If I have a function I want to integrate over the surface area or the volume, but I'm generalizing dimensions:
To start, in $ℝ^2$ the surface area is the arclength (1-dimensional volume) and the 2D volume is the area. In $ℝ^3$ the surface area is the surface area (trivial I guess) and the volume is.. the volume!
So in $ℝ^4$ is the surface area is the 3D volume (n-1 dimensional volume = n-dimensional surface area)?
So what is the general relationship between the n-dimensional surface areal and (n-1)-dimensional volume? And how can I generalize the infinitesimal surface and volume elements as a result of this relation?
(i.e. some functional relation $dS_n = f(dV_{n-1})$)