Editing Volume integral (section)

==In coordinates==

Often the volume integral is represented in terms of a differential volume element <math> dV=dx\, dy\, dz </math>.
<math display="block">\iiint_D f(x,y,z)\,dV.</math>
It can also mean a [[multiple integral|triple integral]] within a region <math>D \subset \R^3</math> of a [[function (mathematics)|function]] <math>f(x,y,z),</math> and is usually written as:
<math display="block">\iiint_D f(x,y,z)\,dx\,dy\,dz.</math>
A volume integral in [[cylindrical coordinates]] is
<math display="block">\iiint_D f(\rho,\varphi,z) \rho \,d\rho \,d\varphi \,dz,</math>
and a volume integral in [[spherical coordinates]] (using the ISO convention for angles with <math>\varphi</math> as the azimuth and <math>\theta</math> measured from the polar axis (see more on [[Spherical coordinate system#Conventions|conventions]])) has the form
<math display="block">\iiint_D f(r,\theta,\varphi) r^2 \sin\theta \,dr \,d\theta\, d\varphi .</math>
The triple integral can be transformed from Cartesian coordinates to any arbitrary coordinate system using the [[Jacobian matrix and determinant]]. Suppose we have a transformation of coordinates from <math> (x,y,z)\mapsto(u,v,w) </math>. We can represent the integral as the following.
<math display="block">\iiint_D f(x,y,z)\,dx\,dy\,dz=\iiint_D f(u,v,w)\left|\frac{\partial (x,y,z)}{\partial (u,v,w)}\right|\,du\,dv\,dw</math>
Where we define the Jacobian determinant to be.
<math display="block">
\mathbf{J}=\frac{\partial (x,y,z)}{\partial (u,v,w)}=
\begin{vmatrix}
\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}& \frac{\partial x}{\partial w}\\
\frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}& \frac{\partial y}{\partial w}\\
\frac{\partial z}{\partial u}& \frac{\partial z}{\partial v}& \frac{\partial z}{\partial w}\\
\end{vmatrix}
</math>