Volume and surface elements in different co-ordinate systems

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This page outlines the value of different volume and surface elements in several different co-ordinate systems.

System Volume Surface
Cartesian dV=dx\,dy\,dz dS=dx\,dy
Cylindrical dV=r\,dr\,d\theta\,dz dS=r\,d\theta\,dz
Spherical dV=r^2\sin\phi\,dr\,d\phi\,d\theta dS=r^2\sin\phi\,d\phi\,d\theta

Note that in the surface area elements dS, the radius r is not a variable but a constant evaluated at the particular value, hence the absence of the dr differential term.

These elements are computed according to following parametrizations:

Spherical coordinates:

x = r \sin \phi \; \cos \theta \quad
y = r \sin \phi \; \sin \theta \quad , \phi \in \left(0,\pi\right),\, \theta \in \left(0,2\pi\right) \quad
z = r \cos \phi \quad


Cylindrical coordinates:

x = r \cos \theta \quad
y = r \sin \theta \quad , \theta \in \left(0,2\pi\right) \quad
z = z \quad


[edit] See also

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