二重积分 $\displaystyle I=\iint_D \frac{x^2-2xy+y^2}{x^2+y^2}\mathrm{d}x\mathrm{d}y$，其中$D$是由平面曲线 $x^2+y^2-\sqrt{x^2+y^2}+x=0$ 所围成的有界闭区域，则$I=$ .
A. $\pi$;
B. $\displaystyle\frac{3}{2}\pi$;
C. $\displaystyle\frac{\pi}{2}$;
D. $0$.
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设 $L$ 是以 $(0,0),(\pi,0),(\pi,\pi)$ 为顶点的三角形的正向边界，利用格林公式计算曲线积分$\displaystyle\oint_{L}y\mathrm{d}x-3x\mathrm{d}y=$ .
A. $2\pi^2$ ;
B. $-2\pi^2$ ;
C. $4\pi^2$ ;
D. $-4\pi^2$ .
三重积分$\displaystyle I=\iiint_\Omega(x^2+y^2+z^2)dv$,其中$\Omega$是由$z^2=x^2+y^2$与$z=-1$围成的区域,则$I$可化为____
A. $ \int_0^{2\pi}d\theta\int_0^1\rho d\rho\int_0^\rho(\rho^2+z^2)dz$
B. $\displaystyle\int_0^{2\pi}d\theta\int_0^1\rho d\rho\int_\rho^{-1}(\rho^2+z^2)dz$
C. $\displaystyle4\int_0^{\frac{\pi}{2}}d\theta\int_0^1\rho d\rho\int_{-1}^{-\rho}(\rho^2+z^2)dz$
D. $\displaystyle\int_0^{\frac{\pi}{2}}d\theta\int_0^1\rho d\rho\int_\rho^{-1}(\rho^2+z^2)dz$
设$P(x,y),Q(x,y)$在单连通区域$D$内存在一阶连续偏导数,则曲线积分$\displaystyle\int_CQ(x,y)dx-P(x,y)dy$在$D$内积分与路经$C$无关的充分必要条件是____
A. $\displaystyle \frac{\partial{Q}}{\partial{x}}=\frac{\partial{P}}{\partial{y}}$
B. $\displaystyle -\frac{\partial{P}}{\partial{x}}=\frac{\partial{Q}}{\partial{y}}$
C. $\displaystyle \frac{\partial{Q}}{\partial{x}}=-\frac{\partial{P}}{\partial{y}}$
D. $\displaystyle \frac{\partial{Q}}{\partial{y}}=\frac{\partial{P}}{\partial{x}}$
设有界单连通闭区域$D$的边界正向曲线为$C$,$P(x,y),Q(x,y)$在该闭区域$D$上存在一阶连续偏导数,则用格林公式将曲线积分$\displaystyle\oint_CP(x,y)dx+Q(x,y)dy$化为二重积分的形式为____
A. $\displaystyle \iint_D(\frac{\partial{Q}}{\partial{x}}+\frac{\partial{P}}{\partial{y}})d\sigma$
B. $\displaystyle \iint_D(\frac{\partial{P}}{\partial{x}}-\frac{\partial{Q}}{\partial{y}})d\sigma$
C. $\displaystyle \iint_D(Q-P)d\sigma$
D. $\displaystyle \iint_D(\frac{\partial{Q}}{\partial{x}}-\frac{\partial{P}}{\partial{y}})d\sigma$
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