local-name="알바니아어" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Gradijent" title="Gradijent – 세르비아어" lang="sr" hreflang="sr" data-title="Gradijent" data-language-autonym="Српски / srpski" data-language-local-name="세르비아어" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gradient_(matematik)" title="Gradient (matematik) – 스웨덴어" lang="sv" hreflang="sv" data-title="Gradient (matematik)" data-language-autonym="Svenska" data-language-local-name="스웨덴어" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82" title="Градиент – 타지크어" lang="tg" hreflang="tg" data-title="Градиент" data-language-autonym="Тоҷикӣ" data-language-local-name="타지크어" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%81%E0%B8%A3%E0%B9%80%E0%B8%94%E0%B8%B5%E0%B8%A2%E0%B8%99%E0%B8%95%E0%B9%8C" title="เกรเดียนต์ – 태국어" lang="th" hreflang="th" data-title="เกรเดียนต์" data-language-autonym="ไทย" data-language-local-name="태국어" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Gradient_(kalkulong_bektor)" title="Gradient (kalkulong bektor) – 타갈로그어" lang="tl" hreflang="tl" data-title="Gradient (kalkulong bektor)" data-language-autonym="Tagalog" data-language-local-name="타갈로그어" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Gradyan" title="Gradyan – 터키어" lang="tr" hreflang="tr" data-title="Gradyan" data-language-autonym="Türkçe" data-language-local-name="터키어" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D0%B8%D0%B5%D0%BD%D1%82_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Градиент (математика) – 타타르어" lang="tt" hreflang="tt" data-title="Градиент (математика)" data-language-autonym="Татарча / tatarça" data-language-local-name="타타르어" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%80%D0%B0%D0%B4%D1%96%D1%94%D0%BD%D1%82" title="Градієнт – 우크라이나어" lang="uk" hreflang="uk" data-title="Градієнт" data-language-autonym="Українська" data-language-local-name="우크라이나어" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Gradiyent" title="Gradiyent – 우즈베크어" lang="uz" hreflang="uz" data-title="Gradiyent" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="우즈베크어" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Gradient" title="Gradient – 베트남어" lang="vi" hreflang="vi" data-title="Gradient" data-language-autonym="Tiếng Việt" data-language-local-name="베트남어" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%A2%AF%E5%BA%A6" title="梯度 – 우어" lang="wuu" hreflang="wuu" data-title="梯度" data-language-autonym="吴语" data-language-local-name="우어" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%A2%AF%E5%BA%A6" title="梯度 – 중국어" lang="zh" hreflang="zh" data-title="梯度" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%A2%AF%E5%BA%A6" title="梯度 – 광둥어" lang="yue" hreflang="yue" data-title="梯度" data-language-autonym="粵語" data-language-local-name="광둥어" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q173582#sitelinks-wikipedia" title="언어 간 링크 편집" class="wbc-editpage">링크 편집</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="이름공간"> <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 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href="mw-data:TemplateStyles:r34311371"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><td class="sidebar-pretitle">관련 문서 둘러보기</td></tr><tr><th class="sidebar-title-with-pretitle" style="padding-bottom:0.25em;"><a href="/wiki/%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="미적분학">미적분학</a></th></tr><tr><td class="sidebar-above" style="padding:0.15em 0.25em 0.3em;font-weight:normal;"> <ul><li><a href="/wiki/%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99%EC%9D%98_%EA%B8%B0%EB%B3%B8%EC%A0%95%EB%A6%AC" class="mw-redirect" title="미적분학의 기본정리">미적분학의 기본정리</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%EA%B7%B9%ED%95%9C" title="함수의 극한">함수의 극한</a></li> <li><a href="/wiki/%EC%97%B0%EC%86%8D_%ED%95%A8%EC%88%98" title="연속 함수">연속 함수</a></li></ul> </div><div class="hlist"> <ul><li><a href="/wiki/%ED%8F%89%EA%B7%A0%EA%B0%92_%EC%A0%95%EB%A6%AC" title="평균값 정리">평균값 정리</a></li> <li><a href="/wiki/%EB%A1%A4%EC%9D%98_%EC%A0%95%EB%A6%AC" title="롤의 정리">롤의 정리</a></li></ul> </div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;display:block;margin-top:0.65em;"><span style="font-size:110%;"><a href="/wiki/%EB%AF%B8%EB%B6%84%ED%95%99" title="미분학">미분</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading"> 정의</th></tr><tr><td class="sidebar-content hlist"> <div class="hlist" style="padding:0.1em 0;line-height:1.2em;"> <ul><li><a href="/wiki/%EB%AF%B8%EB%B6%84" title="미분">미분</a> <ul><li><a href="/w/index.php?title=%EB%AF%B8%EB%B6%84%EC%9D%98_%EC%9D%BC%EB%B0%98%ED%99%94&action=edit&redlink=1" class="new" title="미분의 일반화 (없는 문서)">일반화</a></li> <li><a href="/wiki/%EB%AF%B8%EB%B6%84%EC%86%8C" title="미분소">무한소</a></li> <li><a href="/wiki/%EB%AF%B8%EB%B6%84_(%EC%A3%BC%EC%9A%94_%EB%B6%80%EB%B6%84)" title="미분 (주요 부분)">주요 부분</a></li> <li><a href="/wiki/%EC%A0%84%EB%AF%B8%EB%B6%84" title="전미분">전미분</a></li></ul></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> 개념</th></tr><tr><td class="sidebar-content hlist"> <div class="hlist"> <ul><li><a href="/w/index.php?title=%EB%AF%B8%EB%B6%84_%ED%91%9C%EA%B8%B0%EB%B2%95&action=edit&redlink=1" class="new" title="미분 표기법 (없는 문서)">미분 표기법</a></li> <li><a href="/wiki/%EA%B3%A0%EA%B3%84_%EB%8F%84%ED%95%A8%EC%88%98" class="mw-redirect" title="고계 도함수">고계 도함수</a></li> <li><a href="/wiki/%EB%B3%80%EC%88%98_%EB%B3%80%ED%99%98" title="변수 변환">변수 변환</a></li> <li><a href="/wiki/%ED%85%8C%EC%9D%BC%EB%9F%AC_%EC%A0%95%EB%A6%AC" title="테일러 정리">테일러 정리</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/%EB%AF%B8%EB%B6%84%ED%91%9C" class="mw-redirect" title="미분표">법칙과 항등식</a></th></tr><tr><td class="sidebar-content hlist"> <div class="hlist"> <ul><li><a href="/wiki/%ED%95%A9_%EA%B7%9C%EC%B9%99" title="합 규칙">합 규칙</a></li> <li><a href="/wiki/%EA%B3%B1_%EA%B7%9C%EC%B9%99" title="곱 규칙">곱 규칙</a></li> <li><a href="/wiki/%EB%AA%AB_%EA%B7%9C%EC%B9%99" title="몫 규칙">몫 규칙</a></li> <li><a href="/wiki/%EB%A9%B1_%EA%B7%9C%EC%B9%99" title="멱 규칙">멱 규칙</a></li> <li><a href="/wiki/%EC%97%B0%EC%87%84_%EB%B2%95%EC%B9%99" title="연쇄 법칙">연쇄 법칙</a></li> <li><a href="/wiki/%EC%97%AD%ED%95%A8%EC%88%98%EC%9D%98_%EB%AF%B8%EB%B6%84" class="mw-redirect" title="역함수의 미분">역함수의 미분</a></li> <li><a href="/wiki/%EC%9D%8C%ED%95%A8%EC%88%98%EC%9D%98_%EB%AF%B8%EB%B6%84" class="mw-redirect" title="음함수의 미분">음함수의 미분</a></li></ul> </div></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/%EC%A0%81%EB%B6%84" title="적분">적분</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EC%A0%81%EB%B6%84%ED%91%9C" title="적분표">적분표</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 정의</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EB%B6%80%EC%A0%95%EC%A0%81%EB%B6%84" title="부정적분">부정적분</a></li> <li><a href="/wiki/%EC%A0%81%EB%B6%84" title="적분">적분</a> (<a href="/wiki/%EC%9D%B4%EC%83%81%EC%A0%81%EB%B6%84" class="mw-redirect" title="이상적분">이상적분</a>)</li> <li><a href="/wiki/%EB%A6%AC%EB%A7%8C_%EC%A0%81%EB%B6%84" title="리만 적분">리만 적분</a></li> <li><a href="/wiki/%EB%A5%B4%EB%B2%A0%EA%B7%B8_%EC%A0%81%EB%B6%84" title="르베그 적분">르베그 적분</a></li> <li><a href="/wiki/%EA%B2%BD%EB%A1%9C%EC%A0%81%EB%B6%84%EB%B2%95" title="경로적분법">경로적분</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 적분법</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EB%B6%80%EB%B6%84%EC%A0%81%EB%B6%84" class="mw-redirect" title="부분적분">부분적분</a></li> <li><a href="/w/index.php?title=%EB%94%94%EC%8A%A4%ED%81%AC_%EB%B0%A9%EB%B2%95&action=edit&redlink=1" class="new" title="디스크 방법 (없는 문서)">디스크 방법</a></li> <li><a href="/wiki/%EC%9B%90%ED%86%B5%EC%85%B8_%EB%B0%A9%EB%B2%95" title="원통셸 방법">원통셸 방법</a></li> <li><a href="/wiki/%EC%B9%98%ED%99%98%EC%A0%81%EB%B6%84" class="mw-redirect" title="치환적분">치환적분</a> (<a href="/wiki/%EC%82%BC%EA%B0%81_%EC%B9%98%ED%99%98" title="삼각 치환">삼각 치환</a>)</li> <li><a href="/wiki/%EB%B6%80%EB%B6%84%EB%B6%84%EC%88%98" title="부분분수">부분분수 적분법</a></li> <li><a href="/w/index.php?title=%EC%A0%81%EB%B6%84_%EC%88%9C%EC%84%9C&action=edit&redlink=1" class="new" title="적분 순서 (없는 문서)">적분 순서</a></li> <li><a href="/wiki/%EC%A0%81%EB%B6%84%EC%9D%98_%EC%A0%90%ED%99%94%EC%8B%9D" title="적분의 점화식">적분의 점화식</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/%EC%88%98%EC%97%B4" title="수열">수열</a>과 <a href="/wiki/%EA%B8%89%EC%88%98_(%EC%88%98%ED%95%99)" title="급수 (수학)">급수</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EA%B8%B0%ED%95%98%EA%B8%89%EC%88%98" class="mw-redirect" title="기하급수">기하급수</a> (<a href="/wiki/%EC%82%B0%EC%88%A0-%EA%B8%B0%ED%95%98_%EC%88%98%EC%97%B4" title="산술-기하 수열">산술-기하 수열</a>)</li> <li><a href="/wiki/%EC%A1%B0%ED%99%94%EA%B8%89%EC%88%98" title="조화급수">조화급수</a></li> <li><a href="/wiki/%EA%B5%90%EB%8C%80%EA%B8%89%EC%88%98" title="교대급수">교대급수</a></li> <li><a href="/wiki/%EB%A9%B1%EA%B8%89%EC%88%98" title="멱급수">멱급수</a></li> <li><a href="/wiki/%EC%9D%B4%ED%95%AD%EA%B8%89%EC%88%98" class="mw-redirect" title="이항급수">이항급수</a></li> <li><a href="/wiki/%ED%85%8C%EC%9D%BC%EB%9F%AC_%EA%B8%89%EC%88%98" title="테일러 급수">테일러 급수</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/%EC%88%98%EB%A0%B4%ED%8C%90%EC%A0%95%EB%B2%95" title="수렴판정법">수렴판정법</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EC%9D%BC%EB%B0%98%ED%95%AD_%ED%8C%90%EC%A0%95%EB%B2%95" title="일반항 판정법">일반항 판정법</a></li> <li><a href="/wiki/%EB%B9%84%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="비판정법">비판정법</a></li> <li><a href="/wiki/%EA%B7%BC%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="근판정법">근판정법</a></li> <li><a href="/wiki/%EC%A0%81%EB%B6%84%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="적분판정법">적분판정법</a></li> <li><a href="/wiki/%EB%B9%84%EA%B5%90%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="비교판정법">비교판정법</a></li> <li><a href="/wiki/%EA%B7%B9%ED%95%9C%EB%B9%84%EA%B5%90%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="극한비교판정법">극한비교판정법</a></li> <li><a href="/wiki/%EA%B5%90%EB%8C%80%EA%B8%89%EC%88%98%ED%8C%90%EC%A0%95%EB%B2%95" class="mw-redirect" title="교대급수판정법">교대급수판정법</a></li> <li><a href="/wiki/%EC%BD%94%EC%8B%9C_%EC%9D%91%EC%A7%91%ED%8C%90%EC%A0%95%EB%B2%95" title="코시 응집판정법">코시 응집판정법</a></li> <li><a href="/wiki/%EB%94%94%EB%A6%AC%ED%81%B4%EB%A0%88_%ED%8C%90%EC%A0%95%EB%B2%95" title="디리클레 판정법">디리클레 판정법</a></li> <li><a href="/wiki/%EC%95%84%EB%B2%A8_%ED%8C%90%EC%A0%95%EB%B2%95" title="아벨 판정법">아벨 판정법</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/%EB%B2%A1%ED%84%B0_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="벡터 미적분학">벡터 미적분학</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a class="mw-selflink selflink">기울기</a></li> <li><a href="/wiki/%EB%B0%9C%EC%82%B0_(%EB%B2%A1%ED%84%B0)" title="발산 (벡터)">발산</a></li> <li><a href="/wiki/%ED%9A%8C%EC%A0%84_(%EB%B2%A1%ED%84%B0)" title="회전 (벡터)">회전</a></li> <li><a href="/wiki/%EB%9D%BC%ED%94%8C%EB%9D%BC%EC%8A%A4_%EC%97%B0%EC%82%B0%EC%9E%90" title="라플라스 연산자">라플라시안</a></li> <li><a href="/wiki/%EB%B0%A9%ED%96%A5%EB%8F%84%ED%95%A8%EC%88%98" class="mw-redirect" title="방향도함수">방향도함수</a></li> <li><a href="/w/index.php?title=%EB%B2%A1%ED%84%B0_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%91%9C&action=edit&redlink=1" class="new" title="벡터 미적분표 (없는 문서)">벡터 미적분표</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 정리</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%EB%B0%9C%EC%82%B0%EC%A0%95%EB%A6%AC" class="mw-redirect" title="발산정리">발산정리</a></li> <li><a href="/w/index.php?title=%EA%B8%B0%EC%9A%B8%EA%B8%B0%EC%A0%95%EB%A6%AC&action=edit&redlink=1" class="new" title="기울기정리 (없는 문서)">기울기정리</a></li> <li><a href="/wiki/%EA%B7%B8%EB%A6%B0_%EC%A0%95%EB%A6%AC" title="그린 정리">그린 정리</a></li> <li><a href="/wiki/%EC%BC%88%EB%B9%88-%EC%8A%A4%ED%86%A0%ED%81%AC%EC%8A%A4_%EC%A0%95%EB%A6%AC" class="mw-redirect" title="켈빈-스토크스 정리">켈빈-스토크스 정리</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;"><a href="/wiki/%EB%8B%A4%EB%B3%80%EC%88%98_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" class="mw-redirect" title="다변수 미적분학">다변수 미적분학</a></span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading"> 형식</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/w/index.php?title=%ED%96%89%EB%A0%AC_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99&action=edit&redlink=1" class="new" title="행렬 미적분학 (없는 문서)">행렬</a></li> <li><a href="/wiki/%ED%85%90%EC%84%9C_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="텐서 미적분학">텐서</a></li> <li><a href="/wiki/%EC%99%B8%EB%AF%B8%EB%B6%84" class="mw-redirect" title="외미분">외미분</a></li> <li><a href="/w/index.php?title=%EA%B8%B0%ED%95%98_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99&action=edit&redlink=1" class="new" title="기하 미적분학 (없는 문서)">기하</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 정의</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/%ED%8E%B8%EB%AF%B8%EB%B6%84" title="편미분">편미분</a></li> <li><a href="/wiki/%EC%A4%91%EC%A0%81%EB%B6%84" title="중적분">중적분</a></li> <li><a href="/wiki/%EC%84%A0%EC%A0%81%EB%B6%84" title="선적분">선적분</a></li> <li><a href="/wiki/%EB%A9%B4%EC%A0%81%EB%B6%84" title="면적분">면적분</a></li> <li><a href="/wiki/%EC%82%BC%EC%A4%91%EC%A0%81%EB%B6%84" class="mw-redirect" title="삼중적분">삼중적분</a></li> <li><a href="/wiki/%EC%95%BC%EC%BD%94%EB%B9%84_%ED%96%89%EB%A0%AC" title="야코비 행렬">야코비안</a></li> <li><a href="/wiki/%ED%97%A4%EC%84%B8_%ED%96%89%EB%A0%AC" title="헤세 행렬">헤세 행렬</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;"><span style="font-size:110%;">특수한 경우</span></div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;padding-top:0.15em;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/%EB%B6%84%EC%88%98%EA%B3%84_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="분수계 미적분학">분수계 미적분학</a></li> <li><a href="/w/index.php?title=%EB%A7%90%EB%A6%AC%EC%95%84%EB%B9%88_%EB%AF%B8%EC%A0%81%EB%B6%84&action=edit&redlink=1" class="new" title="말리아빈 미적분 (없는 문서)">말리아빈 미적분</a></li> <li><a href="/wiki/%ED%99%95%EB%A5%A0%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="확률미적분학">확률미적분학</a></li> <li><a href="/wiki/%EB%B3%80%EB%B6%84%EB%B2%95" title="변분법">변분법</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><style data-mw-deduplicate="TemplateStyles:r34311309">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-보기"><a href="/wiki/%ED%8B%80:%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="틀:미적분학"><abbr title="이 틀을 보기">v</abbr></a></li><li class="nv-토론"><a href="/wiki/%ED%8B%80%ED%86%A0%EB%A1%A0:%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="틀토론:미적분학"><abbr title="이 틀에 관해 토론하기">t</abbr></a></li><li class="nv-편집"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%AC%B8%EC%84%9C%ED%8E%B8%EC%A7%91/%ED%8B%80:%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="특수:문서편집/틀:미적분학"><abbr title="이 틀을 편집하기">e</abbr></a></li></ul></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Gradient2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Gradient2.svg/300px-Gradient2.svg.png" decoding="async" width="300" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Gradient2.svg/450px-Gradient2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0f/Gradient2.svg/600px-Gradient2.svg.png 2x" data-file-width="512" data-file-height="255" /></a><figcaption>위의 두 그림에서는 회색의 밝기가 스칼라계의 크기를 뜻한다. 짙은 색일수록 크기가 큰데, 스칼라계의 기울기는 파란색 화살표로 나타냈다.</figcaption></figure> <p><b>기울기</b>(<span lang="en">gradient</span> <small>그레이디언트<sup>[<a href="/wiki/%EC%9C%84%ED%82%A4%EB%B0%B1%EA%B3%BC:%EC%98%81%EC%96%B4%EC%9D%98_%ED%95%9C%EA%B8%80_%ED%91%9C%EA%B8%B0" title="위키백과:영어의 한글 표기">*</a>]</sup></small>) 또는 경도란 <a href="/wiki/%EB%B2%A1%ED%84%B0_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" title="벡터 미적분학">벡터 미적분학</a>에서 <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EC%9E%A5" title="스칼라장">스칼라장</a>의 최대의 증가율을 나타내는 <a href="/wiki/%EB%B2%A1%ED%84%B0%EC%9E%A5" title="벡터장">벡터장</a>을 뜻한다. </p><p>기울기를 나타내는 벡터장을 화살표로 표시할 때 화살표의 방향은 증가율이 최대가 되는 방향이며, 화살표의 크기는 증가율이 최대일 때의 증가율의 크기를 나타낸다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="기울기의_의미"><span id=".EA.B8.B0.EC.9A.B8.EA.B8.B0.EC.9D.98_.EC.9D.98.EB.AF.B8"></span>기울기의 의미</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B8%B0%EC%9A%B8%EA%B8%B0_(%EB%B2%A1%ED%84%B0)&action=edit&section=1" title="부분 편집: 기울기의 의미"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>어느 방안의 공간 온도 분포가 스칼라장 φ로 주어졌다고 가정한다. 이 때, 방안의 어느 한 점(x,y,z)에서의 온도는 φ(x,y,z)로 표시할 수 있다. (온도는 시간에 의해 변화하지 않는다고 가정) 이 경우에 어느 한 지점에서의 기울기는 온도가 가장 빨리 증가하는 방향과 그 증가율을 나타낸다. </p><p>이번에는 산이나 언덕을 가정해보자. 어떤 지점(x,y)에서의 높이를 H(x,y)로 표현하는 경우, 기울기는 가장 (위를 바라보는)경사가 가파른 방향과 그 경사의 크기를 나타낸다. </p><p>기울기를 이용해 다른 방향의 증가율을 구하려면 기울기와 그 방향의 단위 벡터의 내적을 취하면 된다. </p><p>기울기는 무회전성 벡터계이다. 즉, 기울기 벡터계에 대해 선적분을 구하면 결과값은 경로와 상관없이 시작점과 끝점에 따라서만 변화함을 뜻한다. </p> <div class="mw-heading mw-heading2"><h2 id="기울기의_수학적_정의"><span id=".EA.B8.B0.EC.9A.B8.EA.B8.B0.EC.9D.98_.EC.88.98.ED.95.99.EC.A0.81_.EC.A0.95.EC.9D.98"></span>기울기의 수학적 정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B8%B0%EC%9A%B8%EA%B8%B0_(%EB%B2%A1%ED%84%B0)&action=edit&section=2" title="부분 편집: 기울기의 수학적 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>스칼라 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span>의 기울기는 <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 {\boldsymbol {\nabla }}f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0816bdae1d5149cc18c3b3a1eceff237c31cbf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.505ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {\nabla }}f}"></span>로 표현한다. <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> 기호는 벡터 미분 연산자로 <a href="/wiki/%EB%82%98%EB%B8%94%EB%9D%BC" title="나블라">나블라</a>(nabla) 혹은 델(del)연산자라고 부른다. </p><p>기울기는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 각 성분의 <a href="/wiki/%ED%8E%B8%EB%AF%B8%EB%B6%84" title="편미분">편미분</a>으로 구성된 열벡터로 정의하며 다음과 같이 표시한다. </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 {\boldsymbol {\nabla }}f=\left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>f</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}f=\left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/832275c31fbf4723a0ce1df6ff0611bf264ab8eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.443ex; height:6.176ex;" alt="{\displaystyle {\boldsymbol {\nabla }}f=\left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right)}"></span></dd></dl> <p><br /> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nabla }}f=\left({\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>f</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}f=\left({\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8756dde413b97e584b6e36fa69aa6eefc45e10c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.975ex; height:6.176ex;" alt="{\displaystyle {\boldsymbol {\nabla }}f=\left({\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}}\right)}"></span></dd></dl> <p>예를 들어 함수 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y,z)=\ 2x+3y^{2}-\sin(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mtext> </mtext> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y,z)=\ 2x+3y^{2}-\sin(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93337013498b468701a5c6258d863a69b61174d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.711ex; height:3.176ex;" alt="{\displaystyle f(x,y,z)=\ 2x+3y^{2}-\sin(z)}"></span>의 기울기는</dd></dl> <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 {\boldsymbol {\nabla }}f={\begin{pmatrix}{\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}},{\frac {\partial f}{\partial z}}\end{pmatrix}}={\begin{pmatrix}{2},{6y},{-\cos(z)}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mi>y</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}f={\begin{pmatrix}{\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}},{\frac {\partial f}{\partial z}}\end{pmatrix}}={\begin{pmatrix}{2},{6y},{-\cos(z)}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6329aaf10af9b66c36724f014a861aab3d99335a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.662ex; height:4.843ex;" alt="{\displaystyle {\boldsymbol {\nabla }}f={\begin{pmatrix}{\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}},{\frac {\partial f}{\partial z}}\end{pmatrix}}={\begin{pmatrix}{2},{6y},{-\cos(z)}\end{pmatrix}}}"></span>이다.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="성질"><span id=".EC.84.B1.EC.A7.88"></span>성질</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B8%B0%EC%9A%B8%EA%B8%B0_(%EB%B2%A1%ED%84%B0)&action=edit&section=3" title="부분 편집: 성질"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>그래디언트는 <a href="/w/index.php?title=%EB%93%B1%EC%9C%84%EC%84%A0&action=edit&redlink=1" class="new" title="등위선 (없는 문서)">등위선</a>과 직교한다.</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nabla }}f\bot L_{c}(f).\quad L_{c}(f)=\left\{\mathbf {r} (t)|f(\mathbf {r} (t))=c\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>f</mi> <mi mathvariant="normal">⊥</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="1em" /> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}f\bot L_{c}(f).\quad L_{c}(f)=\left\{\mathbf {r} (t)|f(\mathbf {r} (t))=c\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ddcc4bc37dcbcbecfda2c97551bb31358ad6cd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.665ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\nabla }}f\bot L_{c}(f).\quad L_{c}(f)=\left\{\mathbf {r} (t)|f(\mathbf {r} (t))=c\right\}}"></span></dd></dl> <div class="mw-collapsible mw-collapsed" style="border-width: 0.1em 0; border-style: solid; border-color: #aaaaaa; overflow: auto; width: 100%;"> <div style="padding: 0.2em 1em; font-size:small; font-weight: bold; min-height: 1.6em; background-color: transparent;">증명</div> <div class="mw-collapsible-content" style="border-top: 0.1em solid #aaaaaa;"> <p>등위집합은 다음과 같이 정의된다. 함수 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>의 값을 일정하게 하는 곡선상을 매개변수 <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> </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/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>를 이용한 벡터 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>로 나타낸다. </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 L_{c}(f)=\left\{\mathbf {r} (t)=(x_{1},x_{2},...,x_{n})|f(\mathbf {r} (t))=c\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{c}(f)=\left\{\mathbf {r} (t)=(x_{1},x_{2},...,x_{n})|f(\mathbf {r} (t))=c\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6edcdcb477ec050412df84db0d34b50c85a9b0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.808ex; height:2.843ex;" alt="{\displaystyle L_{c}(f)=\left\{\mathbf {r} (t)=(x_{1},x_{2},...,x_{n})|f(\mathbf {r} (t))=c\right\}}"></span></dd></dl> <p>등위집합의 접선벡터는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} '(t)=\lim _{\Delta t\to 0}{\frac {\mathbf {r} (t+\Delta t)-\mathbf {r} (t)}{\Delta t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} '(t)=\lim _{\Delta t\to 0}{\frac {\mathbf {r} (t+\Delta t)-\mathbf {r} (t)}{\Delta t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b985bd0c29420874fe37e7a473c94006c6089aa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:29.143ex; height:6.009ex;" alt="{\displaystyle \mathbf {r} '(t)=\lim _{\Delta t\to 0}{\frac {\mathbf {r} (t+\Delta t)-\mathbf {r} (t)}{\Delta t}}}"></span>이 된다. 한편 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)=f(\mathbf {r} (t))=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t)=f(\mathbf {r} (t))=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38054efc80be126f53a4dbb632e9a534d8d2fe66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.808ex; height:2.843ex;" alt="{\displaystyle g(t)=f(\mathbf {r} (t))=c}"></span>라 할 때 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g'(t)={\frac {\partial f}{\partial x_{1}}}{\frac {dx_{1}}{dt}}+{\frac {\partial f}{\partial x_{2}}}{\frac {dx_{2}}{dt}}+...+{\frac {\partial f}{\partial x_{n}}}{\frac {dx_{n}}{dt}}={\boldsymbol {\nabla }}f\cdot \mathbf {r} '(t)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </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>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>f</mi> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g'(t)={\frac {\partial f}{\partial x_{1}}}{\frac {dx_{1}}{dt}}+{\frac {\partial f}{\partial x_{2}}}{\frac {dx_{2}}{dt}}+...+{\frac {\partial f}{\partial x_{n}}}{\frac {dx_{n}}{dt}}={\boldsymbol {\nabla }}f\cdot \mathbf {r} '(t)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bff3805dae408a7d3a2ff9e41e1c8124f31bcb0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:61.338ex; height:6.009ex;" alt="{\displaystyle g'(t)={\frac {\partial f}{\partial x_{1}}}{\frac {dx_{1}}{dt}}+{\frac {\partial f}{\partial x_{2}}}{\frac {dx_{2}}{dt}}+...+{\frac {\partial f}{\partial x_{n}}}{\frac {dx_{n}}{dt}}={\boldsymbol {\nabla }}f\cdot \mathbf {r} '(t)=0}"></span></dd></dl> <p>따라서 그래디언트와 등위선은 직교한다. </p> </div></div> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%EA%B8%B0%EC%9A%B8%EA%B8%B0_(%EB%B2%A1%ED%84%B0)&action=edit&section=4" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%ED%9A%8C%EC%A0%84_(%EB%B2%A1%ED%84%B0)" title="회전 (벡터)">회전 (벡터)</a></li> <li><a href="/wiki/%EB%B0%9C%EC%82%B0_(%EB%B2%A1%ED%84%B0)" title="발산 (벡터)">발산 (벡터)</a></li></ul> <div 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