标签： 高等数学基础

$\sum _{n=1}^{\mathrm{\infty }}$ $($ $u_n$ $\pm$ $v_n$ $)$ 的敛散性不确定

$\sum _{n=1}^{\mathrm{\infty }}$ $($ $u_n$ $\pm$ $v_n$ $)$ 发散

$\sum _{n=1}^{\mathrm{\infty }}$ $($ $u_n$ $\pm$ $v_n$ $)$ $=$ $s$ $\pm$ $\sigma$

$\sum _{n=1}^{\mathrm{\infty }}$ $u_n$ 与 $\sum _{n=1}^{\mathrm{\infty }}$ $c$ $\cdot$ u_{n}$ 的敛散性相同

$\mathit{r}\mathit{o}\mathit{t}\mathbf{A}$ $=$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\mathit{i}$ $+$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\mathit{j}$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathit{k}$ $=$ $\left|\begin{array}{ccc}\mathit{i}& \mathit{j}& \mathit{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right|$

$\mathrm{div}\mathit{A}$ $=$ $\frac{\partial P}{\partial x}$ $+$ $\frac{\partial Q}{\partial y}$ $+$ $\frac{\partial R}{\partial z}$

${\iint }_{\mathrm{\Sigma }}$ $\mathit{A}$ $\cdot$ $\mathit{n}$ $\mathrm{d}S$

${\oint }_{\mathrm{\Gamma }}$ $P$ $\mathrm{d}x$ $+$ $Q$ $\mathrm{d}y$ $+$ $R$ $\mathrm{d}z$ $=$ ${\iint }_{\mathrm{\Sigma }}$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d}y$ $\mathrm{d}z$ $+$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{d}z\mathrm{d}x$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{d}x\mathrm{d}y$ $=$ ${\iint }_{\mathrm{\Sigma }}\left|\begin{array}{ccc}\mathrm{d}y\mathrm{d}z& \mathrm{d}z\mathrm{d}x& \mathrm{d}x\mathrm{d}y\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right|$ $=$ ${\iint }_{\mathrm{\Sigma }}\left|\begin{array}{ccc}\cos \alpha & \cos \beta & \cos \gamma \\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right|\mathrm{d}S.$

其中，$n$ $=$ $($ $\cos \alpha$, $\cos \beta$, $\cos \gamma$ $)$ 为 $\mathrm{\Sigma }$ 的单位法向量.

$P$ $\mathrm{d}y\mathrm{d}z$ $+$ $Q$ $\mathrm{d}z\mathrm{d}x$ $+$ $R$ $\mathrm{d}x\mathrm{d}y$ $=$ ${\iiint }_{\mathrm{\Omega }}$ $($ $\frac{\partial P}{\partial x}$ $+$ $\frac{\partial Q}{\partial y}$ $+$ $\frac{\partial R}{\partial z}$ $)$ $\mathrm{d}V$

或记作：
$($ $P$ $\cos \alpha$ $+$ $Q$ $\cos \beta$ $+$ $R$ $\cos \gamma$ $)$ $\mathrm{d}S$ $=$ ${\iiint }_{\mathrm{\Omega }}$ $($ $\frac{\partial P}{\partial x}$ $+$ $\frac{\partial Q}{\partial y}$ $+$ $\frac{\partial R}{\partial z}$ $)$ $\mathrm{d}V$

其中，$\partial \mathrm{\Omega }$ 是空间 $\mathrm{\Omega }$ 整个边界曲面的外侧，$\cos \alpha$, $\cos \beta$, $\cos \gamma$ 为 $\partial \mathrm{\Omega }$ 的外法向量的方向余弦。

${\oint }_L$ $P$ $\mathrm{d}x$ $+$ $Q$ $\mathrm{d}y$ 与路径无关 $\iff$ ${\oint }_L$ $P$ $\mathrm{d}x$ $+$ $Q$ $\mathrm{d}y$ $=$ $0$ $\iff$ $\frac{\partial Q}{\partial x}$ $=$ $\frac{\partial P}{\partial y}$, $\mathrm{\forall }(x,y)$ $\in$ $D$ $\iff$ 存在函数 $u(x,y)$, $(x,y)$ $\in$ $D$, 使得 $\mathrm{d}$ $u(x,y)$ $=$ $P$ $\mathrm{d}x$ $+$ $Q$ $\mathrm{d}y$, 此时 $u(x,y)$ $=$ ${\int }_{(x_0,y_0)}^{(x,y)}$ $P$ $\mathrm{d}x$ $+$ $Q$ $\mathrm{d}y$.

${\oint }_L$ $P$ $\mathrm{d}x$ $+$ $Q$ $\mathrm{d}y$ $=$ ${\iint }_D$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{d}x$ $\mathrm{d}y$

格林公式描述了第二类曲线积分和二重积分之间的关系，一般用于二元函数的全微分求积.

$F_x$ $=$ ${\iiint }_{\mathrm{\Omega }}$ $\frac{G{m}_0\rho (x,y,z)(x-x_0)}{{[{(x-x_0)}^2+{(y-y_0)}^2+{(z-z_0)}^2]}^{\frac{3}{2}}}$ $\mathrm{d}v$,
$F_y$ $=$ ${\iiint }_{\mathrm{\Omega }}$ $\frac{G{m}_0\rho (x,y,z)(y-y_0)}{{[{(x-x_0)}^2+{(y-y_0)}^2+{(z-z_0)}^2]}^{\frac{3}{2}}}$ $\mathrm{d}v$,
$F_z$ $=$ ${\iiint }_{\mathrm{\Omega }}$ $\frac{G{m}_0\rho (x,y,z)(z-z_0)}{{[{(x-x_0)}^2+{(y-y_0)}^2+{(z-z_0)}^2]}^{\frac{3}{2}}}$ $\mathrm{d}v$.