## 选项

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

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

[C].   $\operatorname{div} \boldsymbol{A}$ $=$ $\frac{\partial P}{\partial x}$ $\times$ $\frac{\partial Q}{\partial y}$ $\times$ $\frac{\partial R}{\partial z}$

[D].   $\operatorname{div} \boldsymbol{A}$ $=$ $\frac{\partial P}{\partial x}$ $-$ $\frac{\partial Q}{\partial y}$ $-$ $\frac{\partial R}{\partial z}$

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

## 选项

[A].   $\iint_{\Sigma}$ $\boldsymbol{A}$ $\mathrm{~d} S$

[B].   $\iint_{\Sigma}$ $\boldsymbol{A}$ $\cdot$ $\boldsymbol{n}$ $\mathrm{~d} S$

[C].   $\boldsymbol{n}$ $\cdot$ $\iint_{\Sigma}$ $\boldsymbol{A}$ $\mathrm{~d} S$

[D].   $\iint_{\Sigma}$ $\boldsymbol{A}$ $+$ $\boldsymbol{n}$ $\mathrm{~d} S$

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

## 选项

[A].   $\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma}$ $($ $\frac{\partial R}{\partial y}$ $+$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d} y$ $\mathrm{~d} z$ $+$ $($ $\frac{\partial P}{\partial z}$ $+$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{~d} z \mathrm{~d} x$ $+$ $($ $\frac{\partial Q}{\partial x}$ $+$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{~d} x \mathrm{~d} y$

[B].   $\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma}$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d} y$ $\mathrm{~d} z$ $+$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{~d} z \mathrm{~d} x$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{~d} x \mathrm{~d} y$

[C].   $\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $\times$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma}$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d} y$ $\mathrm{~d} z$ $\times$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{~d} z \mathrm{~d} x$ $+$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{~d} x \mathrm{~d} y$

[D].   $\oint_{\Gamma}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $+$ $R$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma}$ $($ $\frac{\partial R}{\partial y}$ $-$ $\frac{\partial Q}{\partial z}$ $)$ $\mathrm{d} y$ $\mathrm{~d} z$ $-$ $($ $\frac{\partial P}{\partial z}$ $-$ $\frac{\partial R}{\partial x}$ $)$ $\mathrm{~d} z \mathrm{~d} x$ $-$ $($ $\frac{\partial Q}{\partial x}$ $-$ $\frac{\partial P}{\partial y}$ $)$ $\mathrm{~d} x \mathrm{~d} y$

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

## 选项

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

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

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

[D].    $P$ $\mathrm{d} y \mathrm{d} z$ $+$ $Q$ $\mathrm{d} z \mathrm{d} x$ $+$ $R$ $\mathrm{d} x \mathrm{d} y$ $=$ $\iiint_{\Omega}$ $($ $\partial P \cdot \partial x$ $+$ $\partial Q \cdot \partial y$ $+$ $\partial R \cdot \partial z$ $)$ $\mathrm{d} V$

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

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

## 选项

[A].   $\oint_{L}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $\neq$ $1$

[B].   $\oint_{L}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $\neq$ $0$

[C].   $\oint_{L}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $=$ $1$

[D].   $\oint_{L}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $=$ $0$

$\oint_{L}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ 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 $u(x, y)$ $=$ $\int_{\left(x_{0}, y_{0}\right)}^{(x, y)}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$.

## 选项

[A].   $\oint_{L}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $=$ $\iint_{D}$ $\big($ $\frac{\partial Q}{\partial x}$ $\times$ $\frac{\partial P}{\partial y}$ $\big)$ $\mathrm{d} x$ $\mathrm{~d} y$

[B].   $\oint_{L}$ $P$ $\mathrm{~d} x$ $+$ $Q$ $\mathrm{~d} y$ $=$ $\iint_{D}$ $\big($ $\frac{\partial Q}{\partial x}$ $\div$ $\frac{\partial P}{\partial y}$ $\big)$ $\mathrm{d} x$ $\mathrm{~d} y$

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

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

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

## 选项

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

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

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

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

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

## 选项

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

[B].   $I_{x}$ $=$ $\iiint_{\Omega}$ $($ $y^{\prime}$ $+$ $z^{\prime}$ $)$ $\rho(x, y, z)$ $\mathrm{d} v$, $I_{y}$ $=$ $\iiint_{\Omega}$ $($ $x^{\prime}$ $+$ $z^{\prime}$ $)$ $\rho(x, y, z)$ $\mathrm{d} v$, $I_{z}$ $=$ $\iiint_{\Omega}$ $($ $x^{\prime}$ $+$ $y^{\prime}$ $)$ $\rho(x, y, z)$ $\mathrm{d} v$

[C].   $I_{x}$ $=$ $\iiint_{\Omega}$ $($ $y$ $+$ $z$ $)$ $\rho(x, y, z)$ $\mathrm{d} v$, $I_{y}$ $=$ $\iiint_{\Omega}$ $($ $x$ $+$ $z$ $)$ $\rho(x, y, z)$ $\mathrm{d} v$, $I_{z}$ $=$ $\iiint_{\Omega}$ $($ $x$ $+$ $y$ $)$ $\rho(x, y, z)$ $\mathrm{d} v$

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

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

## 选项

[A].   $I_{x}$ $=$ $\iint_{D}$ $y^{\prime}$ $\rho(x, y)$ $\mathrm{d} \sigma$, $I_{y}$ $=$ $\iint_{D}$ $x^{\prime}$ $\rho(x, y)$ $\mathrm{d} \sigma$

[B].   $I_{x}$ $=$ $\iint_{D}$ $y$ $\rho(x, y)$ $\mathrm{d} \sigma$, $I_{y}$ $=$ $\iint_{D}$ $x$ $\rho(x, y)$ $\mathrm{d} \sigma$

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

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

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

## 选项

[A].   $\bar{x}$ $=$ $\frac{\iiint_{\Omega} x \rho^{\prime}(x, y, z) \mathrm{d} v}{\iiint_{\Omega} \rho(x, y, z) \mathrm{d} v}$, $\bar{y}$ $=$ $\frac{\iiint_{\Omega} y \rho^{\prime}(x, y, z) \mathrm{d} v}{\iiint_{\Omega} \rho(x, y, z) \mathrm{d} v}$, $\bar{z}$ $=$ $\frac{\iiint_{\Omega} z \rho^{\prime}(x, y, z) \mathrm{d} v}{\iiint_{\Omega} \rho(x, y, z) \mathrm{d} v}$

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

[C].   $\bar{x}$ $=$ $\frac{\iiint_{\Omega} x \rho(x, y, z) \mathrm{d} v}{\iiint_{\Omega} \rho(x, y, z) \mathrm{d} v}$, $\bar{y}$ $=$ $\frac{\iiint_{\Omega} y^{2} \rho(x, y, z) \mathrm{d} v}{\iiint_{\Omega} \rho(x, y, z) \mathrm{d} v}$, $\bar{z}$ $=$ $\frac{\iiint_{\Omega} z^{2} \rho(x, y, z) \mathrm{d} v}{\iiint_{\Omega} \rho(x, y, z) \mathrm{d} v}$

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

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

## 选项

[A].   $\bar{x}$ $=$ $\frac{\iint_{D} x \rho^{\prime}(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$, $\bar{y}$ $=$ $\frac{\iint_{D} y \rho^{\prime}(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$

[B].   $\bar{x}$ $=$ $\frac{\iint_{D} x^{\prime} \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$, $\bar{y}$ $=$ $\frac{\iint_{D} y^{\prime} \rho(x, y) \mathrm{d} \sigma}{\iint_{D} \rho(x, y) \mathrm{d} \sigma}$

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

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

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

## 选项

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

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

[C].   $S$ $=$ $\iint_{D_{x y}}$ $\sqrt{1+\left(\frac{\partial z}{\partial x}\right)+\left(\frac{\partial z}{\partial y}\right)}$ $\mathrm{~d} x \mathrm{~d} y$

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

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

## 选项

[A].   $V$ $=$ $-$ $\iint_{D}$ $z(x, y)$ $\mathrm{d} x \mathrm{~d} y$

[B].   $V$ $=$ $\iint_{D}$ $z(x, y)$ $\mathrm{d} x \mathrm{~d} y$

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

[D].   $V$ $=$ $\iint_{D^{2}}$ $|$ $z(x, y)$ $|$ $\mathrm{d} x \mathrm{~d} y$

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

## 选项

[A].   $\begin{cases} \iint_{\Sigma^{-}} P \mathrm{~d} y \mathrm{~d} z=\iint_{\frac{1}{\Sigma}} P \mathrm{~d} y \mathrm{~d} z, \\ \iint_{\Sigma^{-}} Q \mathrm{~d} z \mathrm{~d} x=\iint_{\frac{1}{\Sigma}} Q \mathrm{~d} z \mathrm{~d} x, \\ \iint_{\Sigma^{-}} R \mathrm{~d} x \mathrm{~d} y=\iint_{\frac{1}{\Sigma}} R \mathrm{~d} x \mathrm{~d} y. \end{cases}$

[B].   $\begin{cases} \iint_{\Sigma^{-}} P \mathrm{~d} y \mathrm{~d} z=-\iint_{\frac{1}{\Sigma}} P \mathrm{~d} y \mathrm{~d} z, \\ \iint_{\Sigma^{-}} Q \mathrm{~d} z \mathrm{~d} x=-\iint_{\frac{1}{\Sigma}} Q \mathrm{~d} z \mathrm{~d} x, \\ \iint_{\Sigma^{-}} R \mathrm{~d} x \mathrm{~d} y=-\iint_{\frac{1}{\Sigma}} R \mathrm{~d} x \mathrm{~d} y. \end{cases}$

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

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

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

## 选项

[A].   $\iint_{\Sigma}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$ $=$ $\iint_{\frac{1}{\Sigma_{1}}}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$ $+$ $\iint_{\frac{1}{\Sigma_{2}}}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$

[B].   $\iint_{\Sigma}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma_{1}}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$ $\times$ $\iint_{\Sigma_{2}}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$

[C].   $\iint_{\Sigma}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$ $=$ $\iint_{\Sigma_{1}}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$ $-$ $\iint_{\Sigma_{2}}$ $P$ $\mathrm{~d} y$ $\mathrm{~d} z$

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

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