# 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:

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

$$\blacksquare$$