題目

10. (10 points) Let

F(x,y,z)=\dfrac{x\mathbf{i}+y\mathbf{j}+z\mathbf{k}}{(x^2+y^2+z^2)^{3/2}}

and let

D=\left\{(x,y,z)\mid \dfrac{x^2}{4}+\dfrac{y^2}{4}+\dfrac{z^2}{9}=1\right\}

. Find the flux of

F

outward across the boundary of

D

解答

此向量場是以原點為中心的反平方徑向場。橢球面包住原點，取原點附近半徑 $\varepsilon$ 的小球後套散度定理；兩個曲面之間散度為 $0$ ，故外橢球的通量等於小球的通量：

\iint_D\mathbf F\cdot\mathbf n\,\mathrm{d}S =\iint_{r=\varepsilon}\frac1{\varepsilon^2}\,\mathrm{d}S =\frac1{\varepsilon^2}(4\pi\varepsilon^2) =\boxed{4\pi}.