P373练习
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设 $\displaystyle f(x,y,z)$ 在 $\displaystyle \mathbb\{R\}^3$ 上有连续的偏导数, 且关于 $\displaystyle x,y$ 各以 $\displaystyle 1$ 为周期, 即:$\displaystyle \forall\ (x,y,z)\in \mathbb\{R\}^3$, 恒有
\begin\{aligned\} f(x+1,y,z)=f(x,y+1,z)=f(x,y,z+1)=f(x,y,z). \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
求证:对任意实数 $\displaystyle \alpha,\beta,\gamma$, 有
\begin\{aligned\} \iiint\_\varOmega \left(\alpha \frac\{\partial f\}\{\partial x\}+\beta\frac\{\partial f\}\{\partial y\}+\gamma\frac\{\partial f\}\{\partial z\}\right)\mathrm\{ d\} x\mathrm\{ d\} y\mathrm\{ d\} z=0, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
其中 $\displaystyle \varOmega=\times \times $ 是单位立方体. (南开大学)
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\begin\{aligned\} \iiint\_\varOmega \alpha \frac\{\partial f\}\{\partial x\}\mathrm\{ d\} x\mathrm\{ d\} y\mathrm\{ d\} z &=\alpha\iint\_\{^2\}\mathrm\{ d\} y\mathrm\{ d\} z\int\_0^1\frac\{\partial f\}\{\partial x\}\mathrm\{ d\} x\\\\ &=\alpha \int\_\{^2\}\left\\mathrm\{ d\} y\mathrm\{ d\} z=0. \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
同理,
\begin\{aligned\} \iiint\_\varOmega \beta\frac\{\partial f\}\{\partial y\}\mathrm\{ d\} x\mathrm\{ d\} y\mathrm\{ d\} z=0 =\iiint\_\varOmega \gamma\frac\{\partial f\}\{\partial z\}\mathrm\{ d\} x\mathrm\{ d\} y\mathrm\{ d\} z=0, \tiny\boxed\{\begin\{array\}\{c\}\mbox\{跟锦数学微信公众号\}\\\\\mbox\{zhangzujin.cn\}\end\{array\}\}\end\{aligned\}
故有结论. 跟锦数学微信公众号. [在线资料](https://mp.weixin.qq.com/s/F-TU-uzeo3EjxI5LzjUvRw) / [公众号](https://mp.weixin.qq.com/s/pdC49P5WZXTEpRBa0JBfow) / [资料目录](https://mbd.pub/o/bread/mbd-YpiZmJdv) / [视频](https://space.bilibili.com/507709073) / [微信群](https://mbd.pub/o/bread/mbd-YZmTkp1w)
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