标签： 高等数学基础

$\overline{x}$ $=$ $\frac{{\iiint }_{\mathrm{\Omega }}x\rho (x,y,z)\mathrm{d}v}{{\iiint }_{\mathrm{\Omega }}\rho (x,y,z)\mathrm{d}v}$, $\overline{y}$ $=$ $\frac{{\iiint }_{\mathrm{\Omega }}y\rho (x,y,z)\mathrm{d}v}{{\iiint }_{\mathrm{\Omega }}\rho (x,y,z)\mathrm{d}v}$, $\overline{z}$ $=$ $\frac{{\iiint }_{\mathrm{\Omega }}z\rho (x,y,z)\mathrm{d}v}{{\iiint }_{\mathrm{\Omega }}\rho (x,y,z)\mathrm{d}v}$

$\overline{x}$ $=$ $\frac{{\iint }_{D}x\rho (x,y)\mathrm{d}\sigma }{{\iint }_D\rho (x,y)\mathrm{d}\sigma }$, $\overline{y}$ $=$ $\frac{{\iint }_{D}y\rho (x,y)\mathrm{d}\sigma }{{\iint }_D\rho (x,y)\mathrm{d}\sigma }$

$S$ $=$ ${\iint }_{D_{xy}}$ $\sqrt{1+{\left(\frac{\partial z}{\partial x}\right)}^2+{\left(\frac{\partial z}{\partial y}\right)}^2}$ $\mathrm{d}x\mathrm{d}y$

$V$ $=$ ${\iint }_D$ $|$ $z(x,y)$ $|$ $\mathrm{d}x\mathrm{d}y$

$\{\begin{array}{l}{\iint }_{{\mathrm{\Sigma }}^{-}}P\mathrm{d}y\mathrm{d}z=-{\iint }_{\mathrm{\Sigma }}P\mathrm{d}y\mathrm{d}z,\\ {\iint }_{{\mathrm{\Sigma }}^{-}}Q\mathrm{d}z\mathrm{d}x=-{\iint }_{\mathrm{\Sigma }}Q\mathrm{d}z\mathrm{d}x,\\ {\iint }_{{\mathrm{\Sigma }}^{-}}R\mathrm{d}x\mathrm{d}y=-{\iint }_{\mathrm{\Sigma }}R\mathrm{d}x\mathrm{d}y.\end{array}$

${\iint }_{\mathrm{\Sigma }}$ $P$ $\mathrm{d}y$ $\mathrm{d}z$ $=$ ${\iint }_{{\mathrm{\Sigma }}_1}$ $P$ $\mathrm{d}y$ $\mathrm{d}z$ $+$ ${\iint }_{{\mathrm{\Sigma }}_2}$ $P$ $\mathrm{d}y$ $\mathrm{d}z$

${\iint }_{\mathrm{\Sigma }}$ $f(x,y,z)$ $\mathrm{d}S$ $=$ ${\iint }_{{\mathrm{\Sigma }}_1}$ $f(x,y,z)$ $\mathrm{d}S$ $+$ ${\iint }_{{\mathrm{\Sigma }}_2}$ $f(x,y,z)$ $\mathrm{d}S$

${\iint }_{\mathrm{\Sigma }}$ $1$ $\mathrm{d}s$ $=$ $S$

${\int }_L$ $[$ $\alpha$ ${\mathit{F}}_1(x,y)$ $+$ $\beta$ ${\mathit{F}}_2(x,y)$ $]$ $\cdot$ $\mathrm{d}\mathit{r}$ $=$ $\alpha$ ${\int }_L$ ${\mathit{F}}_1(x,y)$ $\cdot$ $\mathrm{d}\mathit{r}$ $+$ $\beta$ ${\int }_L$ ${\mathit{F}}_2(x,y)$ $\cdot$ $\mathrm{d}\mathit{r}$

${\int }_L$ $\mathit{F}(x,y)$ $\cdot$ $\mathrm{d}\mathit{r}$ $=$ ${\int }_{L_1}$ $\mathit{F}(x,y)$ $\cdot$ $\mathrm{d}\mathit{r}$ $+$ ${\int }_{L_2}$ $\mathit{F}(x,y)$ $\cdot$ $\mathrm{d}\mathit{r}$

$\{\begin{array}{l}{\int }_{L}P(x,y)\mathrm{d}x=-{\int }_{L^{-}}P(x,y)\mathrm{d}x,\\ {\int }_{L}Q(x,y)\mathrm{d}y=-{\int }_{L^{-}}Q(x,y)\mathrm{d}y.\end{array}$

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

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

${\int }_L$ $f(x,y)$ $\mathrm{d}s$ $=$ $\{\begin{array}{ll}0,& f(x,-y)=-f(x,y),\\ 2{\int }_{L_1}f(x,y)\mathrm{d}s,& f(x,-y)=f(x,y).\end{array}$

$f(-x,y)$ $=$ $-$ $f(x,y)$ $\Rightarrow$ $f(x,y)$ 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.