📝 题目
6.5.4 设 $V$ 为可测闭区域, $\partial V = S$ 为光滑闭曲面,函数 $u\left( {x,y,z}\right) ,v(x,y$ , $z) \in {C}^{2}\left( V\right)$ . 证明:
$$ {\iint }_{S}v\frac{\partial u}{\partial n}\mathrm{\;d}S = {\iiint }_{v}v\left( {\frac{{\partial }^{2}u}{\partial {x}^{2}} + \frac{{\partial }^{2}u}{\partial {y}^{2}} + \frac{{\partial }^{2}u}{\partial {z}^{2}}}\right) \mathrm{d}x\mathrm{\;d}y\mathrm{\;d}z $$
$$ + {\iiint }_{V}\left\lbrack {\frac{\partial u}{\partial x}\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\frac{\partial v}{\partial y} + \frac{\partial u}{\partial z}\frac{\partial v}{\partial z}}\right\rbrack \mathrm{d}x\mathrm{\;d}y\mathrm{\;d}z, $$
其中 $\mathbf{n}$ 为 $S$ 的外法线方向.
💡 答案与解析
6. 5.1 (1) ${4\pi }{R}^{3};\;\left( 2\right) \frac{2}{3}\pi {h}^{3};\;\left( 3\right) 1$ (利用变换求重积分).