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div_grad_curl.html
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div_grad_curl.html
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<!DOCTYPE html>
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<div class="span10 offset1">
<h1>The Gradient, Divergence, and Curl</h1>
<p>The gradient of a scalar function <span class="math">$f:R^n \rightarrow R$</span> is a vector field of partial derivatives. In <span class="math">$R^2$</span>, we have:</p>
<p class="math">\[
~
\nabla{f} = \langle \frac{\partial{f}}{\partial{x}},
\frac{\partial{f}}{\partial{y}} \rangle.
~
\]</p>
<p>It has the interpretation of pointing out the direction of greatest ascent for the surface <span class="math">$z=f(x,y)$</span>.</p>
<p>We move now to two other operations, the divergence and the curl, which combine to give a language to describe vector fields in <span class="math">$R^3$</span>.</p>
<pre class='hljl'>
<span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>CalculusWithJulia</span>
</pre>
<h2>The divergence</h2>
<p>Let <span class="math">$F:R^3 \rightarrow R^3 = \langle F_x, F_y, F_z\rangle$</span> be a vector field. Consider now a small box-like region, <span class="math">$R$</span>, with surface, <span class="math">$S$</span>, on the cartesian grid, with sides of length <span class="math">$\Delta x$</span>, <span class="math">$\Delta y$</span>, and <span class="math">$\Delta z$</span> with <span class="math">$(x,y,z)$</span> being one corner. The outward pointing unit normals are <span class="math">$\pm \hat{i}, \pm\hat{j},$</span> and <span class="math">$\pm\hat{k}$</span>.</p>
<img src="data:image/png;base64,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" />
<p>Consider the sides with outward normal <span class="math">$\hat{i}$</span>. The contribution to the surface integral, <span class="math">$\oint_S (F\cdot\hat{N})dS$</span>, could be <em>approximated</em> by</p>
<p class="math">\[
~
\left(F(x + \Delta x, y, z) \cdot \hat{i}\right) \Delta y \Delta z,
~
\]</p>
<p>whereas, the contribution for the face with outward normal <span class="math">$-\hat{i}$</span> could be approximated by:</p>
<p class="math">\[
~
\left(F(x, y, z) \cdot (-\hat{i}) \right) \Delta y \Delta z.
~
\]</p>
<p>The functions are being evaluated at a point on the face of the surface. For Riemann integrable functions, any point in a partition may be chosen, so our choice will not restrict the generality.</p>
<p>The total contribution of the two would be:</p>
<p class="math">\[
~
\left(F(x + \Delta x, y, z) \cdot \hat{i}\right) \Delta y \Delta z +
\left(F(x, y, z) \cdot (-\hat{i})\right) \Delta y \Delta z =
\left(F_x(x + \Delta x, y, z) - F_x(x, y, z)\right) \Delta y \Delta z,
~
\]</p>
<p>as <span class="math">$F \cdot \hat{i} = F_x$</span>.</p>
<p><em>Were</em> we to divide by <span class="math">$\Delta V = \Delta x \Delta y \Delta z$</span> <em>and</em> take a limit as the volume shrinks, the limit would be <span class="math">$\partial{F}/\partial{x}$</span>.</p>
<p>If this is repeated for the other two pair of matching faces, we get a definition for the <em>divergence</em>:</p>
<blockquote>
<p>The <em>divergence</em> of a vector field <span class="math">$F:R^3 \rightarrow R^3$</span> is given by <span class="math">$~ \text{divergence}(F) = \lim \frac{1}{\Delta V} \oint_S F\cdot\hat{N} dS = \frac{\partial{F_x}}{\partial{x}} +\frac{\partial{F_y}}{\partial{y}} +\frac{\partial{F_z}}{\partial{z}}. ~$</span></p>
</blockquote>
<p>The limit expression for the divergence will hold for any smooth closed surface, <span class="math">$S$</span>, converging on <span class="math">$(x,y,z)$</span>, not just box-like ones.</p>
<h3>General <span class="math">$n$</span></h3>
<p>The derivation of the divergence is done for <span class="math">$n=3$</span>, but could also have easily been done for two dimensions (<span class="math">$n=2$</span>) or higher dimensions <span class="math">$n>3$</span>. The formula in general would be: for <span class="math">$F(x_1, x_2, \dots, x_n): R^n \rightarrow R^n$</span>:</p>
<p class="math">\[
~
\text{divergence}(F) = \sum_{i=1}^n \frac{\partial{F_i}}{\partial{x_i}}.
~
\]</p>
<hr />
<p>In <code>Julia</code>, the divergence can be implemented different ways depending on how the problem is presented. Here are two functions from the <code>CalculusWithJulia</code> package for when the problem is symbolic or numeric:</p>
<pre class='hljl'>
<span class='hljl-nf'>divergence</span><span class='hljl-p'>(</span><span class='hljl-n'>F</span><span class='hljl-oB'>::</span><span class='hljl-nf'>Vector</span><span class='hljl-p'>{</span><span class='hljl-n'>Sym</span><span class='hljl-p'>},</span><span class='hljl-t'> </span><span class='hljl-n'>vars</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>sum</span><span class='hljl-p'>(</span><span class='hljl-n'>diff</span><span class='hljl-oB'>.</span><span class='hljl-p'>(</span><span class='hljl-n'>F</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>vars</span><span class='hljl-p'>))</span><span class='hljl-t'>
</span><span class='hljl-nf'>divergence</span><span class='hljl-p'>(</span><span class='hljl-n'>F</span><span class='hljl-oB'>::</span><span class='hljl-n'>Function</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>pt</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>sum</span><span class='hljl-p'>(</span><span class='hljl-nf'>diag</span><span class='hljl-p'>(</span><span class='hljl-n'>ForwardDiff</span><span class='hljl-oB'>.</span><span class='hljl-nf'>jacobian</span><span class='hljl-p'>(</span><span class='hljl-n'>F</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>pt</span><span class='hljl-p'>)))</span>
</pre>
<p>The latter being a bit inefficient, as all <span class="math">$n^2$</span> partial derivatives are found, but only the <span class="math">$n$</span> diagonal ones are used.</p>
<h2>The curl</h2>
<p>Before considering the curl for <span class="math">$n=3$</span>, we derive a related quantity in <span class="math">$n=2$</span>. The "curl" will be a measure of the microscopic circulation of a vector field. To that end we consider a microscopic box-region in <span class="math">$R^2$</span>:</p>
<img src="data:image/png;base64,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" />
<p>Let <span class="math">$F=\langle F_x, F_y\rangle$</span>. For small enough values of <span class="math">$\Delta{x}$</span> and <span class="math">$\Delta{y}$</span> the line integral, <span class="math">$\oint_C F\cdot d\vec{r}$</span> can be <em>approximated</em> by <span class="math">$4$</span> terms:</p>
<p class="math">\[
~
\begin{align}
\left(F(x,y) \cdot \hat{i}\right)\Delta{x} &+
\left(F(x+\Delta{x},y) \cdot \hat{j}\right)\Delta{y} +
\left(F(x,y+\Delta{y}) \cdot (-\hat{i})\right)\Delta{x} +
\left(F(x,y) \cdot (-\hat{j})\right)\Delta{x}\\
&=
F_x(x,y) \Delta{x} + F_y(x+\Delta{x},y)\Delta{y} +
F_x(x, y+\Delta{y}) (-\Delta{x}) + F_y(x,y) (-\Delta{y})\\
&=
(F_y(x + \Delta{x}, y) - F_y(x, y))\Delta{y} -
(F_x(x, y+\Delta{y})-F_x(x,y))\Delta{x}.
\end{align}
~
\]</p>
<p>The Riemann approximation allows a choice of evaluation point for Riemann integrable functions, and the choice here lends itself to further analysis. Were the above divided by <span class="math">$\Delta{x}\Delta{y}$</span>, the area of the box, and a limit taken, partial derivatives appear to suggest this formula:</p>
<p class="math">\[
~
\lim \frac{1}{\Delta{x}\Delta{y}} \oint_C F\cdot d\vec{r} =
\frac{\partial{F_y}}{\partial{x}} - \frac{\partial{F_x}}{\partial{y}}.
~
\]</p>
<p>The scalar function on the right hand side is called the (two-dimensional) curl of <span class="math">$F$</span> and the left-hand side lends itself as a measure of the microscopic circulation of the vector field, <span class="math">$F:R^2 \rightarrow R^2$</span>.</p>
<hr />
<p>Consider now a similar scenario for the <span class="math">$n=3$</span> case. Let <span class="math">$F=\langle F_x, F_y,F_z\rangle$</span> be a vector field and <span class="math">$S$</span> a box-like region with side lengths <span class="math">$\Delta x$</span>, <span class="math">$\Delta y$</span>, and <span class="math">$\Delta z$</span>, anchored at <span class="math">$(x,y,z)$</span>.</p>
<img 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CsQQIBFIgKloS0NRVzcLapQSYYNkxo1kpo1K7j5OwB7IYAAizQLM7ToJn81C+MMMG87eFBavPiXr//zH2nrVuvmAeB9BBBgkWynqe9TTWU7TatHsZ1nninfYwB8FwEEWOTAWeny5U4dOGv1JPZy+LD03nvFH1+3ToqN9f48AKxBAAGwlRdekFzF73UqSZo82buzALAOAQTANrKzpaVLS1++Zo2UlOS9eQBYhwACYBsu14Xv+2UYkskhWYAtEEAAbCM4WJo+XQoJKXnZlCkFd4cH4Pu4GSpgkfZRhszRAVaPYTujR0t9+kgxMe6PHzokNWxozUwAvI89QABsx7+E//QL5IKUgK0QQIBFDp411eUDpw6e5aATAPA2AgiwyDmntDXR1Dmn1ZMAgP0QQAAAwHYIIAAAYDsEEAAAsB0CCLBI0xBpYQ8/NS3hmjQAAM/iOkCARSJrGBp2uWH1GABgS+wBAiySlGVq5v58JWVxGjwAeBsBBFgk7pw0dotLceesngQA7IcAAgAAtkMAAQAA2yGAAACA7RBAgEVCA6SeDQ2FckN4APA6ToMHLHJ5bUOf9uYlCABWYA8QYJF8l6m0XFP5Lk6DBwBvI4AAi+xJlmovcGpPstWTAID9EEAAAMB2CCAAAGA7BBAAALAdAggA4NNGjBghwzBUv359OZ1Oq8dBFUEAARa5KlJKHOavqyKtngTwXWlpaVq5cqVq166tU6dOac2aNVaPhCqCAAIsEuAwVLemoQCHYfUogM9aunSpMjMzNXXqVAUFBWnu3LlWj4QqggACLPJDmqm+nzr1QxrXAQI8Ze7cuYqMjNSQIUPUp08fffzxx0pISHBb57777pNhGKV+3HfffdYMD4/iMrSARVJzpQ+Pm5rUwepJAN+0b98+xcbG6qGHHlJgYKCGDx+ulStXasGCBXr66aeL1uvfv7+aNm1a7Pnr1q3Tli1bFBwc7MWp4S0EEADAJxW+3TVs2DBJUu/evVWnTh3NmzevWAD179/f7bk7duzQyy+/rKZNm2ry5MneGxpew1tgAACfk5ubq0WLFqlZs2bq1q2bJCkgIED33nuvDh8+rC+++KLU58bHx6tv377y9/fXhx9+qLp163prbHgRAQQA8DmrV6/WmTNnNHToULfHhw8fLkmaN29eic/LzMzUXXfdpVOnTmnp0qVq27atx2eFNQggwCINg6V//dahhhxeAFS6wsApfPurUJcuXdSiRQutWLFCaWlpbstM09SwYcP09ddf65VXXtGdd97ptXnhfQQQYJF6wYb+fLWf6gVzGjxQmeLi4rRu3Tp17NhRrVq1KrZ82LBhyszM1LJly9wenzBhgt5//3098MADevzxx701LixCAAEWSckxteKISyk5nAYPVKb58+fL5XIV2/tTqPDxX18T6J133tGUKVPUo0cPzZo1yytzwlqcBQZY5Gi6dO/6fO28218RQVZPA/gG0zQ1f/58+fv76/e//32J67Rs2VKdO3fW1q1b9c033yg1NVWjR49Wy5YttXLlSgUEBHh5aliBAAIA+Iz169fr2LFjioiI0IQJE0pdLysrS1LBXqBPP/1Uubm56tixo6ZPn15s3Xbt2hU7TR7VHwEEAPAZhW9rpaSklOu2F4sWLSra47N06dIS1xk5ciQB5IMIIACAz1i6dGmpIQP8GgdBAxap6SddW6fgMwDAuwggwCJXRBjaNSBAV0RwGjx8Q1xcnCZNmqQNGzZYPQpQJgIIAFApHnjgAU2dOlU333yzevfubfU4wAURQIBFvj5tKmhunr4+zXWAUP3NmTNHO3bs0M6dOyWJO6ijyiOAAIuYknJdBZ+B6iwuLk5PPPGE3n77bTVp0kSbN2/WBx98oOXLl1s9GlAqzgIDAFySxo0b6+zZs0VfX3/99crLy7NwIqBs7AECAFSaY8eOyTAM3XfffVaPAlwQAQQAAGyHt8AAi1wRLn1zj7+ah1k9CQDYDwEEWKSmv6ErI62eAgDsibfAAIv8mG7qgS+c+jGd88AAwNsIIMAiZ3KkuQdNncmxehIAsB8CCAAA2A4BBAAAbIcAAioJN4IEgOqDAAIqycXeCLJeTenpaxyqV9MLwwEA3BBAQCWoyI0gG9Yy9GInPzWsZXh6PADAeQgg4BJV9EaQ6bmmNp50KT2X0+DhO7KzsyVJQUFBFk8CXBgXQgQuUUVvBHk4TbppTb523u2v9lGenBDwnu+//16S1KhRI4snAS6MPUBAJeJGkLCrQ4cO6emnn9bDDz8sh8Ohfv36WT0ScEEEEADgkh04cEDTpk1TRESEVq5cqauvvtrqkYAL4i0wAMAl69+/v7Kysqweo1Q//CDl5UmtW1s9CaoK9gABFglwSA1rFXwG4FljxkhXXCHdc4+0b5/V06AqYA8QYJGrIg2dGBJQ5nqmKe3fX/DL28/PC4PBlo4ckX51LL/lsuPaSGdrFX195EC4dl3CDqYzZwo+f/CBtGqVdPPN0oQJ0i23XOKgqLYIIKCKMk1pzRpp0iRp505p717pqqusngq+6PRp6fLLJZfL6kl+7X23r554s3K2mp9f8Pnzzws+kpKkKM7CtCUCCLDIvmRTvT5xau0d/roq8peLIZ4fPjfcIK1fT/zAc6KipMOHq9YeoLvfvVvHz/5Y9PX/9XxFNze7ucLbu/9+ac8eyeGQgoKkESOkP/6R+LEzAgiwSJ5Lij9X8FkqOXw+/1zq0UMyuFg0PKx5c6sncFdjywEp+FDR183bnFX7Kyq+vcaNC97mGz9eevRRKTy8EoZEtUYAARYzTemjjwgfwJOWLi14PdWqVfa6sAcCCLCIaUraa2j4TOnbPYQP4EkhIVZPgKqGAAK8rPCtricnStrjrxqdTcIHALyMAAIqUdOmTWWaJd/c9PxjfLpeL7263KVRfQ2Fcd9IAPAqAgjwsNIPbjZksMsHACzBNWgBDyk8uLljR+muu6Tg4ILw2bRJuukm6WSmqb9uz1f8uZL3GAEAPIcAAipZWeFTuNPnVJY0ZY9Lp6ru7ZMAwGcRQEAlKW/4AACsRwABl4jwAYDqhwACKojwAYDqiwACLlJlhU+dIGlUK0N1OAUeALyOAALKqbL3+FwWauit7v66LJRdRQDgbQQQUAZPvdWV5TS1P9lUlpPT4AHA2wggoBSePsbn27NS25VOfXu2cuYFAJQfAQSch4ObAcD3EUDAzwgfALAPAgi2R/gAgP0QQLAtq8PHkBToKPgMAPAuAgi2Y3X4FLo2ylDOqABdG0UCAYC3EUCwjaoSPgAA6xFA8HlVNXy+TTHVflWevk3hOkAA4G0EEHxWVQ2fQln50tdnCj4DALyLAILPqerhAwCwHgGEaistTVqwQHI6C74mfAAA5eVv9QBARZimNHKktHp1wdd16kiTJkk7d0o33FAQPj16ED0AgJIRQKiWZsz4JX7GjJFyc6tf+DQLlZbf4qdmoVZPAgD2QwCh2omNlcaN++Xr3Fzp6aelf/6zeoRPoYggQwObV6OBAcCHcAwQqhWXS7rxxoLPkuTnV/B59erqFT+SdCrT1Kt783Uqk9PgAcDb2AOEasXhKDigWZIaNPgleq64wrqZKio+U/rLNpd6xDhUL9jqaQDAXgggVDtr1lg9AQCguuMtMAAAYDsEEAAAsB0CCLBI7UDpriaGagdaPQkA2A/HAAEWaRFm6D+38xIEACuwBwiwSJ7LVFKWqTwXp8EDgLcRQIBF9iVL0Yuc2pds9SQAYD8EEAAAsB0CCAAA2A4BBAAAbIcAAgAAtsM5uIBFromUUkf6qxavQgDwOn71AhbxcxgK4yKIAGAJ3gIDLHI41dTtHzt1OJXrAAGAtxFAgEXS86T/xptKz7N6EgCwHwIIAADYDgEEAABshwACAAC2QwABFmlcS5rR1aHGtayeBADsh9PgAYvUrWnoT1f6WT0GANgSe4AAiyRnm1p02KXkbE6DBwBvI4AAixzLkIZvzNexDKsnAQD7IYAAAIDtEEAAAMB2CCAAAGA7BBBgkVr+Uudog7vBA4AF+NULWKRVuKGv+vESBAArsAcIAADYDgEEWGTXaVPGnDztOs11gADA2wggAABgOwQQAACwHQIIAADYDgEEAABsh3NwAYu0CZcO3+uvRrWsngQA7IcAAixSw99Qy9pWTwFUPbHxsTqdedrqMeDjeAsMsMjRNFPDNjh1NI3T4AGpIHz6LOmjTm91UnJWstsyh8GfK1Qu/h8FWCQlV1r8vamUXKsnAaz16/BZc3hNseWGDHVo0MGCyeDLeAsMAGCJ2PhYTd40ucToKRQVHKXX73xdjWs39uJksAMCCLCQYbqsHgHwuvKGz5Ndn9TDHR9WrUDOFEDlI4AALzNd+cr+ZqvC163S8Px2ikvrohCHUbQ8NDBEDULqKSc/Vz+mxhV7/m8iW0iSjqedULYzx21Z/VrRCgsK1dnsVCWedxBpsH9NNQqLUb4rXz+cPVZsu83DL5O/w1/x6Qk6l5fptiyqZh1F1gxXek6GEs6dclsW6Beopj//1/mh5COS3I9puiyskYL8g/RTRqLSctPdlkXUCFfd4DrKzMvUifQEt2V+hp9aRDSVJP2Qckz5Zr7b8kahDRQcEKykzDNKyT7rtiwsMFT1Q6KV48zRj2knzvtJDUWoebGf/4eUo0rxc6lBrXrFlknSD2ePKtflVMPQBnK6nDpy9sdi67QIbyo/h59OpJ1UpjPLbVl0cJTCa9RWWk66fjqX6Lashn+QmoQ1kiQdSv6h2HYvq91YQX6BSsg4pfTcDLdldWpGqE7NSJ3Ly1T8ef+GAY4ANQtv8vPPV9K/YYyCA2oqMfO0zmanui2rHRSmerXqKtuZo+Pn/RsaMnR5ZMG/4bHU48rNz3Nb3iCknkIDQ4r9HIQPqhICCPCic9v+q7T/LlX+mQQFNm2r0CaJevWrv7itc1vTGzWx25+VlHlGo9f+udg2Ng39QJL04lfTdeD0Qbdlf+v6uHo266ENP36pqTvedFvWsUE7vXLzZGXn55S43Q/ueUfhNWprxs652hIf67bs4fb3a9AV/bTjpz2a9OXLbssuj2iut3q/VrDep08oz+V0W/72nf9Ws/Ameuebd7Xmh8/clg1tc4/GXDtCB5N/0LjPJrotq1uzjt4bME+S9NSG55SUdcZt+dRbX9C19a7S+wfXaPGBlW7L7mxxq57s/IhOZpwq9rMGOPy19Fb39SXp8fXPKDAsTZOuf1Jto35T4vIeTa7Xiz0mKiP3XIn/hh/fu1S1HMGaumO2YhN2uy0bd90Y3d3qTm09uVP/2PKa27I2Ua30+u0F/64lbXdx3zfUKLSB5u5ZrHXHNrktu++q3+sPVw/W/qTv9MSGyW7LGobU15J+s3+e/+9KzUlzWz6z50tqW7e1ln/7gVZ89x+3Zf0v76XHOz2k42knis0U7F9TawctkyQ9u/llHTsv1P954wR1a/Tboq8JH1RFhmmanIJiM0lJUnS0+2OJiVLdutbM4+ucKUlyBNWQIzhUqWsXypkUr9AedyuwSSudyUrWmawUt/XZA1TAo3uA8psXew18dfCoIqMK9gBlO7MU/Yr7Cl89sE2NwxqzB0jl3wNE+KAqI4BsiADyjtzjB5W+8X1l7f5CYXcMV1jPwTJNU4ZhlP1keFRZr4Gkc0nFAihxfKLq1uJFUh6ED6oD3gIDKlnO0QNK/XCuco/sl1+dBgrv/6CCf9tTkogf+DTCB9UJAQRUAldOllxpKfKvGyPlFxwDU+f+v6tG284yHH4WTwd4FuGD6ogAAi6BMyVJGZs/0LktaxUQ00zRj76ioJZXK/rRf1k9GuBxhA+qMwIIqID8tGSdXf2msnZ/ISOwpmp17aWQG/pZPRbgFYQPfAEBBJST6cpX3smjCmzUUkZQsJynTxYc39PpNjlqBFs9HuBxhA98CQEElMGVk6XMbf9V+qbVyk89rZjnlsgRHKp6f55u9WiAVxA+8EUEEFAK0zSV9tF8ZfxvjczcLNVs112hPZ6WIzjU6tEAryB84MsIIECSy+XStddeq4YNG2r17GkKiGkuw89P+WnJqtW1t0Ju6Cv/iOp/DZjvv/9erVu31vTp0/Xwww9bPQ6qKMIHduCwegCgKpg/b5727t2rR68IV+K/HlH2t9slSZFDxyu87yifiB+0qZ0jAAAR1UlEQVRJatmypYYOHapJkyYpLS2t7CfAVmLjY9VnSR91eqtTqfETFRyll299WcceO6Ynuj1B/KDaIoBge+nbP9Oz48epc8MItb8spuD6PW06WT2WxzzxxBNKSkrS9Okcw4QChA/siLfAYEvOlCTJkPzD62rtpi8Vn3pOf5/wV0U/+jerR/O4tm3b6pprrtGcOXM0YcIEORz8d5Bd8VYX7IzffLCV3OMHdeadKfrp+ZFK/2y5JOndrftkGIYGjflTsfW7dOkiwzAUG+t+d/SUlBRdeeWVqlGjhjZt2lTseRcyadIkGYahJUuWlLh89uzZMgxDr732WonLS3PllVfKMIxSP1566aWide+9914dP35c69evv6jvAd/AHh+APUCwidwT3+vsqtfd78/V6TaZpqmNGzeqdevWCg8PL/a8KVOmqEePHnrmmWe0du1aSVJ2drb69u2r7777TsuXL9eNN954UbN07dpVkrRt2zYNGTLEbVlKSoomTpyo1q1ba+zYsRe13cGDB8vpdLo9lpOTo6lTpyonJ0c33HBD0eNdunSRJH3++ee67bbbLur7oPpijw/wCwIIPsuVkyVn0kkFNmohI6imJKPY/bkOHDig5ORk9erVq8Rt3HjjjerVq5fWrl2rLVu2qHPnzho6dKi+/PJLzZo1S/fcc89Fz9W5c2c5HA5t37692LJnn31Wp0+f1sKFCxUQEHBR2504caLb19nZ2erfv79yc3M1d+7covCSpOuuu06StGXLloueH9UP4QMURwDB5/z6/lx+oeGqN+EtBdRtqOhHXym27okTJyRJ9erVK3V7L774oj755BM988wzat26tVatWqVnnnlGf/zjHys0X1hYmNq0aaPdu3crLy+vKHT279+v119/XX369NEdd9xRoW0XyszMVN++fbVx40a9/fbbGj58uNvy0NBQ1ahRo+jnh28ifIDSEUDwGa7MDKW8N+NX9+cquH6PYRilPufMmTOSpIiIiFLXueaaazRkyBAtXrxY69ev15gxYzR58uRLmrVr16765ptvtHfvXnXo0EGS9Oijj8rhcOjVV1+9pG2fO3dOffr00ebNm7Vw4UINHjy4xPUiIyN1+vTpS/peqJoIH6BsBBCqNdOVr9yj3yqoRVsZNYLlyswoOL7ntz3lCKpZ5vNr1ixYJysr64LrRUVFSZJq166tf//735c8d9euXfXmm29q+/bt6tChg1auXKnPP/9cTz75pC6//PIKbzc9PV29e/fW1q1btWzZMv3ud78rdd2srCwFB3MPM19C+ADlx1lgqJZcOVnK+OID/fSPB5T07/Fynk6Q4XCo7kMvKKR7v3LFjyTVrVtwgcPk5ORS15k2bZqmTZumevXqKTU1VYsWLbrk+QsPQt62bZuys7M1fvx41a9fv9hxPBcjLS1Nt99+u7Zt26YVK1ZcMH5cLpdSU1OLfn5Ub5zVBVw89gCh2kn7ZLHSN6765f5cI5+Wf1SDCm3ryiuvlMPh0OHDh0tcvmzZMj3++OO69dZbtWDBArVu3VqTJk3SkCFDVKNGjQr/DL/5zW8UFRWl7du36+WXX9axY8c0f/58hYZW7D5jZ8+e1e233649e/Zo1apV6tOnzwXXP3z4sFwul6666qoKfT9UDezxASqOPUCoFnKPH5Ir+5wkyczPU62uvVT/7wtUZ8TTCmzSqsLbDQ8P19VXX60dO3bINE23ZZ999plGjhypdu3aadWqVYqJidFjjz2muLg4zZw5s8Tt9ejRQ4ZhaOPGjWV+786dO+vgwYOaMmWKOnbsqJEjR1Zom8nJybrlllu0d+9evf/++2XGj1Sw50nSRZ/Cj6rBij0+I0aMkGEYql+/frHLLQDVEXuAUGWZrnxlf7NV6RtXKffIfoUPfEQh3e5U7Tvvq9Tv079/f02aNEmxsbHq1KngFhi7du3SgAED1KhRI61du7Zoz8z48eM1c+ZMvfjiixo9erTCwsLctuVyuSRJ/v5lv7S6du2qjz76SNnZ2Zo+fXqpB2uXtc3Bgwdr165duummm7Rt27aiuCkUExOjMWPGuD22bt06+fn5lSuWUHVYtccnLS1NK1euVO3atXXq1CmtWbNG/fr1q5RtA5YxYTuJiaYpuX8kJlo9lbvMvVvMk8/dZ8Y9drt5atqfzcw9X5qufKdHvteJEydMPz8/85FHHjFN0zS///57s169embdunXNw4cPF1v/H//4hynJnDhxotvjLpfLrFOnjtm0aVMzLy+vzO+7ZMkSU5I5fPjwUtcpa5v5+flmcHCwKanUj4EDB7o959y5c2ZISIjZv3//Mmf0VWW9BhIzEk1NkttHYoZ1L5LtJ7abdy6+s9hMv/6IejnKfPnLl82MnIxK//5vvPGGKcmcP3++GRQUZN51112V/j0AbyOAbKiqBlBecqKZm/CjaZqmmbl/m3l6wYtmzo/feeV7Dx482KxTp46ZkVHxPx779u0zJZkzZ84sc12Xy2V27NjRDAkJMePj4ytlm+U1d+5cU5K5adOmSttmdVNdAsjq8CnUsWNHMzIy0szJyTHvuece08/Pzzx58qTbOs8//7wpyRw7dmyx5z/77LOmJHPcuHEemxG4WBwDBMv9+v5caR8vkCTVbNPpko/vuRj/+Mc/lJGRUeqxPeWxefNm1atXT/fff3+Z686cOVOxsbGaPHmyYmJiKmWb5eF0OvXPf/5Tffv2Vffu3Stlm6h8Vemsrn379ik2Nlb33nuvAgMDNXz4cOXn52vBggVu602YMEHdu3fXjBkz9OGHHxY9/r///U8vvPCCrr76ak2ZMsUjMwIVYZjmeUd+wuclJUnR0e6PJSZK3j4jOu9UnFLenVp0f67QG/uX+/o9nvDuu+/q9OnT+tOfit8UtTKcOHFCixYt0qFDh7Rw4UJ169ZNn3/+uVfvxn7s2LGiK0O3aNHCa9+3qinrNZB0LknRr7ivkDg+UXVrefZFUhXP6ho3bpymTZumL7/8Ut26dVNeXp4aNGigyMhIHTp0yG3duLg4XXPNNfLz89PevXsVHBysdu3a6dSpU9qxY4fatGnj8XmB8iKAbMjKAHJlZyov/oiCWrSVKzNdZxa8qJBud7rdn8tXzZs3T6NGjVJkZKT69OmjqVOnXvAK1PCc0l4DUVGmpnw5RQv3LtS3p791Wz6g9QC9edebqhNcp9LnqYrhI0m5ubmKiYlRWFiYjhw5UvT4ww8/rNdff12bNm0qtidx5cqV+t3vfqdbb71VUVFRWrZsmWbNmlXhW8cAnkIA2ZAVAfTr+3MZDocaPLdYhn+g574hcAGlvQa+Sv6P+i0r/eymEdeM0IL+C0pdfrGqavgUWr58uQYNGqSJEyfq+eefL3r8q6++UteuXTVy5Ei9/fbbxZ43evRovfXWW5Kkfv36afXq1d4aGSg3ToOHR5nOPCUv+Vfx+3MRP6iCjqcev6Tl5VXVw6fQvHnzJEnDhg1ze7xLly5q0aKFVqxYoenTpxe7HMSAAQOKAshTbykDl4qDoFHpTFe+sr8tuLCg4R8gwz9A4f0fVIPJixTed5T8I7j9AqqmOy+/U35G6W/F9mt1ade+qUoHN5clLi5O69atU8eOHdWqVfGTEYYNG6bMzEwtW7bM7fHk5GSNGTNGISEhCgoK0tixY3Xu3DlvjQ2UGwGESvPr+3Odnj1RuccKjqGIHPKXi7o/F2CVZhHNNPKakq/IHV0rWg92eLBC261O4VNo/vz5crlcxfb+FCp8fO7cuW6Pjx49WidOnNCMGTM0ZcoUHTp0SI899pjH5wUuFscA2ZAnjgFK/3yF0v677Jf7c/W422unsAMX60KvgSMpR9RievEz5J6/6XlN7H5xN6utLm91nc80TTVv3lwnTpxQfHy8os//x/pZly5dtHXrVu3bt09t27bVnDlzNGbMGA0aNEjLli2TaZrq1auXPv300zJv0At4GwFkQ5UVQLnHD8ovrI78wqOUsfk/cqYkKuSGfrzFhSqvrNfAqA9Gad7ueUXLHIZDqU+lKiQopFzbr67hU+izzz7TbbfdpoiICA0YMKDU9Xbs2KE9e/Zo3Lhxeuihh9S+fXtFRUVpz549Cg8PlyT99NNPuvrqq+V0OrVnzx41btzYWz8GcEEEkA1dSgCdf3+usDuGK+yOoZ4ZFPCQsl4Dcalxuur1q5SakypJerTTo5rWa1qZ263u4VNo8ODBxY7tuZDw8HA1bdpUe/fu1YYNG4qdGv/RRx/prrvuUvfu3bVhwwavXvsKKA0BZEMVDaDs73YqZcUM5Z9JUGDztgrtcbctrt8D31Oe10Bcapw++f4TXVXvKnVu1PmC2/OV8AHshNPgcUHOlCS50pIVeFkrOULCFXhZK4WO9N4tKgCrNK7dWKM7jL7gOoQPUH0RQChR7vGDSt+wSll7NivwstaKfuxVBTZqoTojnrZ6NMByhA9Q/RFAcONMSVLywilF9+cK7/+ggn/b0+qxgCqB8AF8BwEEN35hEfILCVed+//O8T3AzwgfwPcQQHBj+Pmrzv1/t3oMoEogfADfRQBBknT6tNUTAN5T1v/fXaZLj619TDNiZ5S6DuEDVG8EECRJbdpYPQFQdWw6tqnU+CF8AN9AAAHAebKcWcUeI3wA38KFEG3I6ZSioqTUVKsnAaoGw5Dy8iS/n4/5d5kuDV01VMu+WaaY0BiN++04wgfwMQSQTS1aJI0dSwQBhiGNGye9+mrxZRm5GQoJLN/9vwBULwSQjTmdUkqK1VMA1oqM/GXPDwD7IIAAAIDtcEteAABgOwQQAACwHQIIAADYDgEEAABshwACAAC2QwABAADbIYAAAIDtEEAAAMB2CCAAAGA7BBAAALAdAggAANgOAQQAAGyHAAIAALZDAAEeNmLECBmGofr168vpdFo9DgBABBDgUWlpaVq5cqVq166tU6dOac2aNVaPBAAQAQR41NKlS5WZmampU6cqKChIc+fOtXokAIAIIMCj5s6dq8jISA0ZMkR9+vTRxx9/rISEhKLl6enpCg0N1ZVXXlni8/Pz8xUTE6O6desqNzfXW2MDgM8jgAAP2bdvn2JjY3XvvfcqMDBQw4cPV35+vhYsWFC0TmhoqAYPHqwDBw5oy5YtxbaxZs0aJSQkaOTIkQoMDPTm+ADg0wggwEMK3+4aNmyYJKl3796qU6eO5s2b57bemDFjJElvvfVWqdt44IEHPDkqANiOYZqmafUQgK/Jzc1VTEyMwsLCdOTIkaLHH374Yb3++uvatGmTunfvXvR4hw4ddPDgQSUkJCg0NFSS9NNPP6lx48bq3LmzNm/e7PWfAQB8GXuAAA9YvXq1zpw5o6FDh7o9Pnz4cEkqthfowQcf1Llz57R06dKix95++205nU72/gCAB7AHCPCAO+64Q59++qm+++47tWrVym1Zy5YtlZCQoISEBIWFhUmSMjIy1KBBA11xxRXavn27JOk3v/mNEhMTdfLkSQUHB3v9ZwAAX8YeIKCSxcXFad26derYsWOx+JEKjgnKzMzUsmXLih4LCQnRkCFDFBsbq71792rjxo06fPiwhg4dSvwAgAcQQEAlmz9/vlwuV9HBz+crfPz8awI9+OCDkgoOhubgZwDwLN4CAyqRaZpq3ry5Tpw4ofj4eEVHR5e4XpcuXbR161bt27dPbdu2LXr8uuuu05EjR5Sdna02bdpox44d3hodAGzF3+oBAF+yfv16HTt2TBEREZowYUKp62VlZUkq2Av02muvFT3+4IMPFp0Wz94fAPAc9gABlWjw4MFux/aUJSoqSvHx8UUXOczIyFBkZKQCAgLcDpIGAFQujgECKtHSpUtlmma5P5KSktyu8HzgwAHl5eVp0KBBxA8AeBABBFQhr7zyiiTpoYcesngSAPBtHAMEWOz48eNasmSJ9u/frxUrVuiOO+5Qp06drB4LAHwaxwABFtu4caNuuukmhYSE6Oabb9bs2bNVv359q8cCAJ9GAAEAANvhGCAAAGA7BBAAALAdAggAANgOAQQAAGyHAAIAALZDAAEAANshgAAAgO0QQAAAwHYIIAAAYDsEEAAAsB0CCAAA2A4BBAAAbIcAAgAAtkMAAQAA2/l/M1NpHWU2yCgAAAAASUVORK5CYII=" />
<p>The box-like volume in space with the top area, with normal <span class="math">$\hat{k}$</span>, designated as <span class="math">$S_1$</span>. The curve <span class="math">$C_1$</span> traces around <span class="math">$S_1$</span> in a counter clockwise manner, consistent with the right-hand rule pointing in the outward normal direction. The face <span class="math">$S_1$</span> with unit normal <span class="math">$\hat{k}$</span> looks like:</p>
<img src="data:image/png;base64,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" />
<p>Now we compute the <em>line integral</em>. Consider the top face, <span class="math">$S_1$</span>, connecting <span class="math">$(x,y,z+\Delta z), (x + \Delta x, y, z + \Delta z), (x + \Delta x, y + \Delta y, z + \Delta z), (x, y + \Delta y, z + \Delta z)$</span>, Using the <em>right hand rule</em>, parameterize the boundary curve, <span class="math">$C_1$</span>, in a counter clockwise direction so the right hand rule yields the outward pointing normal (<span class="math">$\hat{k}$</span>). Then the integral <span class="math">$\oint_{C_1} F\cdot \hat{T} ds$</span> is <em>approximated</em> by the following Riemann sum of <span class="math">$4$</span> terms:</p>
<p class="math">\[
~
F(x,y, z+\Delta{z}) \cdot \hat{i}\Delta{x} +
F(x+\Delta x, y, z+\Delta{z}) \cdot \hat{j} \Delta y +
F(x, y+\Delta y, z+\Delta{z}) \cdot (-\hat{i}) \Delta{x} +
F(x, y, z+\Delta{z}) \cdot (-\hat{j}) \Delta{y}.
~
\]</p>
<p>(The points <span class="math">$c_i$</span> are chosen from the endpoints of the line segments.)</p>
<p class="math">\[
~
\oint_{C_1} F\cdot \hat{T} ds \approx
(F_y(x+\Delta x, y, z+\Delta{z}) - F_y(x, y, z+\Delta{z})) \Delta{y} -
(F_x(x,y + \Delta{y}, z+\Delta{z}) - F_x(x, y, z+\Delta{z})) \Delta{x}
~
\]</p>
<p>As before, were this divided by the <em>area</em> of the surface, we have after rearranging and cancellation:</p>
<p class="math">\[
~
\frac{1}{\Delta{S_1}} \oint_{C_1} F \cdot \hat{T} ds \approx
\frac{F_y(x+\Delta x, y, z+\Delta{z}) - F_y(x, y, z+\Delta{z})}{\Delta{x}}
-
\frac{F_x(x, y+\Delta y, z+\Delta{z})-F_x(x, y, z+\Delta{z})}{\Delta{y}}.
~
\]</p>
<p>In the limit, as <span class="math">$\Delta{S} \rightarrow 0$</span>, this will converge to <span class="math">$\partial{F_y}/\partial{x}-\partial{F_x}/\partial{y}$</span>.</p>
<p>Had the bottom of the box been used, a similar result would be found, up to a minus sign.</p>
<p>Unlike the two dimensional case, there are other directions to consider and here the other sides will yield different answers. Consider now the face connecting <span class="math">$(x,y,z), (x+\Delta{x}, y, z), (x+\Delta{x}, y, z + \Delta{z})$</span>, and $ (x,y,z+\Delta{z})$ with outward pointing normal <span class="math">$-\hat{j}$</span>. Let <span class="math">$S_2$</span> denote this face and <span class="math">$C_2$</span> describe its boundary. Orient this curve so that the right hand rule points in the <span class="math">$-\hat{j}$</span> direction (the outward pointing normal). Then, as before, we can approximate:</p>
<p class="math">\[
~
\begin{align}
\oint_{C_2} F \cdot \hat{T} ds
&\approx
F(x,y,z) \cdot \hat{i} \Delta{x} +
F(x+\Delta{x},y,z) \cdot \hat{k} \Delta{z} +
F(x,y,z+\Delta{z}) \cdot (-\hat{i}) \Delta{x} +
F(x, y, z) \cdot (-\hat{k}) \Delta{z}\\
&= (F_z(x+\Delta{x},y,z) - F_z(x, y, z))\Delta{z} -
(F_x(x,y,z+\Delta{z}) - F(x,y,z)) \Delta{x}.
\end{align}
~
\]</p>
<p>Dividing by <span class="math">$\Delta{S}=\Delta{x}\Delta{z}$</span> and taking a limit will give:</p>
<p class="math">\[
~
\lim \frac{1}{\Delta{S}} \oint_{C_2} F \cdot \hat{T} ds =
\frac{\partial{F_z}}{\partial{x}} - \frac{\partial{F_x}}{\partial{z}}.
~
\]</p>
<p>Had, the opposite face with outward normal <span class="math">$\hat{j}$</span> been chosen, the answer would differ by a factor of <span class="math">$-1$</span>.</p>
<p>Similarly, let <span class="math">$S_3$</span> be the face with outward normal <span class="math">$\hat{i}$</span> and curve <span class="math">$C_3$</span> bounding it with parameterization chosen so that the right hand rule points in the direction of <span class="math">$\hat{i}$</span>. This will give</p>
<p class="math">\[
~
\lim \frac{1}{\Delta{S}} \oint_{C_3} F \cdot \hat{T} ds =
\frac{\partial{F_z}}{\partial{y}} - \frac{\partial{F_y}}{\partial{z}}.
~
\]</p>
<p>In short, depending on the face chosen, a different answer is given, but all have the same type.</p>
<blockquote>
<p>Define the <em>curl</em> of a <span class="math">$3$</span>-dimensional vector field <span class="math">$F=\langle F_x,F_y,F_z\rangle$</span> by: <span class="math">$~ \text{curl}(F) = \langle \frac{\partial{F_z}}{\partial{y}} - \frac{\partial{F_y}}{\partial{z}}, \frac{\partial{F_x}}{\partial{z}} - \frac{\partial{F_z}}{\partial{x}}, \frac{\partial{F_y}}{\partial{x}} - \frac{\partial{F_x}}{\partial{y}} \rangle. ~$</span></p>
</blockquote>
<p>If <span class="math">$S$</span> is some surface with closed boundary <span class="math">$C$</span> oriented so that the unit normal, <span class="math">$\hat{N}$</span>, of <span class="math">$S$</span> is given by the right hand rule about <span class="math">$C$</span>, then</p>
<p class="math">\[
~
\hat{N} \cdot \text{curl}(F) = \lim \frac{1}{\Delta{S}} \oint_C F \cdot \hat{T} ds.
~
\]</p>
<p>The curl has a formal representation in terms of a <span class="math">$3\times 3$</span> determinant, similar to that used to compute the cross product, that is useful for computation:</p>
<p class="math">\[
~
\text{curl}(F) = \det\left[
\begin{array}{}
\hat{i} & \hat{j} & \hat{k}\\
\frac{\partial}{\partial{x}} & \frac{\partial}{\partial{y}} & \frac{\partial}{\partial{z}}\\
F_x & F_y & F_z
\end{array}
\right]
~
\]</p>
<hr />
<p>In <code>Julia</code>, the curl can be implemented different ways depending on how the problem is presented. We will use the Jacobian matrix to compute the required partials. If the Jacobian is known, this function from the <code>CalculusWithJulia</code> package will combine the off-diagonal terms appropriately:</p>
<pre class='hljl'>
<span class='hljl-k'>function</span><span class='hljl-t'> </span><span class='hljl-nf'>curl</span><span class='hljl-p'>(</span><span class='hljl-n'>J</span><span class='hljl-oB'>::</span><span class='hljl-n'>Matrix</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>Mx</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Nx</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Px</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>My</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Ny</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Py</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Mz</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Nz</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Pz</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>J</span><span class='hljl-t'>
</span><span class='hljl-p'>[</span><span class='hljl-n'>Py</span><span class='hljl-oB'>-</span><span class='hljl-n'>Nz</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Mz</span><span class='hljl-oB'>-</span><span class='hljl-n'>Px</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>Nx</span><span class='hljl-oB'>-</span><span class='hljl-n'>My</span><span class='hljl-p'>]</span><span class='hljl-t'> </span><span class='hljl-cs'># ∇×VF</span><span class='hljl-t'>
</span><span class='hljl-k'>end</span>
</pre>
<p>The computation of the Jacobian differs whether the problem is treated numerically or symbolically. Here are two functions:</p>
<pre class='hljl'>
<span class='hljl-nf'>curl</span><span class='hljl-p'>(</span><span class='hljl-n'>F</span><span class='hljl-oB'>::</span><span class='hljl-nf'>Vector</span><span class='hljl-p'>{</span><span class='hljl-n'>Sym</span><span class='hljl-p'>},</span><span class='hljl-t'> </span><span class='hljl-n'>vars</span><span class='hljl-oB'>=</span><span class='hljl-nf'>free_symbols</span><span class='hljl-p'>(</span><span class='hljl-n'>F</span><span class='hljl-p'>))</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>curl</span><span class='hljl-p'>(</span><span class='hljl-n'>F</span><span class='hljl-oB'>.</span><span class='hljl-nf'>jacobian</span><span class='hljl-p'>(</span><span class='hljl-n'>vars</span><span class='hljl-p'>))</span><span class='hljl-t'>
</span><span class='hljl-nf'>curl</span><span class='hljl-p'>(</span><span class='hljl-n'>F</span><span class='hljl-oB'>::</span><span class='hljl-n'>Function</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>pt</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>curl</span><span class='hljl-p'>(</span><span class='hljl-n'>ForwardDiff</span><span class='hljl-oB'>.</span><span class='hljl-nf'>jacobian</span><span class='hljl-p'>(</span><span class='hljl-n'>F</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>pt</span><span class='hljl-p'>))</span>
</pre>
<h3>The <span class="math">$\nabla$</span> (del) operator</h3>
<p>The divergence, gradient, and curl all involve partial derivatives. There is a notation employed that can express the operations more succinctly. Let the <a href="https://en.wikipedia.org/wiki/Del">Del operator</a> be defined in Cartesian coordinates by the formal expression:</p>
<blockquote>
<p class="math">\[
~
\nabla = \langle
\frac{\partial}{\partial{x}},
\frac{\partial}{\partial{y}},
\frac{\partial}{\partial{z}}
\rangle.
~
\]</p>
</blockquote>
<p>This is a <em>vector differential operator</em> that acts on functions and vector fields through the typical notation to yield the three operations:</p>
<p class="math">\[
~
\begin{align}
\nabla{f} &= \langle
\frac{\partial{f}}{\partial{x}},
\frac{\partial{f}}{\partial{y}},
\frac{\partial{f}}{\partial{z}}
\rangle, \quad\text{the gradient;}\\
\nabla\cdot{F} &= \langle
\frac{\partial}{\partial{x}},
\frac{\partial}{\partial{y}},
\frac{\partial}{\partial{z}}
\rangle \cdot F =
\langle
\frac{\partial}{\partial{x}},
\frac{\partial}{\partial{y}},
\frac{\partial}{\partial{z}}
\rangle \cdot
\langle F_x, F_y, F_z \rangle =
\frac{\partial{F_x}}{\partial{x}} +
\frac{\partial{F_y}}{\partial{y}} +
\frac{\partial{F_z}}{\partial{z}},\quad\text{the divergence;}\\
\nabla\times F &= \langle
\frac{\partial}{\partial{x}},
\frac{\partial}{\partial{y}},
\frac{\partial}{\partial{z}}
\rangle \times F =
\det\left[
\begin{array}{}
\hat{i} & \hat{j} & \hat{k} \\
\frac{\partial}{\partial{x}}&
\frac{\partial}{\partial{y}}&
\frac{\partial}{\partial{z}}\\
F_x & F_y & F_z
\end{array}
\right],\quad\text{the curl}.
\end{align}
~
\]</p>
<div class="alert alert-info" role="alert">
<div class="markdown"><p>Mathematically operators have not been seen previously, but the concept of an operation on a function that returns another function is a common one when using <code>Julia</code>. We have seen many examples (<code>plot</code>, <code>D</code>, <code>quadgk</code>, etc.). In computer science such functions are called <em>higher order</em> functions, as they accept arguments which are also functions.</p>
</div>
</div>
<hr />
<p>In the <code>CalculusWithJulia</code> package, the constant <code>\nabla[\tab]</code>, producing <span class="math">$\nabla$</span> implements this operator for functions and symbolic expressions.</p>
<pre class='hljl'>
<span class='hljl-nf'>f</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-oB'>*</span><span class='hljl-n'>y</span><span class='hljl-oB'>*</span><span class='hljl-n'>z</span><span class='hljl-t'>
</span><span class='hljl-nf'>f</span><span class='hljl-p'>(</span><span class='hljl-n'>v</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>f</span><span class='hljl-p'>(</span><span class='hljl-n'>v</span><span class='hljl-oB'>...</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>F</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-nf'>F</span><span class='hljl-p'>(</span><span class='hljl-n'>v</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>F</span><span class='hljl-p'>(</span><span class='hljl-n'>v</span><span class='hljl-oB'>...</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-t'>
</span><span class='hljl-nf'>∇</span><span class='hljl-p'>(</span><span class='hljl-nf'>f</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>))</span><span class='hljl-t'> </span><span class='hljl-cs'># symbolic operation on the symbolic expression f(x,y,z)</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}y z\\x z\\x y\end{array} \right] \]</div>
<p>This usage of <code>∇</code> takes partial derivatives according to the order given by:</p>
<pre class='hljl'>
<span class='hljl-nf'>free_symbols</span><span class='hljl-p'>(</span><span class='hljl-nf'>f</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>))</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}x\\y\\z\end{array} \right] \]</div>
<p>which may <strong>not</strong> be as desired. In this case, the variables can be specified using a tuple to pair up the expression with the variables to differentiate against:</p>
<pre class='hljl'>
<span class='hljl-nf'>∇</span><span class='hljl-p'>(</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-nf'>f</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>])</span><span class='hljl-t'> </span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}y z\\x z\\x y\end{array} \right] \]</div>
<p>For numeric expressions, we have:</p>
<pre class='hljl'>
<span class='hljl-nf'>∇</span><span class='hljl-p'>(</span><span class='hljl-n'>f</span><span class='hljl-p'>)(</span><span class='hljl-ni'>1</span><span class='hljl-p'>,</span><span class='hljl-ni'>2</span><span class='hljl-p'>,</span><span class='hljl-ni'>3</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-cs'># a numeric computation. Also can call with a point [1,2,3]</span>
</pre>
<pre class="output">
3-element Array{Int64,1}:
6
3
2
</pre>
<p>(The extra parentheses are unfortunate. Here <code>∇</code> is called like a function.)</p>
<p>The divergence can be found symbolically:</p>
<pre class='hljl'>
<span class='hljl-n'>∇</span><span class='hljl-t'> </span><span class='hljl-oB'>⋅</span><span class='hljl-t'> </span><span class='hljl-nf'>F</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}3\end{equation*}</div>
<p>Or numerically:</p>
<pre class='hljl'>
<span class='hljl-p'>(</span><span class='hljl-n'>∇</span><span class='hljl-t'> </span><span class='hljl-oB'>⋅</span><span class='hljl-t'> </span><span class='hljl-n'>F</span><span class='hljl-p'>)(</span><span class='hljl-ni'>1</span><span class='hljl-p'>,</span><span class='hljl-ni'>2</span><span class='hljl-p'>,</span><span class='hljl-ni'>3</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-cs'># a numeric computation. Also can call (∇ ⋅ F)([1,2,3])</span>
</pre>
<pre class="output">
3.0
</pre>
<p>Similarly, the curl. Symbolically:</p>
<pre class='hljl'>
<span class='hljl-n'>∇</span><span class='hljl-t'> </span><span class='hljl-oB'>×</span><span class='hljl-t'> </span><span class='hljl-nf'>F</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}0\\0\\0\end{array} \right] \]</div>
<p>and numerically:</p>
<pre class='hljl'>
<span class='hljl-p'>(</span><span class='hljl-n'>∇</span><span class='hljl-t'> </span><span class='hljl-oB'>×</span><span class='hljl-t'> </span><span class='hljl-n'>F</span><span class='hljl-p'>)(</span><span class='hljl-ni'>1</span><span class='hljl-p'>,</span><span class='hljl-ni'>2</span><span class='hljl-p'>,</span><span class='hljl-ni'>3</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-cs'># numeric. Also can call (∇ × F)([1,2,3])</span>
</pre>
<pre class="output">
3-element Array{Float64,1}:
0.0
0.0
0.0
</pre>
<p>There is a subtle difference in usage. Symbolically the evaluation of <code>F(x,y,z)</code> first is desired, numerically the evaluation of <code>∇ ⋅ F</code> or <code>∇ × F</code> first is desired. As <code>⋅</code> and <code>×</code> have lower precedence than function evaluation, parentheses must be used in the numeric case.</p>
<div class="alert alert-info" role="alert">
<div class="markdown"><p>As mentioned, for the symbolic evaluations, a specification of three variables (here <code>x</code>, <code>y</code>, and <code>z</code>) is necessary. This use takes <code>free_symbols</code> to identify three free symbols which may not always be the case. (It wouldn't be for, say, <code>F(x,y,z) = [a*x,b*y,0]</code>, <code>a</code> and <code>b</code> constants.) In those cases, the notation accepts a tuple to specify the function or vector field and the variables, e.g. (<code>∇( (f(x,y,z), [x,y,z]) )</code>, as illustrated; <code>∇ × (F(x,y,z), [x,y,z])</code>; or <code>∇ ⋅ (F(x,y,z), [x,y,z])</code> where this is written using function calls to produce the symbolic expression in the first positional argument, though a direct expression could also be used. In these cases, the named versions <code>gradient</code>, <code>curl</code>, and <code>divergence</code> may be preferred.</p>
</div>
</div>
<h2>Interpretation</h2>
<p>The divergence and curl measure complementary aspects of a vector field. The divergence is defined in terms of flow out of an infinitesimal box, the curl is about rotational flow around an infinitesimal area patch.</p>
<p>Let <span class="math">$F(x,y,z) = [x, 0, 0]$</span>, a vector field pointing in just the <span class="math">$\hat{i}$</span> direction. The divergence is simply <span class="math">$1$</span>. If <span class="math">$V$</span> is a box, as in the derivation, then the divergence measures the flow into the side with outward normal <span class="math">$-\hat{i}$</span> and through the side with outward normal <span class="math">$\hat{i}$</span> which will clearly be positive as the flow passes through the region <span class="math">$V$</span>, increasing as <span class="math">$x$</span> increases, when <span class="math">$x > 0$</span>.</p>
<p>The radial vector field <span class="math">$F(x,y,z) = \langle x, y, z \rangle$</span> is also an example of a divergent field. The divergence is:</p>
<pre class='hljl'>
<span class='hljl-nf'>F</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-t'> </span><span class='hljl-n'>real</span><span class='hljl-oB'>=</span><span class='hljl-kc'>true</span><span class='hljl-t'>
</span><span class='hljl-n'>∇</span><span class='hljl-t'> </span><span class='hljl-oB'>⋅</span><span class='hljl-t'> </span><span class='hljl-nf'>F</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}3\end{equation*}</div>
<p>There is a constant outward flow, emanating from the origin. Here we picture the field when <span class="math">$z=0$</span>:</p>
<img src="data:image/png;base64,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" />
<p>Consider the limit definition of the divergence:</p>
<p class="math">\[
~
\nabla\cdot{F} = \lim \frac{1}{\Delta{V}} \oint_S F\cdot\hat{N} dA.
~
\]</p>
<p>In the vector field above, the shape along the curved edges has constant magnitude field. On the left curved edge, the length is smaller and the field is smaller than on the right. The flux across the left edge will be less than the flux across the right edge, and a net flux will exist. That is, there is divergence.</p>
<p>Now, were the field on the right edge less, it might be that the two balance out and there is no divergence. This occurs with the inverse square laws, such as for gravity and electric field:</p>
<pre class='hljl'>
<span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-t'> </span><span class='hljl-n'>real</span><span class='hljl-oB'>=</span><span class='hljl-kc'>true</span><span class='hljl-t'>
</span><span class='hljl-n'>R</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-n'>Rhat</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>R</span><span class='hljl-oB'>/</span><span class='hljl-nf'>norm</span><span class='hljl-p'>(</span><span class='hljl-n'>R</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>VF</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-ni'>1</span><span class='hljl-oB'>/</span><span class='hljl-nf'>norm</span><span class='hljl-p'>(</span><span class='hljl-n'>R</span><span class='hljl-p'>)</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>*</span><span class='hljl-t'> </span><span class='hljl-n'>Rhat</span><span class='hljl-t'>
</span><span class='hljl-n'>∇</span><span class='hljl-t'> </span><span class='hljl-oB'>⋅</span><span class='hljl-t'> </span><span class='hljl-n'>VF</span><span class='hljl-t'> </span><span class='hljl-oB'>|></span><span class='hljl-t'> </span><span class='hljl-n'>simplify</span>
</pre>
<div class="well well-sm">\begin{equation*}0\end{equation*}</div>
<hr />
<p>The vector field <span class="math">$F(x,y,z) = \langle -y, x, 0 \rangle$</span> is an example of a rotational field. It's curl can be computed symbolically through:</p>
<pre class='hljl'>
<span class='hljl-nf'>curl</span><span class='hljl-p'>([</span><span class='hljl-oB'>-</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-ni'>0</span><span class='hljl-p'>],</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>])</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}0\\0\\2\end{array} \right] \]</div>
<p>This vector field rotates as seen in this figure showing slices for different values of <span class="math">$z$</span>:</p>
<img src="data:image/png;base64,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" />
<p>The field has a clear rotation about the <span class="math">$z$</span> axis (illustrated with a line), the curl is a vector that points in the direction of the <em>right hand</em> rule as the right hand fingers follow the flow with magnitude given by the amount of rotation.</p>
<p>This is a bit misleading though, the curl is defined by a limit, and not in terms of a large box. The key point for this field is that the strength of the field is stronger as the points get farther away, so for a properly oriented small box, the integral along the closer edge will be less than that along the outer edge.</p>
<p>Consider a related field where the strength gets smaller as the point gets farther away but otherwise has the same circular rotation pattern</p>
<pre class='hljl'>
<span class='hljl-n'>R</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-oB'>-</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>0</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-n'>VF</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-n'>R</span><span class='hljl-t'> </span><span class='hljl-oB'>/</span><span class='hljl-t'> </span><span class='hljl-nf'>norm</span><span class='hljl-p'>(</span><span class='hljl-n'>R</span><span class='hljl-p'>)</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'>
</span><span class='hljl-nf'>curl</span><span class='hljl-p'>(</span><span class='hljl-n'>VF</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>])</span><span class='hljl-t'> </span><span class='hljl-oB'>.|></span><span class='hljl-t'> </span><span class='hljl-n'>simplify</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}0\\0\\0\end{array} \right] \]</div>
<p>Further, the curl of <code>R/norm(R)^3</code> now points in the <em>opposite</em> direction of the curl of <code>R</code>. This example isn't typical, as dividing by <code>norm(R)</code> with a power greater than <span class="math">$1$</span> makes the vector field discontinuous at the origin.</p>
<p>The curl of the vector field <span class="math">$F(x,y,z) = \langle 0, 1+y^2, 0\rangle$</span> is <span class="math">$0$</span>, as there is clearly no rotation as seen in this slice where <span class="math">$z=0$</span>:</p>
<img src="data:image/png;base64,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" />
<p>Algebraically, this is so:</p>
<pre class='hljl'>
<span class='hljl-nf'>curl</span><span class='hljl-p'>(</span><span class='hljl-n'>Sym</span><span class='hljl-p'>[</span><span class='hljl-ni'>0</span><span class='hljl-p'>,</span><span class='hljl-ni'>1</span><span class='hljl-oB'>+</span><span class='hljl-n'>y</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-p'>,</span><span class='hljl-ni'>0</span><span class='hljl-p'>],</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>])</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}0\\0\\0\end{array} \right] \]</div>
<p>Now consider a similar field <span class="math">$F(x,y,z) = \langle 0, 1+x^2, 0,\rangle$</span>. A slice is somewhat similar, in that the flow lines are all in the <span class="math">$\hat{j}$</span> direction:</p>
<img src="data:image/png;base64,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" />
<p>However, this vector field has a curl:</p>
<pre class='hljl'>
<span class='hljl-nf'>curl</span><span class='hljl-p'>([</span><span class='hljl-ni'>0</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>1</span><span class='hljl-oB'>+</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-p'>,</span><span class='hljl-ni'>0</span><span class='hljl-p'>],</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>])</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}0\\0\\2 x\end{array} \right] \]</div>
<p>The curl points in the <span class="math">$\hat{k}$</span> direction (out of the figure). A useful visualization is to mentally place a small paddlewheel at a point and imagine if it will turn. In the constant field case, there is equal flow on both sides of the axis, so it any forces on the wheel blades will balance out. In the latter example, if <span class="math">$x > 0$</span>, the force on the right side will be greater than the force on the left so the paddlewheel would rotate counter clockwise. The right hand rule for this rotation will point in the upward, or <span class="math">$\hat{k}$</span> direction, as seen algebraically in the curl.</p>
<p>Following Strang, in general the curl can point in any direction, so the amount the paddlewheel will spin will be related to how the paddlewheel is oriented. The angular velocity of the wheel will be <span class="math">$(1/2)(\nabla\times{F})\cdot\hat{N}$</span>, <span class="math">$\hat{N}$</span> being the normal for the paddlewheel.</p>
<p>If <span class="math">$\vec{a}$</span> is some vector and <span class="math">$\hat{r} = \langle x, y, z\rangle$</span> is the radial vector, then <span class="math">$\vec{a} \times \vec{r}$</span> has a curl, which is given by:</p>
<pre class='hljl'>
<span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-t'> </span><span class='hljl-n'>a1</span><span class='hljl-t'> </span><span class='hljl-n'>a2</span><span class='hljl-t'> </span><span class='hljl-n'>a3</span><span class='hljl-t'>
</span><span class='hljl-n'>a</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>a1</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>a2</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>a3</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-n'>r</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-p'>]</span><span class='hljl-t'>
</span><span class='hljl-nf'>curl</span><span class='hljl-p'>(</span><span class='hljl-n'>a</span><span class='hljl-t'> </span><span class='hljl-oB'>×</span><span class='hljl-t'> </span><span class='hljl-n'>r</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-p'>])</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}2 a_{1}\\2 a_{2}\\2 a_{3}\end{array} \right] \]</div>
<p>The angular velocity then is <span class="math">$\vec{a} \cdot \hat{N}$</span>. The curl is constant. As the dot product involves the cosine of the angle between the two vectors, we see the turning speed is largest when <span class="math">$\hat{N}$</span> is parallel to <span class="math">$\vec{a}$</span>. This gives a similar statement for the curl like the gradient does for steepest growth rate: the maximum rotation rate of <span class="math">$F$</span> is <span class="math">$(1/2)\|\nabla\times{F}\|$</span> in the direction of <span class="math">$\nabla\times{F}$</span>.</p>
<p>The curl of the radial vector field, <span class="math">$F(x,y,z) = \langle x, y, z\rangle$</span> will be <span class="math">$\vec{0}$</span>:</p>
<pre class='hljl'>
<span class='hljl-nf'>curl</span><span class='hljl-p'>([</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>],</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-n'>z</span><span class='hljl-p'>])</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}0\\0\\0\end{array} \right] \]</div>
<p>We will see that this can be anticipated, as <span class="math">$F = (1/2) \nabla(x^2+y^2+z^2)$</span> is a gradient field.</p>
<p>In fact, the curl of any radial field will be <span class="math">$\vec{0}$</span>. Here we represent a radial field as a scalar function of <span class="math">$\vec{r}$</span> time <span class="math">$\hat{r}$</span>:</p>
<pre class='hljl'>
<span class='hljl-n'>H</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>SymFunction</span><span class='hljl-p'>(</span><span class='hljl-s'>"H"</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>R</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>sqrt</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>Rhat</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-p'>]</span><span class='hljl-oB'>/</span><span class='hljl-n'>R</span><span class='hljl-t'>
</span><span class='hljl-nf'>curl</span><span class='hljl-p'>(</span><span class='hljl-nf'>H</span><span class='hljl-p'>(</span><span class='hljl-n'>R</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>*</span><span class='hljl-t'> </span><span class='hljl-n'>Rhat</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-p'>[</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-p'>])</span>
</pre>
<div class="well well-sm">\[ \left[ \begin{array}{r}0\\0\\0\end{array} \right] \]</div>
<p>Were one to represent the curl in <a href="https://en.wikipedia.org/wiki/Del_in_cylindrical_and_spherical_coordinates">spherical</a> coordinates (below), this follows algebraically from the formula easily enough. To anticipate this, due to symmetry, the curl would need to be the same along any ray emanating from the origin and again by symmetry could only possible point along the ray. Mentally place a paddlewheel along the <span class="math">$x$</span> axis oriented along <span class="math">$\hat{i}$</span>. There will be no rotational forces that could make the wheel spin around the <span class="math">$x$</span>-axis, hence the curl must be <span class="math">$0$</span>.</p>
<h2>The Maxwell equations</h2>
<p>The divergence and curl appear in <a href="https://en.wikipedia.org/wiki/Maxwell%27s_equations">Maxwell</a>'s equations describing the relationships of electromagnetism. In the formulas below the notation is <span class="math">$E$</span> is the electric field; <span class="math">$B$</span> is the magnetic field; <span class="math">$\rho$</span> is the charge <em>density</em> (charge per unit volume); <span class="math">$J$</span> the electric current density (current per unit area); and <span class="math">$\epsilon_0$</span>, <span class="math">$\mu_0$</span>, and <span class="math">$c$</span> are universal constants.</p>
<p>The equations in differential form are:</p>
<blockquote>
<p>Gauss's law: <span class="math">$\nabla\cdot{E} = \rho/\epsilon_0$</span>.</p>
</blockquote>
<p>That is, the divergence of the electric field is proportional to the density. We have already mentioned this in <em>integral</em> form.</p>
<blockquote>
<p>Gauss's law of magnetism: <span class="math">$\nabla\cdot{B} = 0$</span></p>
</blockquote>
<p>The magnetic field has no divergence. This says that there no magnetic charges (a magnetic monopole) unlike electric charge, according to Maxwell's laws.</p>
<blockquote>
<p>Faraday's law of induction: <span class="math">$\nabla\times{E} = - \partial{B}/\partial{t}$</span>.</p>
</blockquote>
<p>The curl of the <em>time-varying</em> electric field is in the direction of the partial derivative of the magnetic field. For example, if a magnet is in motion in the in the <span class="math">$z$</span> axis, then the electric field has rotation in the <span class="math">$x-y$</span> plane <em>induced</em> by the motion of the magnet.</p>
<blockquote>
<p>Ampere's circuital law: <span class="math">$\nabla\times{B} = \mu_0J + \mu_0\epsilon_0 \partial{E}/\partial{t}$</span></p>
</blockquote>
<p>The curl of the magnetic field is related to the sum of the electric current density and the change in time of the electric field.</p>
<hr />
<p>In a region with no charges (<span class="math">$\rho=0$</span>) and no currents (<span class="math">$J=\vec{0}$</span>), such as a vacuum, these equations reduce to two divergences being <span class="math">$0$</span>: <span class="math">$\nabla\cdot{E} = 0$</span> and <span class="math">$\nabla\cdot{B}=0$</span>; and two curl relationships with time derivatives: <span class="math">$\nabla\times{E}= -\partial{B}/\partial{t}$</span> and <span class="math">$\nabla\times{B} = \mu_0\epsilon_0 \partial{E}/\partial{t}$</span>.</p>
<p>We will see later how these are differential forms are consequences of related integral forms.</p>
<h2>Algebra of vector calculus</h2>
<p>The divergence, gradient, and curl satisfy several algebraic <a href="https://en.wikipedia.org/wiki/Vector_calculus_identities">properties</a>.</p>
<p>Let <span class="math">$f$</span> and <span class="math">$g$</span> denote scalar functions, <span class="math">$R^3 \rightarrow R$</span> and <span class="math">$F$</span> and <span class="math">$G$</span> be vector fields, <span class="math">$R^3 \rightarrow R^3$</span>.</p>
<h3>Linearity</h3>
<p>As with the sum rule of univariate derivatives, these operations satisfy:</p>
<p class="math">\[
~
\begin{align}
\nabla(f + g) &= \nabla{f} + \nabla{g}\\
\nabla\cdot(F+G) &= \nabla\cdot{F} + \nabla\cdot{G}\\
\nabla\times(F+G) &= \nabla\times{F} + \nabla\times{G}.
\end{align}
~
\]</p>
<h3>Product rule</h3>
<p>The product rule <span class="math">$(uv)' = u'v + uv'$</span> has related formulas:</p>
<p class="math">\[
~
\begin{align}
\nabla{(fg)} &= (\nabla{f}) g + f\nabla{g} = g\nabla{f} + f\nabla{g}\\
\nabla\cdot{fF} &= (\nabla{f})\cdot{F} + f(\nabla\cdot{F})\\
\nabla\times{fF} &= (\nabla{f})\times{F} + f(\nabla\times{F}).
\end{align}
~
\]</p>
<h3>Rules over cross products</h3>
<p>The cross product of two vector fields is a vector field for which the divergence and curl may be taken. There are formulas to relate to the individual terms:</p>
<p class="math">\[
~
\begin{align}
\nabla\cdot(F \times G) &= (\nabla\times{F})\cdot G - F \cdot (\nabla\times{G})\\
\nabla\times(F \times G) &= F(\nabla\cdot{G}) - G(\nabla\cdot{F} + (G\cdot\nabla)F-(F\cdot\nabla)G\\
&= \nabla\cdot(BA^t - AB^t).
\end{align}
~
\]</p>
<p>The curl formula is more involved.</p>
<h3>Vanishing properties</h3>
<p>Surprisingly, the curl and divergence satisfy two vanishing properties. First</p>
<blockquote>
<p>The curl of a gradient field is <span class="math">$\vec{0}$</span> <span class="math">$~ \nabla \times \nabla{f} = \vec{0}, ~$</span></p>
</blockquote>
<p>if the scalar function <span class="math">$f$</span> is has continuous second derivatives (so the mixed partials do not depend on order).</p>
<p>Vector fields where <span class="math">$F = \nabla{f}$</span> are conservative. Conservative fields have path independence, so any line integral, <span class="math">$\oint F\cdot \hat{T} ds$</span>, around a closed loop will be <span class="math">$0$</span>. But the curl is defined as a limit of such integrals, so it too will be <span class="math">$\vec{0}$</span>. In short, conservative fields have no rotation.</p>
<p>What about the converse? If a vector field has zero curl, then integrals around infinitesimally small loops are <span class="math">$0$</span>. Does this <em>also</em> mean that integrals around larger closed loops will also be <span class="math">$0$</span>, and hence the field is conservative? The answer will be yes, <em>under assumptions</em>. But the discussion will wait for later.</p>
<p>The combination <span class="math">$\nabla\cdot\nabla{f}$</span> is defined and is called the Laplacian. This is denoted <span class="math">$\Delta{f}$</span>. The equation <span class="math">$\Delta{f} = 0$</span> is called Laplace's equation. It is <em>not</em> guaranteed for any scalar function <span class="math">$f$</span>, but the <span class="math">$f$</span> for which it holds are important.</p>
<p>Second,</p>
<blockquote>
<p>The divergence of a curl field is <span class="math">$0$</span>: <span class="math">$~ \nabla \cdot(\nabla\times{F}) = 0. ~$</span></p>
</blockquote>
<p>This is not as clear, but can be seen algebraically as terms cancel. First:</p>
<p class="math">\[
~
\nabla\cdot(\nabla\times{F}) =
\langle
\frac{\partial}{\partial{x}},
\frac{\partial}{\partial{y}},
\frac{\partial}{\partial{z}}\rangle \cdot
\langle
\frac{\partial{F_z}}{\partial{y}} - \frac{\partial{F_y}}{\partial{z}},
\frac{\partial{F_x}}{\partial{z}} - \frac{\partial{F_z}}{\partial{x}},
\frac{\partial{F_y}}{\partial{x}} - \frac{\partial{F_x}}{\partial{y}}
\rangle
=
\left(\frac{\partial^2{F_z}}{\partial{y}\partial{x}} - \frac{\partial^2{F_y}}{\partial{z}\partial{x}}\right) +
\left(\frac{\partial^2{F_x}}{\partial{z}\partial{y}} - \frac{\partial^2{F_z}}{\partial{x}\partial{y}}\right) +
\left(\frac{\partial^2{F_y}}{\partial{x}\partial{z}} - \frac{\partial^2{F_x}}{\partial{y}\partial{z}}\right)
~
\]</p>
<p>Focusing on one component function, <span class="math">$F_z$</span> say, we see this contribution:</p>
<p class="math">\[
~
\frac{\partial^2{F_z}}{\partial{y}\partial{x}} -
\frac{\partial^2{F_z}}{\partial{x}\partial{y}}.
~
\]</p>
<p>This is zero under the assumption that the second partial derivatives are continuous.</p>
<p>From the microscopic picture of a box this can also be seen. Again we focus on just the appearance of the <span class="math">$F_z$</span> component function. Let the faces with normals <span class="math">$\hat{i}, \hat{j},-\hat{i}, -\hat{j}$</span> be labeled <span class="math">$A, B, C$</span>, and <span class="math">$D$</span>. This figure shows <span class="math">$A$</span> (enclosed in blue) and <span class="math">$B$</span> (enclosed in green):</p>
<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAkAAAAGACAIAAADK+EpIAAAABmJLR0QA/wD/AP+gvaeTAAAcNklEQVR4nO3de3hU5Z3A8d+E3EgCuUC4iNwSQAQBJbFWvGC7iau1F7UmdqUqpT5AV1crtiSWalV0zXSfPurWqmT7VK3tbg3x0dq6LmSoGpWLZvCCCAoZLspFwclAArnP2T8ODBM4IcnczvvOfD9/Tc4hyavPmfebOe/kjcMwDAEAQDdJdg8AAIBQEDAAgJYIGABASwQMAKAlAgYA0BIBAwBoiYABALREwAAAWiJgAAAtETAAgJYIGABASwQMAKAlAgYA0BIBAwBoiYABALREwAAAWiJgAAAtETAAgJYIGABASwQMAKAlAgYA0BIBAwBoiYABALREwAAAWiJgAAAtETAAgJYIGABASwQMAKAlAgYA0BIBAwBoiYABALREwAAAWiJgAAAtETAAgJYIGABAS8l2DwDoB59PGhrsHgQQaWeeKVOn2j0IjREwqK25WZ54QpxOaWqyeyhA5IwdK3fdJRdfbPc49EbAoCqfTx57TB59VHw+u4cCRM64cXL33bJggaSm2j0U7REwqMcyXZMmyYQJtg0JGJCjR2XdOjGMHgdJV6QRMKjE8oZhQYFUVMiCBZLM5QrltbfLs8/K/ff3qJd5w3DRIklPt29kcYgZAWogXdBdIF179544SLqiiXkBdiNd0F2o6drp2/nwWw/fPOvmOWPnxGKccYfZAfYhXdBdeOn6w3t/6PJ3zZsxLxZDjUfMEbAD6YLuQk3Xdu/2B+sf/POmP3f5u2IxzrjGTIHYIl3QHelSBvMFYoV0QXcRTVdKUkqnvzOq4417zBqIPtIF3UVirStw8KKxFy3/5vL7Xr+vfld9VEcd95g7EE2kC7qLQrq+MeEbInKf3BfFYScGZhBERyTSteeI8aftRsUs/mYC7BC1dCFSCBgiLXKvug60yf80+gkYYo10aYKAIXK4YQjdkS6tMKcgEkgXdEe6NMTMgvCQLuiOdGmL+QWhIl3QHenSHLMMBi5W6cpPlxsm8Q4ORAHpigsEDAMR21ddYzIdS2c6Ivs1kehIVxwhYOgfbhhCd6Qr7jDvoC+kC7ojXXGK2Qe9sztdnx8x/rjN+MW5LIMhVKQrrhEwWLE7XaaDbVLj8RMwhIJ0JQAChp7USBcQOtKVMJiPcBzpgu5IV4JhVgLpgv5IV0JibkpspAu6I10JjBkqUemQrhGD5cbJvIMDvSBdCU+JeQoxpUO6TGdkOO6awU4cOAXpgogQsMSiT7oAa6QLQZizEgPpgu5IF07BzBXvdE7XZ0eMZz417jmPZbDERrrQC6XnL4RF53SZvmqTF3b4CVjiIl04LQ1mMQyY/ulCoiNd6AfmsvhCuqA70oV+Y0aLF6QLuiNdGCDmNf2RLuiOdCEkzG46i/d0jRzsmD+Fd3DENdKFMGg/xyWoeE+XaXSG/PQcAhanSBfCFiczXQJJjHQhnpEuRAjznT5IF3RHuhBRzHo6SNR0fXbE+P1W//1Fg+weCMJGuhAFcTv3xYlETZfpqzb56y7j/iK7x4FwkC5ETZzPgBpL7HQhHpAuRBnzoHpIF3RHuhATzIYqIV3QHelCDDEnqoF0QXekCzHHzGg30tW7kYMdt0zlF5mVR7pgk4SeH21GuvoyOkNum0bAFEa6YCtmSTuQLuiOdEEBzJWxRbqgO9IFZTBjxgrpGrjdLUb1Vv+DxezEoQbSBcUwb0Yf6QqVt13+vtt4sNjucYB0QUnMntFEuhBj27bJsGGSlxexL0i6oDDm0OggXbDFLbfIxo1y662ydGm4GSNdUB4zaaSRLthlzRqprxcRcTrld78LPWOkC5pgPo0c0gV7ORwyfbps3iwi0tIiTqc8+aTcfrvceWd/M0a6oBWHYRh2j0F/zc2yeLG8+KK0tp44OGKEXH65nHuuJPGruCFq7pBNXmPOKIfdA9GHYcgHH8jq1bJ//4mDaWnyve/Jk0+eLmPt7fLEE/LrX/f4xHHj5O67ZcECSU3t7fNIV8jmPjO3fle9iLwx/41Lx19q93C0RMAiweeTYcPE77d7HEDvsrJOd1Oxs1POOkt27Dj2Ia+6oo+AhY/7WpGQkyNDh4rPZ/c4gN61tMhTT0l+vtx1l8XZlBSpqJDFi0kXNELAImT2bFm//jR3WoDY6eyUtjbp7j5xJDlZfvEL+elPJTe318/60Y8kJUV++ENuGEIXBCxC1qyxewRxaFeL8dQW/8PnsxNH//j98sorcu+98v77Jw6adw4rKk6XLlNqqixY0NtJ0gUFETCoq6ldXv3MePh8u8ehi8cekyVLTnyYnS133NHHq65+IF1QFgED4sWCBfLAA+LzDeBV12mRLiiOgAHxIjtbli2T5ubwX3WZbnjhhnWfrwt8eHnh5b+a+6s5Y+eE/5WBiCBgIdq9W5591u5BxLv9R+WLnUnLN9s9Dp38TJJFHh/AJ1x9tcyYYXH8muevOdh60HzMqy6oiYCFaOdOufdeuwcR/xwiDv43R9X48dYBW7V9VWtXq4g8cNkD98y9J9bDAvqBTSIAnM6U4VPsHgJgjVdg4Ro3Tm66ye5BAKGaOdPuEQChImDhGj9eli+3exAAkHi4hQgA0BIBg7p2NhtL3+nu+98BSEgEDOrydcjqz/lrCQCsETAAgJYIGABASwQMSGjbttk9AiBUBAxIXFu2yBVXSEeH3eMAQkLAoK4zMhz/Np1LNIqWLxePR/70J7vHAYSE2QHqGjFYfnwWl2i0uN3y/PMiIsuWSWen3aMBBo7ZAUhQFRXi94uI7N8vzz1n92iAgSNgQCLaskVee+3Ehw88wEoY9EPAoK6dzcaS9ezEERXLlx97+WXatYuVMOiHgEFdvg75x1524oi8LVukpubkgw89xEoYNEPAgISzfLl0n/LK1uNhJQyaIWBAYtmy5dibDwOSjk8DrIRBLwQMSCyB1a85c44dSUqSESNEWAmDbggYkED27JGVK0VEHA65+eYTx5csOfbgkUdsGBUQGgIGdZ2R4bhzBpdoJI0ZI++9J2Vl8p3vyFlnnTh+221y5pmycKGsXm3f4IABSrZ7AECvRgyWmycTsAg75xypqZHOTlm79sTBzEzxeCQlxb5hAQPH7AAkolNbRb2gHQIGANASAYO6djQbd6xjJw4A1ggY1HWoQ97Yx04cAKwRMACAlggYAEBLBAwAoCUCBgDQEgGDusZkOn4+k0sUgDVmB6grP13mTeISBWCN2QEAoCUCBgDQEgGDujzNxq1r2YkDgDUCBnUd7pC397MTBwBrBAwAoCUCBgDQEgEDAGiJgAEAtETAoK4zMx2Vs7hEAVhjdoC6hqfLDwq5RAFYY3YAAGiJgAEAtETAoK7Gw8bit9iJA4A1AgZ1NXfK+i/ZiQOANQIGANASAQMAaImAAQC0RMAAAFoiYFDX2CzHsvO4RAFYY3aAuoalSdlELlEA1pgdAABaImAAAC0RMKhr+2HjljfZiQOANQIGdbV0SsMBduIAYI2AAQC0RMAAAFoiYAAALREwAICWCBjUNS7Lce9sLlEA1pgdoK68NLl2ApcoAGvMDgAALREwAICWCBjUtf2w8aN6duIAYI2AQV0tnfLeQXbiAGCNgAEAtETAAABaImAAAC0RMACAlggY1DUuy/FAEZcoAGvMDlBXXpp8dzyXKABrzA4AAC0RMACAlggY1LXtkHHT6+zEAcAaAYO6jnTJh1524gBgjYABALREwAAAWiJgAAAtETAAgJYIGNQ1Psvx0PmD7B4FAEURMKgrN02uGuuwexQAFEXAAABaImAAAC0RMKjr00PGvNfYiQOANQIGdR3tks1N7MQBwBoBAwBoiYABALREwAAAWiJgAAAtETCoa8IQR9XX2IkDgDUCBnXlpMoVZ7ITBwBrBAwAoCUCBgDQEgGDuj45ZFz/D3biAGCNgEFdrV3yiY+dOABYI2AAAC0RMACAlggYAEBLBAwAoCUCBnVNHOL4jwvYiQOANQIGdWWnSukYduIAYI2AAQC0RMAAAFoiYFDXJ4eM69awEwcAawQM6mrtku2H2IkDgDUCBgDQEgEDAGiJgAEAtETAAABaImBQ18QhjkcuZCcOANYIGNSVnSrfGM1OHACsETAAgJYIGABASwQM6trqM66pYycOANYIGNTV1i07mtmJA4A1AgYA0BIBAwBoiYABALREwAAAWiJgUFfBUMd/zmEnDgDWCBjUNTRFLh3FThwArBEwAICWCBgAQEsEDOra4jO+u5qdOABYI2BQV3u37G5hJw4A1ggYAEBLBAwAoCUCBgDQEgEDAGiJgEFdhUMdT1zEThwArBEwqGtIiswZyU4cAKwRMACAlggYAEBLBAzq+thnfGtVl92jAKAoAgZ1dXTL3iN2DwKAqggYAEBLBAwAoCUCBgDQEgEDAGiJgEFdk4Y6VlzMThwArBEwqCsrRS4YwU4cAKwRMACAlggYAEBLBAzq+thnXP4qO3EAsEbAoK6Obvmy1e5BAFAVAQMAaImAAQC0RMAAAFoiYAAALREwqGvSUMfvL2UnDgDWCBjUlZUixcPZiQOANQIGANASAQMAaImAQV2bm4x/+l924gBgjYBBXZ1++arN7kEAUBUBAwBoiYABALREwAAAWiJgAAAtETCoa3K245m57MQBwBoBg7oyk+XcYezEAcAaAQMAaImAAQC0RMCgro+ajMteYScOANYIGNTV5Rdfu92DAKAqAgYA0BIBAwBoiYABALREwAAAWiJgUNeUbMdzl7ETBwBrBAzqykiWGXnsxAHAGgEDAGiJgAEAtETAoK5NXuOSv7MTB+KQt9V78OhBu0ehvWS7BwD0qtuQ5g67BwFElLfV++j6Rx/b8Njh9sN2j0V7BAwAYsEyXdPzp4/NHmvjqLRGwAAgupo7mp9494mqt6p8bb7AwWn50youqpg3c94gB78rEiICBgDRQrqiioBBXY6O1sGdjWv3tAUfnJA99oysUebjrV9t8wbNCyIyMXvc6KyR5uMtX33a1HYo+GxBzvhRmSPMxx8f/NTX3uNsYc74kcfPbj649VB7c/DZSbkTR2QMNx9/dGDr4Y4eZyfnTsw/fnbTgS3NHS3BZ6fkFgzPGHb87MfNHUd6nM0rHD44z3z8wZebj3QeDT47NW9S3uBc8/H7X350tLM1+OzZwybnpuccO/vFpqNdbT3PTslNzzYfv/fFptaeZ5s7polkmo837v+wrbvHzv+GGIHH7v0ftHf3WI08Z/jUoWlDzMcN+97v8HcGn52Rf/aQ1Czz8bv73uv093gnzsz8aVmpx77vO/s2dvm7g8/OGjE9MyXDfLxhr7vb8AefPXfEORkpg83H6/c2+A0j+Ox5I2cMTk43H6/b0xD8nyAis0fOSD9+du2ed6Wn2SNnpienmf/h6/Y0nHS2aNSstEGpIuI3jPV7j511iOPCMcXSC9IVAwQMKvK3HW1e/d/DN6y+/pKvvbytx4T+rcLSQMA27HVv+erT4LPfnvTPgYCt29PwqXd78NnvTr4yELC1e97Z3uQJPnvNlKsCAXv783c8vp3BZ79/1ncCAXvr8/U7D+0OPls29XuBgNV/tu6zw58Hn/3B2dcEAvb67rf3NO8LPnvDtO8HAvb67rf3tewPPjtkelkgYK/teuuLI18Gn81JGxoI2Jpdbx7o+ca2vPScQMBcO9/4qtUbfPactjGBgK3e+bqv548CwWlYteO1k95xMCpzRCBgq3b846Rgn5E1KhCwVz1rjvZM8tghZwQC9sr2uvae4ZyQPTYQsFca6zp6hnNi9rhAwP6+fXVXzzQW5kwIBOzlbf9nSI/4TcktMANmiPHytlelp6l5k8yA+Q3/qWenDZ9iBqzb3xU421vASFfMOIyeP8Kgn+rrZe5cEZFLLpH6ertHE3eMrs7Dq/6ceeGVyXkj7R5L3HrjDbnsMhGR5GTp7Dz5bMZDGa1drSLyl+v+cv3062M9OD2RrhjjFRjUYBitm9YlZWWnFUwXEUdySvZV8+0eE9BfpMsWBAw2M9pbj2xY3VL/UlJWdva3FwSfcjqdLpfL5XKJSElJSU5Ojoj4fD6v11tcXFxVVZWbm2vPoIHjwk8X13nIuIUYIm4hhs/fcqj5tdoj61elTZ415LJrUyecbfnPHA7H7Nmz3W538MHS0tKGhgaPx8NzO2TcQgxTZF91cZ2HgK2kYBuju0u6u0fc9dth85f1Vi/z+VxeXn7S8YULF/p8vurq6qiPEjhFc0ez823nuEfGVboqA/Walj/t2auf/fAnH94066aB1ovrPDTcQkQMGUbrpnUpo8cn548RkUHZw7KvXnj6zwjcVznpuMfjERGv12vxOUDURGmti+s8NAQMsRC80JVz7b/2/xPNJ3ZRUZHl8VN/YgWiJKpv0+A6Dw0BQ3R1H/qq5Y0Xj2xYnTZ5Vt4Pl/Z2q7A3Lpfr1B9L6+rqXC5XVVXVqU94IOJi8A5DrvPQEDBEV3fTAfH7R9z12xB+o8tcGAh+Yjc2NlZXV9fW1tbU1JSVlUVyoMApYvPmeK7zkBEwRJphtH60Lm3SzKTBWSKSOmFq6oSpoX0l8/5JQ0NDRUWFeaSgoKCkpMTpdEZqsIClWP5eF9d5yAgYIiZ4oStlxFgzYOEwn9grV67s7R80NTXV1NRUV1ef9OZjIGSx/5XkPq/zgKKiIi71YAQMEdDtO9jy5l/N3+gKYaGrNy6Xa/bs2b2ddbvd5jPf5/P19m+A/rNrN43TX+cBTqdz48aNURqDpggYIqB9x2ajuzu0ha7enLowcJKioqKioqL+/NwKnJ6NG0H1eZ2b6urqCgoKojcMTREwhMQw2ra8m372+eJwiEjGeXMzzpsb2e/Q22/GABFk+x6G/bnOm5qaNm7cuHBhH780mYAIGAamx0LXmZMGDc2L0jeqqakRkdLS0ih9fSQ429Nl6s91Xl1dXVFR0dTUFJshaYSAob+6DzcdWftKy5svp044O7f89rQp50XpG5WVlXk8HvN2f2lpaUlJSeDdWUD4FElXP6/zlStXXnfddbEZknYIGPrr6IZV/rajkV3ossSyFqJEkXSZ+nOdNzY2er3ewsLCGIxHRwQMvTOM9h0fm3+gS0SGlP7A3uEAIVMqXf3n8/k8Ho/5ysx8t21FRUVxcTG/3WwiYLAQtNCVk3+r05GSaveIgBBpmi6T+VZb87G5PUdlZSV/WiWAgKGHmC10hc98PrtcLo/Hs2jRooKCApbKEEzrdJ1kxYoVtbW1IrJw4cLy8nJegZkIGHo4/MozjsGZMVjoCl9hYSF77cBSPKXLtGjRokWLFtk9CuUQsIRnGJ37d6WMnmB+lPsvd9o6GiAs8ZcunAYBS1yBha7kYaOH/+Tf7R4OEBbSlYAIWCLSaKEL6BPpSlgELBF5n6tKGVOoxUIXcBqkK8ERsMRgGF1NXwZylX8r732A3kgXhIDFvcBCV+r4s/NuXGr3cIBwkS4EELC4xUIX4gzpwkkIWHwyursO/PZn6dMvYKELcYB0wRIBiyOG4T9yOCkrW0Qcg5JH3f1fkpRk95iAsJAunAYBiweBha70cy7Mufr4X72jXtAZ6UKfCJjeWOhC/CFd6CcCpjH/kcNf/PonGcXfZKELkdLa1fq3T/5WPr08+ODaz9bmZ+ZPzpsc7e9OujAgBEw3hmF0tDnSBotIUubQ0b/6I3/rBJHy+DuPP/TmQ/tb9o8ZOiZw0G/4f/zyj7d7t8+bMc9Z6hyZGZUflUgXQkDAtBFY6Mqcc9WQbx77E+PUCxG0cd/G/S37ReT+1+8PHHx799tbD24VkZe2vvToFY9G/JuSLoSMgGmAhS7ExrJLlz334XNd/q46T13aoDTz4ItbXzQfLLlwSU56TgS/HelCmAiY6jr37Tzw+NKM80tY6EK0FeYW3jjzxqfff1pEOv2d5sG9zXtFJDst+/YLbo/UNyJdiAgCpiTDMPzdjkHJIpIyesKoe55JSs+we0xICIEXYX7DH3w8Ui+/SBciiICpJbDQNfRbN2fMvsw8SL0QM8EvwgJy0nPu+PodYX5l0oWII2CqYKELigi8CAscufPrd2anZYf8BUkXooSAKaFtq9v7nJOFLqjgpBdhGSkZIa9+kS5EFQGzj2GIiDgcIpJWOIOFLqhj2aXLnnn/GUMMESmbVhbC6hfpQgwQMBsEFrry5v08deI0EXGkpPIbXVBHYW7h/HPnP/3+09lp2QP93S/ShZghYDF10kKXWS9AQeZK2IDefEi6EGMELHaOvrvG99IKFrqghcLcwtu+dls/V79IF2xBwGInfdr5o2ZcyEIXdPGby3+T5Ojjj/KQLtiIgEXLsYWuN1/Ov7VqUE6+iCRlDrV7UMAAnL5epAu2I2CRF7zQlXfDErNeQNwgXVAEAYuw5tdeaHY9z0IX4hLpglIIWIQNnnVx5oVXstCFOEO6oCACFmG86kKcIV1QFgEDYI10QXEEDIC10j+WbtizIfDh9Pzp98y9p2xaWZ/vrQdig4CF64sv5Omn+/5ngGo++aSPf3DH1++44YUbhFddUJXDMLeUxQDV18vcuXYPAoiE5GTp7LQ47jf8ZSvLyqeX86oLauIVGJDQfvlLufZa61NJjqQXyl+I7XCAASBgIRo1SubPt3sQQNiuvFLO44+nQk8ELERTpsjUqU6Xy+VyuUSkpKQkJydHRHw+n9frLS4urqqqys3NtXuYQAQ4nVzqUBFrYOFyOByzZ892u93BB0tLSxsaGjweD09sxA0udaiGhdmwmE/m8vLyk44vXLjQ5/NVV1fbMSgg8rjUoSACFpbATZWTjns8HhHxer02jAmIAi51KIiAhcV8VhcVFVkeP/XHVUBTXOpQEAELi8vlOvVn0rq6OpfLVVVVdeqzHdAUlzoUxLsQQ2euCgQ/qxsbG6urq2tra2tqasrKyuwbGhBJXOpQEwELnXnzpKGhoaKiwjxSUFBQUlLidDptHRcQYVzqUBNvow9daWmpy+Xq7X9gU1NTVVWVHF/lrq6u5n3G0NTpL/XGxsba2tqcnByfz9fQ0FBZWckdRcQGAQud5a/FBCxatGjFihWBxy6Xq7GxMYajAyLm9Jd6WVlZQUGB+WrM6XRWVVU1NTXFdoBIULyJI0SnrgqcpLq6uq6uzny8dOlSj8fT2/MfUFmfl3plZWXgXYher7egoCBGI0PCYw0sRL39WkzAU089VVxcHMMRAVHR56UefMOwtrZ26dKlsRgWwC3EkBUVFW3cuLGf//cqKipqa2u5hQgd9fNSdzqdNTU15eXlgTd6ANHGLcQBKysrM5/SIlJaWtrnG7HcbndtbW1DQ0NMRgdEzIAu9YqKCnPDX96aiJjhFVh0NTY2Ll68uKamhrcgIhG43e7i4uKGhgbeiIgY4BVYFDU2NlZWVtbV1eXm5rrdbm4hIv40NTWZl7f5ofmXVrjfgNggYNFivvaqrKx0u91ut7u6ujovL8/uQQERlpubm5eXF3jnocvlysnJYWtExAa3EKMlNzfX5/MFH+F/NeKS2+2uqakxfz5raGioqqoqLCy0e1BICAQMAKAlbiECALREwAAAWiJgAAAtETAAgJYIGABASwQMAKAlAgYA0BIBAwBoiYABALREwAAAWiJgAAAtETAAgJYIGABASwQMAKAlAgYA0BIBAwBoiYABALREwAAAWiJgAAAtETAAgJYIGABASwQMAKCl/wcTIvg3ABhvfwAAAABJRU5ErkJggg==" />
<p>We will get from the <em>approximate</em> surface integral of the <em>approximate</em> curl the following terms:</p>
<pre class='hljl'>
<span class='hljl-nd'>@vars</span><span class='hljl-t'> </span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-t'> </span><span class='hljl-n'>Δx</span><span class='hljl-t'> </span><span class='hljl-n'>Δy</span><span class='hljl-t'> </span><span class='hljl-n'>Δz</span><span class='hljl-t'>
</span><span class='hljl-n'>p1</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>p2</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>p3</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>p4</span><span class='hljl-oB'>=</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>Δx</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>Δx</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>Δy</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-p'>),</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>y</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-n'>Δy</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-n'>z</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>F_z</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-nf'>SymFunction</span><span class='hljl-p'>(</span><span class='hljl-s'>"F_z"</span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-n'>ex</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-oB'>-</span><span class='hljl-nf'>F_z</span><span class='hljl-p'>(</span><span class='hljl-n'>p2</span><span class='hljl-oB'>...</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-nf'>F_z</span><span class='hljl-p'>(</span><span class='hljl-n'>p3</span><span class='hljl-oB'>...</span><span class='hljl-p'>))</span><span class='hljl-oB'>*</span><span class='hljl-n'>Δz</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-cs'># face A</span><span class='hljl-t'>
</span><span class='hljl-p'>(</span><span class='hljl-oB'>-</span><span class='hljl-nf'>F_z</span><span class='hljl-p'>(</span><span class='hljl-n'>p3</span><span class='hljl-oB'>...</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-nf'>F_z</span><span class='hljl-p'>(</span><span class='hljl-n'>p4</span><span class='hljl-oB'>...</span><span class='hljl-p'>))</span><span class='hljl-oB'>*</span><span class='hljl-n'>Δz</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-cs'># face B</span><span class='hljl-t'>
</span><span class='hljl-p'>(</span><span class='hljl-nf'>F_z</span><span class='hljl-p'>(</span><span class='hljl-n'>p1</span><span class='hljl-oB'>...</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-nf'>F_z</span><span class='hljl-p'>(</span><span class='hljl-n'>p4</span><span class='hljl-oB'>...</span><span class='hljl-p'>))</span><span class='hljl-oB'>*</span><span class='hljl-n'>Δz</span><span class='hljl-t'> </span><span class='hljl-oB'>+</span><span class='hljl-t'> </span><span class='hljl-cs'># face C</span><span class='hljl-t'>
</span><span class='hljl-p'>(</span><span class='hljl-nf'>F_z</span><span class='hljl-p'>(</span><span class='hljl-n'>p2</span><span class='hljl-oB'>...</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-t'> </span><span class='hljl-nf'>F_z</span><span class='hljl-p'>(</span><span class='hljl-n'>p1</span><span class='hljl-oB'>...</span><span class='hljl-p'>))</span><span class='hljl-oB'>*</span><span class='hljl-n'>Δz</span><span class='hljl-t'> </span><span class='hljl-cs'># face D</span>
</pre>
<div class="well well-sm">\begin{equation*}Δz \left(- \operatorname{F_{z}}{\left (x,y,z \right )} + \operatorname{F_{z}}{\left (x + Δx,y,z \right )}\right) + Δz \left(\operatorname{F_{z}}{\left (x,y,z \right )} - \operatorname{F_{z}}{\left (x,y + Δy,z \right )}\right) + Δz \left(\operatorname{F_{z}}{\left (x,y + Δy,z \right )} - \operatorname{F_{z}}{\left (x + Δx,y + Δy,z \right )}\right) + Δz \left(- \operatorname{F_{z}}{\left (x + Δx,y,z \right )} + \operatorname{F_{z}}{\left (x + Δx,y + Δy,z \right )}\right)\end{equation*}</div>
<p>The term for face <span class="math">$A$</span>, say, should be divided by <span class="math">$\Delta{y}\Delta{z}$</span> for the curl approximation, but this will be multiplied by the same amount for the divergence calculation, so it isn't written.</p>
<p>The expression above simplifies to:</p>
<pre class='hljl'>
<span class='hljl-nf'>simplify</span><span class='hljl-p'>(</span><span class='hljl-n'>ex</span><span class='hljl-p'>)</span>
</pre>
<div class="well well-sm">\begin{equation*}0\end{equation*}</div>
<p>This is because of how the line integrals are oriented so that the right-hand rule gives outward pointing normals. For each up stroke for one face, there is a downstroke for a different face, and so the corresponding terms cancel each other out. So providing the limit of these two approximations holds, the vanishing identity can be anticipated from the microscopic picture.</p>
<h5>Example</h5>
<p>The <a href="https://en.wikipedia.org/wiki/Maxwell%27s_equations#Charge_conservation">invariance of charge</a> can be derived as a corollary of Maxwell's equation. The divergence of the curl of the magnetic field is <span class="math">$0$</span>, leading to:</p>
<p class="math">\[
~
0 = \nabla\cdot(\nabla\times{B}) =
\mu_0(\nabla\cdot{J} + \epsilon_0 \nabla\cdot{\frac{\partial{E}}{\partial{t}}}) =
\mu_0(\nabla\cdot{J} + \epsilon_0 \frac{\partial}{\partial{t}}(\nabla\cdot{E}))
= \mu_0(\nabla\cdot{J} + \frac{\partial{\rho}}{\partial{t}}).
~
\]</p>
<p>That is <span class="math">$\nabla\cdot{J} = -\partial{\rho}/\partial{t}$</span>. This says any change in the charge density in time (<span class="math">$\partial{\rho}/\partial{t}$</span>) is balanced off by a divergence in the electric current density (<span class="math">$\nabla\cdot{J}$</span>). That is, charge can't be created or destroyed in an isolated system.</p>
<h2>Fundamental theorem of vector calculus</h2>
<p>The divergence and curl are complementary ideas. Are there other distinct ideas to sort a vector field by? The Helmholtz decomposition says not really. It states that vector fields that decay rapidly enough can be expressed in terms of two pieces: one with no curl and one with no divergence.</p>
<p>From <a href="https://en.wikipedia.org/wiki/Helmholtz_decomposition">Wikipedia</a> we have this formulation:</p>
<p>Let <span class="math">$F$</span> be a vector field on a <strong>bounded</strong> domain <span class="math">$V$</span> which is twice continuously differentiable. Let <span class="math">$S$</span> be the surface enclosing <span class="math">$V$</span>. Then <span class="math">$F$</span> can be decomposed into a curl-free component and a divergence-free component:</p>
<p class="math">\[
~
F = -\nabla(\phi) + \nabla\times A.
~
\]</p>
<p>Without explaining why, these values can be computed using volume and surface integrals:</p>
<p class="math">\[
~
\begin{align}
\phi(\vec{r}') &=
\frac{1}{4\pi} \int_V \frac{\nabla \cdot F(\vec{r})}{\|\vec{r}'-\vec{r} \|} dV -
\frac{1}{4\pi} \oint_S \frac{F(\vec{r})}{\|\vec{r}'-\vec{r} \|} \cdot \hat{N} dS\\
A(\vec{r}') &= \frac{1}{4\pi} \int_V \frac{\nabla \times F(\vec{r})}{\|\vec{r}'-\vec{r} \|} dV +
\frac{1}{4\pi} \oint_S \frac{F(\vec{r})}{\|\vec{r}'-\vec{r} \|} \times \hat{N} dS.
\end{align}
~
\]</p>
<p>If <span class="math">$V = R^3$</span>, an unbounded domain, <em>but</em> <span class="math">$F$</span> <em>vanishes</em> faster than <span class="math">$1/r$</span>, then the theorem still holds with just the volume integrals:</p>
<p class="math">\[
~
\begin{align}
\phi(\vec{r}') &=\frac{1}{4\pi} \int_V \frac{\nabla \cdot F(\vec{r})}{\|\vec{r}'-\vec{r} \|} dV\\
A(\vec{r}') &= \frac{1}{4\pi} \int_V \frac{\nabla \times F(\vec{r})}{\|\vec{r}'-\vec{r}\|} dV.
\end{align}
~
\]</p>
<h2>Change of variable</h2>
<p>The divergence and curl are defined in a manner independent of the coordinate system, though the method to compute them depends on the Cartesian coordinate system. If that is inconvenient, then it is possible to develop the ideas in different coordinate systems.</p>
<p>Some details are <a href="https://en.wikipedia.org/wiki/Curvilinear_coordinates">here</a>, the following is based on <a href="https://www.jfoadi.me.uk/documents/lecture_mathphys2_05.pdf">some lecture notes</a>.</p>
<p>We restrict to <span class="math">$n=3$</span> and use <span class="math">$(x,y,z)$</span> for Cartesian coordinates and <span class="math">$(u,v,w)$</span> for an <em>orthogonal</em> curvilinear coordinate system, such as spherical or cylindrical. If <span class="math">$\vec{r} = \langle x,y,z\rangle$</span>, then</p>
<p class="math">\[
~
\begin{align}
d\vec{r} &= \langle dx,dy,dz \rangle = J \langle du,dv,dw\rangle\\
&=
\left[ \frac{\partial{\vec{r}}}{\partial{u}} \vdots
\frac{\partial{\vec{r}}}{\partial{v}} \vdots
\frac{\partial{\vec{r}}}{\partial{w}} \right] \langle du,dv,dw\rangle\\
&= \frac{\partial{\vec{r}}}{\partial{u}} du +
\frac{\partial{\vec{r}}}{\partial{v}} dv
\frac{\partial{\vec{r}}}{\partial{w}} dw.
\end{align}
~
\]</p>
<p>The term <span class="math">${\partial{\vec{r}}}/{\partial{u}}$</span> is tangent to the curve formed by <em>assuming</em> <span class="math">$v$</span> and <span class="math">$w$</span> are constant and letting <span class="math">$u$</span> vary. Similarly for the other partial derivatives. Orthogonality assumes that at every point, these tangent vectors are orthogonal.</p>
<p>As <span class="math">${\partial{\vec{r}}}/{\partial{u}}$</span> is a vector it has a magnitude and direction. Define the scale factors as the magnitudes:</p>
<p class="math">\[
~
h_u = \| \frac{\partial{\vec{r}}}{\partial{u}} \|,\quad
h_v = \| \frac{\partial{\vec{r}}}{\partial{v}} \|,\quad
h_w = \| \frac{\partial{\vec{r}}}{\partial{w}} \|.
~
\]</p>
<p>and let <span class="math">$\hat{e}_u$</span>, <span class="math">$\hat{e}_v$</span>, and <span class="math">$\hat{e}_w$</span> be the unit, direction vectors.</p>
<p>This gives the following notation:</p>
<p class="math">\[
~
d\vec{r} = h_u du \hat{e}_u + h_v dv \hat{e}_v + h_w dw \hat{e}_w.
~
\]</p>
<p>From here, we can express different formulas.</p>
<p>For line integrals, we have the line element:</p>
<p class="math">\[
~
dl = \sqrt{d\vec{r}\cdot d\vec{r}} = \sqrt{(h_ud_u)^2 + (h_vd_v)^2 + (h_wd_w)^2}.
~
\]</p>
<p>Consider the surface for constant <span class="math">$u$</span>. The vector <span class="math">$\hat{e}_v$</span> and <span class="math">$\hat{e}_w$</span> lie in the surface's tangent plane, and the surface element will be:</p>
<p class="math">\[
~
dS_u = \| h_v dv \hat{e}_v \times h_w dw \hat{e}_w \| = h_v h_w dv dw \| \hat{e}_v \| = h_v h_w dv dw.
~
\]</p>
<p>This uses orthogonality, so <span class="math">$\hat{e}_v \times \hat{e}_w$</span> is parallel to <span class="math">$\hat{e}_u$</span> and has unit length. Similarly, <span class="math">$dS_v = h_u h_w du dw$</span> and <span class="math">$dS_w = h_u h_v du dv$</span> .</p>
<p>The volume element is found by <em>projecting</em> <span class="math">$d\vec{r}$</span> onto the <span class="math">$\hat{e}_u$</span>, <span class="math">$\hat{e}_v$</span>, <span class="math">$\hat{e}_w$</span> coordinate system through <span class="math">$(d\vec{r} \cdot\hat{e}_u) \hat{e}_u$</span>, <span class="math">$(d\vec{r} \cdot\hat{e}_v) \hat{e}_v$</span>, and <span class="math">$(d\vec{r} \cdot\hat{e}_w) \hat{e}_w$</span>. Then forming the triple scalar product to compute the volume of the parallelepiped:</p>
<p class="math">\[
~
\left[(d\vec{r} \cdot\hat{e}_u) \hat{e}_u\right] \cdot
\left(
\left[(d\vec{r} \cdot\hat{e}_v) \hat{e}_v\right] \times
\left[(d\vec{r} \cdot\hat{e}_w) \hat{e}_w\right]
\right) =
(h_u h_v h_w) ( du dv dw ) (\hat{e}_u \cdot (\hat{e}_v \times \hat{e}_w) =
h_u h_v h_w du dv dw,
~
\]</p>
<p>as the unit vectors are orthonormal, their triple scalar product is <span class="math">$1$</span> and <span class="math">$d\vec{r}\cdot\hat{e}_u = h_u du$</span>, etc.</p>
<h3>Example</h3>
<p>We consider spherical coordinates with</p>
<p class="math">\[
~
F(r, \theta, \phi) = \langle
r \sin(\phi) \cos(\theta),
r \sin(\phi) \sin(\theta),
r \cos(\phi)
\rangle.
~
\]</p>
<p>The following figure draws curves starting at <span class="math">$(r_0, \theta_0, \phi_0)$</span> formed by holding <span class="math">$2$</span> of the <span class="math">$3$</span> variables constant. The tangent vectors are added in blue. The surface <span class="math">$S_r$</span> formed by a constant value of <span class="math">$r$</span> is illustrated.</p>
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