Suppose we're working in $\mathbb{R}^3$.
For $1\leq p<\infty$ we define the $p$-norm $||\cdot||_p$ as $||(x,y,z)||_p$.
Now, let $f$ be a scalar field in $\mathbb{R}^3$.
I want to compute the volume integral of the field over the ball and/or the surface integral over the border of the ball. Is there a parametrization of the surface of the ball? Or some coordinates in which I can compute the volume integral?
Of course, the case $p=2$ is clear and easy, as one may use spherical coordinates, and the case $p=\infty$ is even easier, as the ball is simply a cube. But what about doing it for a general $p$? Is it even possible? I feel like this should be reasonably easy to compute but I don't really know what to do.