月度归档： 2022年10月

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

$\iint_{\Sigma}$ $\boldsymbol{A}$ $\cdot$ $\boldsymbol{n}$ $\mathrm{~d} S$

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

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

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

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

其中，$\partial \Omega$ 是空间 $\Omega$ 整个边界曲面的外侧，$\cos \alpha$, $\cos \beta$, $\cos \gamma$ 为 $\partial \Omega$ 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 $\Leftrightarrow$ $\oint_{L}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $=$ $0$ $\Leftrightarrow$ $\frac{\partial Q}{\partial x}$ $=$ $\frac{\partial P}{\partial y}$, $\forall(x, y)$ $\in$ $D$ $\Leftrightarrow$ 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 $u(x, y)$ $=$ $\int_{\left(x_{0}, y_{0}\right)}^{(x, y)}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$.

$\oint_{L}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $=$ $\iint_{D}$ $\big($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $\big)$ $\mathrm{d} x$ $\mathrm{~d} y$

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.

$F_{x}$ $=$ $\iiint_{\Omega}$ $\frac{G m_{0} \rho(x, y, z)\left(x-x_{0}\right)}{\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right]^{\frac{3}{2}}}$ $\mathrm{~d} v$,
$F_{y}$ $=$ $\iiint_{\Omega}$ $\frac{G m_{0} \rho(x, y, z)\left(y-y_{0}\right)}{\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right]^{\frac{3}{2}}}$ $\mathrm{~d} v$,
$F_{z}$ $=$ $\iiint_{\Omega}$ $\frac{G m_{0} \rho(x, y, z)\left(z-z_{0}\right)}{\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}+\left(z-z_{0}\right)^{2}\right]^{\frac{3}{2}}}$ $\mathrm{~d} v$.

$I_{x}$ $=$ $\iiint_{\Omega}$ $($ $y^{2}$ $+$ $z^{2}$ $)$ $\rho(x, y, z)$ $\mathrm{d} v$, $I_{y}$ $=$ $\iiint_{\Omega}$ $($ $x^{2}$ $+$ $z^{2}$ $)$ $\rho(x, y, z)$ $\mathrm{d} v$, $I_{z}$ $=$ $\iiint_{\Omega}$ $($ $x^{2}$ $+$ $y^{2}$ $)$ $\rho(x, y, z)$ $\mathrm{d} v$

$I_{x}$ $=$ $\iint_{D}$ $y^{2}$ $\rho(x, y)$ $\mathrm{d} \sigma$, $I_{y}$ $=$ $\iint_{D}$ $x^{2}$ $\rho(x, y)$ $\mathrm{d} \sigma$

$\bar{x}$ $=$ $\frac{\iiint_{\Omega} x \rho(x, y, z) \mathrm{d} v}{\iiint_{\Omega} \rho(x, y, z) \mathrm{d} v}$, $\bar{y}$ $=$ $\frac{\iiint_{\Omega} y \rho(x, y, z) \mathrm{d} v}{\iiint_{\Omega} \rho(x, y, z) \mathrm{d} v}$, $\bar{z}$ $=$ $\frac{\iiint_{\Omega} z \rho(x, y, z) \mathrm{d} v}{\iiint_{\Omega} \rho(x, y, z) \mathrm{d} v}$

$\bar{x}$ $=$ $\frac{\iint_{D} x \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$, $\bar{y}$ $=$ $\frac{\iint_{D} y \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$

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

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

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

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