# 向量、向量微积分和力学Vectors, vector calculus and mechanics MA10236

The following statements are equivalent for a simply connected region $R$ in $\mathbb{R}^{2}$ :
(a) $\mathrm{f}(x, y)=P(x, y) \mathbf{i}+Q(x, y) \mathbf{j}$ has a smooth potential $F(x, y)$ in $R$
(b) $\int \mathbf{f} \cdot d \mathbf{r}$ is independent of the path for any curve $C$ in $R$

(c) $\oint_{C} \mathbf{f} \cdot d \mathbf{r}=0$ for every simple closed curve $C$ in $R$
(d) $\frac{\partial P}{\partial y}=\frac{\partial Q}{\partial x}$ in $R \quad$ (in this case, the differential form $P d x+Q d y$ is exact)

Let $\Sigma$ be a surface in $\mathbf{R}^{3}$ parametrized by $x=x(u, v), y=y(u, v)$, $z=z(u, v)$, for $(u, v)$ in some region $R$ in $\mathbf{R}^{2}$. Let $\mathbf{r}(u, v)=x(u, v) \mathbf{i}+y(u, v) \mathbf{j}+z(u, v) \mathbf{k}$ be the position vector for any point on $\Sigma$, and let $f(x, y, z)$ be a real-valued function defined on some subset of $\mathbb{R}^{3}$ that contains $\Sigma$. The surface integral of $f(x, y, z)$ over $\Sigma$ is
$$\iint_{\Sigma} f(x, y, z) d \sigma=\iint_{R} f(x(u, v), y(u, v), z(u, v))\left|\frac{\partial \mathbf{r}}{\partial u} \mathbf{x} \frac{\partial \mathbf{r}}{\partial v}\right| d u d v .$$
In particular, the surface area $S$ of $\Sigma$ is
$$S=\iint_{\Sigma} 1 d \sigma$$