分类： 考研数学

$V$ $=$ $\iint_{D}$ $|$ $z(x, y)$ $|$ $\mathrm{d} x \mathrm{~d} y$

$\begin{cases} \iint_{\Sigma^{-}} P \mathrm{~d} y \mathrm{~d} z=-\iint_{\Sigma} P \mathrm{~d} y \mathrm{~d} z, \\ \iint_{\Sigma^{-}} Q \mathrm{~d} z \mathrm{~d} x=-\iint_{\Sigma} Q \mathrm{~d} z \mathrm{~d} x, \\ \iint_{\Sigma^{-}} R \mathrm{~d} x \mathrm{~d} y=-\iint_{\Sigma} R \mathrm{~d} x \mathrm{~d} y. \end{cases}$

$\iint_{\Sigma}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma_{1}}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$ $+$ $\iint_{\Sigma_{2}}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$

$\iint_{\Sigma}$ $f(x, y, z)$ $\mathrm{d} S$ $=$ $\iint_{\Sigma_{1}}$ $f(x, y, z)$ $\mathrm{d} S$ $+$ $\iint_{\Sigma_{2}}$ $f(x, y, z)$ $\mathrm{d} S$

$\iint_{\Sigma}$ $1$ $\mathrm{~d} s$ $=$ $S$

$\int_{L}$ $\big[$ $\alpha$ $\boldsymbol{F}_{1}(x, y)$ $+$ $\beta$ $\boldsymbol{F}_{2}(x, y)$ $\big]$ $\cdot$ $\mathrm{d} \boldsymbol{r}$ $=$ $\alpha$ $\int_{L}$ $\boldsymbol{F}_{1}(x, y)$ $\cdot$ $\mathrm{d} \boldsymbol{r}$ $+$ $\beta$ $\int_{L}$ $\boldsymbol{F}_{2}(x, y)$ $\cdot$ $\mathrm{d} \boldsymbol{r}$

$\int_{L}$ $\boldsymbol{F}(x, y)$ $\cdot$ $\mathrm{d} \boldsymbol{r}$ $=$ $\int_{L_{1}}$ $\boldsymbol{F}(x, y)$ $\cdot$ $\mathrm{d} \boldsymbol{r}$ $+$ $\int_{L_{2}}$ $\boldsymbol{F}(x, y)$ $\cdot$ $\mathrm{d} \boldsymbol{r}$

$\begin{cases} \int_{L} P(x, y) \mathrm{d} x=-\int_{L^{-}} P(x, y) \mathrm{d} x, \\ \int_{L} Q(x, y) \mathrm{d} y=-\int_{L^{-}} Q(x, y) \mathrm{d} y. \end{cases}$

$\int_{\Gamma}$ $f(x, y, z)$ $\mathrm{d} s$ $=$ $\int_{\Gamma}$ $f(z, x, y)$ $\mathrm{d} s$ $=$ $\int_{\Gamma}$ $f(y, z, x)$ $\mathrm{d} s$ $=$ $\frac{1}{3}$ $\cdot$ $\int_{\Gamma}$ $[$ $f(x, y, z)$ $+$ $f(z, x, y)$ $+$ $f(y, z, x)$ $]$ $\mathrm{d} s$

$\int_{L}$ $f(x, y)$ $\mathrm{d} s$ $=$ $\int_{L}$ $f(y, x)$ $\mathrm{d} s$ $=$ $\frac{1}{2}$ $\cdot$ $\int_{L}$ $[$ $f(x, y)$ $+$ $f(y, x)$ $]$ $\mathrm{d} s$

$\int_{L}$ $f(x, y)$ $\mathrm{d} s$ $=$ $\left\{\begin{array}{ll} 0, & f(x, -y)=-f(x, y), \\2 \int_{L_{1}} f(x, y) \mathrm{d} s, & f(x, -y)=f(x, y). \end{array}\right.$

$f(-x, y)$ $=$ $-$ $f(x, y)$ $\Rightarrow$ $f(x, y)$ 是关于 $x$ 的奇函数.
$f(-x, y)$ $=$ $f(x, y)$ $\Rightarrow$ $f(x, y)$ 是关于 $x$ 的偶函数.

$\int_{L}$ $f(x, y)$ $\mathrm{d} s$ $=$ $\left\{\begin{array}{ll} 0, & f(-x, y)=-f(x, y), \\2 \int_{L_{1}} f(x, y) \mathrm{d} s, & f(-x, y)=f(x, y). \end{array}\right.$

$f(-x, y)$ $=$ $-$ $f(x, y)$ $\Rightarrow$ $f(x, y)$ 是关于 $x$ 的奇函数.
$f(-x, y)$ $=$ $f(x, y)$ $\Rightarrow$ $f(x, y)$ 是关于 $x$ 的偶函数.

$\int_{L}$ $1$ $\mathrm{d} s$ $=$ $H$

$\big|$ $\int_{L}$ $f(x, y)$ $\mathrm{d} s$ $\big|$ $\leqslant$ $\int_{L}$ $|$ $f(x, y)$ $|$ $\mathrm{d} s$

$\int_{L}$ $f(x, y)$ $\mathrm{d} s$ $\leqslant$ $\int_{L}$ $g(x, y)$ $\mathrm{d} s$