8. Multivariate functions' surface

Definition

A surface of dimension $k$ in $\mathbb{R}^{n}$ is a subset $S \subset \mathbb{R}^{n}$ each point of which has a neighborhood in $S$ homeomorphic to $\mathbb{R}^{k}$.

surface integrals of the first kind

Suppose that $\text{supp}f\subset U$, $(U,\varphi)$is a local coordinate system, the surface integrals of the first kind function $f$ over $S$ is defined as:

$$\int_{S} f \mathrm{d} V=\int_{\varphi^{-1}(U)} f \circ \varphi(t) \sqrt{\operatorname{det}\left(\left\langle\frac{\partial \varphi}{\partial t^{i}}, \frac{\partial \varphi}{\partial t^{j}}\right\rangle\right)} \mathrm{d} t^{1} \ldots \mathrm{d} t^{k}$$

$\blacksquare$