Hyper-volume by integration

Question

I would like to know what the hyper-volume of a hyper-object is by integration in cartesian and polar coordinates, too!
I have a theory about it of the cartesian coordinate which is the following:
$$V_N = \int ... \int_{V_N} \sqrt{ 1+ \sum_{k=1}^N ({df \over dx_k})^2 } dx_1 ... dx_N$$ So V1 = Length, V2 = Area ... etc. I just tried to generalize the formulas. Do you find it correct? I really would like to know the other part of this formula in polar coordinates.
Thank you!

Answer 1

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}$Your formula is correct. Precisely, if $D \subset \Reals^{N}$ is the closure of a bounded open set, and if $f:D \to \Reals$ is continuously differentiable, then the graph of $f$ is an $N$-manifold with boundary in $\Reals^{N} \times \Reals$ whose $N$-dimensional volume is $$ \int \cdots \int_{D}\ \Bigl[1 + \sum_{k=1}^{N} f_{k}^{2}\Bigr]^{1/2} dx_{1} \cdots dx_{N},\qquad f_{k} = \frac{\dd f}{\dd x_{k}}. $$ To show this, parametrize the graph by $$ \Phi\bigl(x_{1}, \dots, x_{N}, f(x_{1}, \dots, x_{N})\bigr). $$ The volume density is the magnitude of the generalized cross product $(-f_{1}, -f_{2}, \dots, -f_{N}, 1)$ of the partial derivatives: \begin{align*} \Phi_{1} &= (1, 0, \dots, 0, f_{1}), \\ \Phi_{2} &= (0, 1, \dots, 0, f_{2}), \\ &\ \vdots \\ \Phi_{N} &= (0, 0, \dots, 1, f_{N}). \end{align*}

Your question about polar coordinates is less easy to answer, both because there is not a single convention for polar coordinates in $\Reals^{N}$, and because the formula is not as simple.