Volume integral

In coordinates

Often the volume integral is represented in terms of a differential volume element ${\displaystyle dV=dxdydz}$. $${\displaystyle {\iiint }_{D}f(x,y,z)dV.}$$ It can also mean a triple integral within a region ${\displaystyle D\subset {\mathbb{R}}^3}$ of a function ${\displaystyle f(x,y,z),}$ and is usually written as: $${\displaystyle {\iiint }_{D}f(x,y,z)dxdydz.}$$ A volume integral in cylindrical coordinates is $${\displaystyle {\iiint }_{D}f(\rho ,\phi ,z)\rho d\rho d\phi dz,}$$ and a volume integral in spherical coordinates (using the ISO convention for angles with ${\displaystyle \phi }$ as the azimuth and ${\displaystyle \theta }$ measured from the polar axis (see more on conventions)) has the form $${\displaystyle {\iiint }_{D}f(r,\theta ,\phi )r^2\sin \theta drd\theta d\phi .}$$ 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 ${\displaystyle (x,y,z)\mapsto (u,v,w)}$. We can represent the integral as the following. $${\displaystyle {\iiint }_{D}f(x,y,z)dxdydz={\iiint }_{D}f(u,v,w)\left|\frac{\mathrm{\partial }(x,y,z)}{\mathrm{\partial }(u,v,w)}\right|dudvdw}$$ Where we define the Jacobian determinant to be. $${\displaystyle \mathbf{J}=\frac{\mathrm{\partial }(x,y,z)}{\mathrm{\partial }(u,v,w)}=\left|\begin{array}{ccc}\frac{\mathrm{\partial }x}{\mathrm{\partial }u}& \frac{\mathrm{\partial }x}{\mathrm{\partial }v}& \frac{\mathrm{\partial }x}{\mathrm{\partial }w}\\ \frac{\mathrm{\partial }y}{\mathrm{\partial }u}& \frac{\mathrm{\partial }y}{\mathrm{\partial }v}& \frac{\mathrm{\partial }y}{\mathrm{\partial }w}\\ \frac{\mathrm{\partial }z}{\mathrm{\partial }u}& \frac{\mathrm{\partial }z}{\mathrm{\partial }v}& \frac{\mathrm{\partial }z}{\mathrm{\partial }w}\end{array}\right|}$$