{"id":155912,"date":"2022-03-26T01:10:21","date_gmt":"2022-03-25T19:40:21","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/vector-calculus-definition-formulas-and-identities\/"},"modified":"2022-12-29T01:07:39","modified_gmt":"2022-12-28T19:37:39","slug":"vector-calculus-definition-formulas-and-identities","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/vector-calculus\/","title":{"rendered":"Vector Calculus \u2013 Definition, Formulas and Identities"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/vector-calculus\/#Vector_Calculus_Formulas\" title=\"Vector Calculus Formulas\">Vector Calculus Formulas<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/vector-calculus\/#Vector_Calculus_Definition\" title=\"Vector Calculus Definition\">Vector Calculus Definition<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/vector-calculus\/#Vector_Calculus_Definition-2\" title=\"Vector Calculus Definition\">Vector Calculus Definition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/vector-calculus\/#Vector_Calculus_Formulas-2\" title=\"Vector Calculus Formulas\">Vector Calculus Formulas<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/vector-calculus\/#_%E2%88%AF_SF_nd%CF%83\" title=\"= \u222f  SF. nd\u03c3\">= \u222f  SF. nd\u03c3<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Vector_Calculus_Formulas\"><\/span>Vector Calculus Formulas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Vector Calculus \u2013 Definition: <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/vector-calculus\/\">Vector calculus<\/a> is the study of the calculus of vectors and vector fields. It is a powerful tool for solving problems in physics and engineering.<\/p>\n<p>The basic vector calculus formulas are:<\/p>\n<p>\u2022 Derivative of a vector:<\/p>\n<p>dv = lim dx\u21920 (v + dx) \u2212 v<\/p>\n<p>dv = (v \u00b7 dx) + (dx \u00b7 dv)<\/p>\n<p>\u2022 Gradient of a vector:<\/p>\n<p>\u2207v = (v \u00b7 \u2207)v<\/p>\n<p>\u2022 Divergence of a vector:<\/p>\n<p>div v = (v \u00b7 div)v<\/p>\n<p>\u2022 Curl of a vector:<\/p>\n<p>curl v = (v \u00b7 curl)v<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-155911 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/vector-calculus-definition-formulas-and-identities.jpg\" alt=\"Vector Calculus \u2013 Definition, Formulas and Identities\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/vector-calculus-definition-formulas-and-identities.jpg?v=1648237218 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/vector-calculus-definition-formulas-and-identities-300x212.jpg?v=1648237218 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Vector_Calculus_Definition\"><\/span>Vector Calculus Definition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Vector calculus is a branch of mathematics that deals with the properties and behavior of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Vector\">vectors<\/a>, vector fields, and tensors in three-dimensional space. It incorporates the principles of calculus and linear algebra to investigate the properties of vectors and vector fields, and to find solutions to problems involving them.<\/p>\n<p>Vector calculus is used in a wide range of applications, including physics, engineering, and mathematics. It is particularly important in the study of fluids and electromagnetism.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Vector_Calculus_Definition-2\"><\/span><b>Vector Calculus Definition<\/b><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">It is also known as vector analysis which deals with the differentiation and the integration of the vector field in the three-dimensional Euclidean space. Vector fields represent the distribution of a given vector to each point in the subset of the space. In the Euclidean space, the vector field on a domain represented in the form of a vector-valued function which compares the n-tuple of the real numbers to each point on the domain.<\/span><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Vector analysis is a type of analysis that deals with the quantities which have both the magnitude and the direction. Vector calculus also deals with two integrals known as the line integrals and the surface integrals.<\/span><\/p>\n<p><b>Line Integral<\/b><\/p>\n<p><span style=\"font-weight: 400;\">According to vector calculus, the line integral of a vector field is known as the integral of some particular function along a curve. In simple words, the line integral said to be integral in which the function that is to be integrated is calculated along with the curve. You can integrate some particular type of the vector-valued functions along with the curve. For example, you can also integrate the scalar-valued function along the curve. Sometimes, the line integral is also called the path integral, or the curve integral or the curvilinear integrals.<\/span><\/p>\n<p><b>Surface Integral<\/b><\/p>\n<p><span style=\"font-weight: 400;\">In calculus, the surface integral is known as the generalization of different integrals to the integrations over the surfaces. It means that you can think about the double integral being related to the line integral. For a specific given surface, you can integrate the scalar field over the surface, or the vector field over the surface.<\/span><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Vector_Calculus_Formulas-2\"><\/span><b>Vector Calculus Formulas<\/b><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Let us now learn about the different vector calculus formulas in this vector calculus pdf. The important vector calculus formulas are as follows:<\/span><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">From the fundamental theorems, you can take,<\/span><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">F(x,y,z)=P(x,y,z)i+Q(x,y,z)j+R(x,y,z)k<\/span><\/p>\n<p><b>Fundamental Theorem of the Line Integral<\/b><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Consider<\/span> <span style=\"font-weight: 400;\">F=\u25bdf <\/span><span style=\"font-weight: 400;\">and a curve C that has the endpoints A and B.<\/span><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Then you would get<\/span><\/p>\n<p><span data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mo&gt;&amp;#x222B;&lt;\/mo&gt;&lt;mi&gt;c&lt;\/mi&gt;&lt;mi&gt;F&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mi&gt;d&lt;\/mi&gt;&lt;mi&gt;r&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mi&gt;f&lt;\/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;\/mo&gt;&lt;mi&gt;B&lt;\/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;\/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;\/mo&gt;&lt;mi&gt;f&lt;\/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;\/mo&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;\/mo&gt;&lt;\/math&gt;\">\u222bcF.dr=f(B)\u2212f(A)\u222bcF.dr=f(B)\u2212f(A)<\/span><\/p>\n<p><b>Circulation Curl Form<\/b><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">According to the Green\u2019s theorem,<\/span><\/p>\n<p><span data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x222C;&lt;\/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;D&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x2202;&lt;\/mi&gt;&lt;mi&gt;Q&lt;\/mi&gt;&lt;\/mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x2202;&lt;\/mi&gt;&lt;mi&gt;x&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/mfrac&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;mo&gt;&amp;#x2212;&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x2202;&lt;\/mi&gt;&lt;mi&gt;P&lt;\/mi&gt;&lt;\/mrow&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x2202;&lt;\/mi&gt;&lt;mi&gt;y&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/mfrac&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;mi&gt;d&lt;\/mi&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mo&gt;&amp;#x222E;&lt;\/mo&gt;&lt;mi&gt;C&lt;\/mi&gt;&lt;mi&gt;F&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mi&gt;d&lt;\/mi&gt;&lt;mi&gt;r&lt;\/mi&gt;&lt;\/math&gt;\">\u222cD(\u2202Q\u2202x)\u2212(\u2202P\u2202y)dA=\u222eCF.dr\u222cD(\u2202Q\u2202x)\u2212(\u2202P\u2202y)dA=\u222eCF.dr<\/span><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">According to the Stoke\u2019s theorem,<\/span><\/p>\n<p><span data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x222C;&lt;\/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;D&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/msub&gt;&lt;mo&gt;&amp;#x25BD;&lt;\/mo&gt;&lt;mo&gt;&amp;#x00D7;&lt;\/mo&gt;&lt;mi&gt;F&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mi&gt;n&lt;\/mi&gt;&lt;mi&gt;d&lt;\/mi&gt;&lt;mi&gt;&amp;#x03C3;&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mo&gt;&amp;#x222E;&lt;\/mo&gt;&lt;mi&gt;C&lt;\/mi&gt;&lt;mi&gt;F&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mi&gt;d&lt;\/mi&gt;&lt;mi&gt;r&lt;\/mi&gt;&lt;\/math&gt;\">\u222cD\u25bd\u00d7F.nd\u03c3=\u222eCF.dr\u222cD\u25bd\u00d7F.nd\u03c3=\u222eCF.dr<\/span><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Here, C refers to the edge curve of S.<\/span><\/p>\n<p><b>Flux Divergence Form<\/b><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">According to the Green\u2019s theorem,<\/span><\/p>\n<p><span data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x222C;&lt;\/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;D&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/msub&gt;&lt;mo&gt;&amp;#x25BD;&lt;\/mo&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mi&gt;F&lt;\/mi&gt;&lt;mi&gt;d&lt;\/mi&gt;&lt;mi&gt;A&lt;\/mi&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mo&gt;&amp;#x222E;&lt;\/mo&gt;&lt;mi&gt;C&lt;\/mi&gt;&lt;mi&gt;F&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mi&gt;n&lt;\/mi&gt;&lt;mi&gt;d&lt;\/mi&gt;&lt;mi&gt;s&lt;\/mi&gt;&lt;\/math&gt;\">\u222cD\u25bd.FdA=\u222eCF.nds\u222cD\u25bd.FdA=\u222eCF.nds<\/span><\/p>\n<p style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">According to the Divergence theorem,<\/span><\/p>\n<p><span data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;block&quot;&gt;&lt;mo&gt;&amp;#x222B;&lt;\/mo&gt;&lt;mo&gt;&amp;#x222B;&lt;\/mo&gt;&lt;msub&gt;&lt;mo&gt;&amp;#x222B;&lt;\/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;D&lt;\/mi&gt;&lt;\/mrow&gt;&lt;\/msub&gt;&lt;mi class=&quot;MJX-variant&quot;&gt;&amp;#x25BD;&lt;\/mi&gt;&lt;mo&gt;.&lt;\/mo&gt;&lt;mi&gt;F&lt;\/mi&gt;&lt;mi&gt;d&lt;\/mi&gt;&lt;mi&gt;V&lt;\/mi&gt;&lt;\/math&gt;\">\u222b\u222b\u222bD\u25bd.FdV\u222b\u222b\u222bD\u25bd.FdV<\/span><\/p>\n<h3><span style=\"font-weight: 400;\">= \u222f  SF. nd\u03c3<\/span><\/h3>\n","protected":false},"excerpt":{"rendered":"<p>Vector Calculus Formulas Vector Calculus \u2013 Definition: Vector calculus is the study of the calculus of vectors and vector fields. 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