how to solve this problem about the surface integral

Question

The force per unit area acting on a sphere with radius $r$ is $F=r^2\vec r+cos\theta \vec \theta$, what is the net force on of the sphere?

To solve this problem, i will use the surface integral and convert the force field in terms of standard basis in cartesian coordinate and do the surface integral but doing in this way completely ignored the meaning of spherical coordinates.

My question is how to do the surface integral computating with the spherical coordinate instead of converting to the standard cartesian coordinate.

Answer 1

Well, you can observe that since the integral is additive you can evaluate the $r$ part and the $\theta$ part independently. Now, the $r$ part will give zero because of the spherical symmetry.

For the similar symmetry reason but this time polar, the $\theta$ part will give a vector pointing into the $z$ (i.e. up/down) direction. This should allow you to transform the problem into a $1$-dimensional integral which is easily solved.