## 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.

Let

and let

We are interested in a formula for evaluating a surface integral where r is a function of angular variables

We have

Let consider

We want to find the expression of

We have

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

We have

Now let us see another example.

Consider the torus of equation

We have

and we get

Then

So

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