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    <name>
      <text>27.4.3.1 Jacobian in Three Variables</text>
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      <text><![CDATA[<h3>The Jacobian in 3 dimensions</h3>

<p>When changing the coordinate system of a triple integral \(\displaystyle I=\displaystyle\int\!\!\displaystyle\int\!\!\displaystyle\int_{V}f(x,y,z)\;\mathrm{d}V\) we need to extend the above definition of the Jacobian to 3 dimensions. The volume element \(\mathrm{d}V=\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\) becomes \(\mathrm{d}V=|J(u,v,w)|\; \mathrm{d}u\,\mathrm{d}v\,\mathrm{d}w\). As before the limits and integrand must also be transformed.</p>

<div class="HELM_keypoint">
<h4>Key Point</h4>
<h6>Jacobian in Three Variables</h6>
For given transformations \(x=x(u,v,w)\), \(y=y(u,v,w)\)
and \(z=z(u,v,w)\) the Jacobian is
\[\displaystyle J(u,v,w)=\left|
\begin{array}{ccc}
\displaystyle \frac{\partial x}{\partial u}&\displaystyle \frac{\partial x}{\partial v}&\displaystyle \frac{\partial x}{\partial w}\\
&&\\
\displaystyle \frac{\partial y}{\partial
u}&\displaystyle \frac{\partial y}{\partial
v}&\displaystyle \frac{\partial y}{\partial w}\\
&&\\
\displaystyle \frac{\partial z}{\partial
u}&\displaystyle \frac{\partial z}{\partial
v}&\displaystyle \frac{\partial z}{\partial w}
\end{array}\right|\]
The same pattern persists as in the 2-dimensional case.
Across a row of the determinant the numerators are the same and down a
column the denominators are the same.
</div>
<br>
The volume element \(\mathrm{d}V=\mathrm{d}x\,\mathrm{d} y\,\mathrm{d}z\) becomes \(\mathrm{d}V=|J(u,v,w)|\; \,\mathrm{d}u\,\mathrm{d} v\,\mathrm{d} w\). As before the
limits and integrand must also be transformed.]]></text>
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