Let's say we have a closed domain $D \subset R^m$. We then define a function over $D$ explicitly $f:D\rightarrow \mathbb{R}^n$.
We can also state that is our function $f$ is differentiable $k$ times.
Is there a way to describe the surface area covered by $f$ in $\mathbb{R}^n$?
I had an intuition that we can use the integral of the gradient over the space of $D$ in some way to calculate this object. But I'm not sure exactly how to express this.
Any ideas on how we can construct the surface area object?