Surface integral in spherical coordinates
Let us consider a surface integral
where is a surface which have a parameterization described in terms of angles and in spherical coordinates.
We are interested in a formula for evaluating a surface integral where r is a function of angular variables
We want to find the expression of
We use orthogonal coordinatesAs are orthonormal vectors, so are so the norm of is the square root of sum of squares of its components.
Finally we have
We now evaluate the surface of sphere
We have in this situation so so
Now let us see another example.
Consider the torus of equation
and we get
This result is consistent with area of surface of rotation equals to the product of length of the curve by the length of the mass center curve