[静力学] 阿基米德原理

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Divergence theorem
21. If $f$ is a scalar function with continuous first partial derivatives, prove that
\[
\iint_S f \mathbf{n} d S=\iiint_D \nabla f d V .
\]
[Hint: Use (2) on $f\bf a$, where $\bf a$ is a constant vector, and Problem 27 in Exercises 9.7.]
22. The buoyancy force on a floating object is $\mathbf{B}=-\iint_S p \mathbf{n} d S$, where $p$ is the fluid pressure. The pressure $p$ is related to the density of the fluid $\rho(x, y, z)$ by a law of hydrostatics: $\nabla p=\rho(x, y, z) \mathrm{g}$, where $g$ is the constant acceleration of gravity. If the weight of the object is $\mathbf{W}=m \mathrm{~g}$, use the result of Problem 21 to prove Archimedes' principle, $\mathbf{B}+\mathbf{W}=\mathbf{0}$.

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$f$是所给的标量场, $\mathbf a$是一个常向量场.
对于标量场$f$、向量场$v$有公式$\operatorname{div}(fv)=(∇f)·v+f(\operatorname{div}v)$,
此处$\bf a$是一个常向量场, 即$\operatorname{div}a=0$, 我们有$\operatorname{div}(f\mathbf a)=(∇f)·\bf a$,
将$\mathbf F=f\mathbf a$代入散度定理$$\iint_{S}(f\mathbf a\cdot \mathbf{n}) d S=\iiint_{D} \operatorname{div}(f\mathbf a)d V$$
即$$\mathbf a\cdot \iint_{S}f\mathbf{n} d S=\mathbf a\cdot\iiint_{D}∇f\,d V$$由$\bf a$的任意性,$$\iint_S f \mathbf{n} d S=\iiint_D \nabla f d V .$$同样地, 将$\mathbf F=\mathbf c\times\mathbf P$代入散度定理, 可以证明: 对于一个向量场$\mathbf P(x,y,z)$,$$\iint_S \mathbf P\times \mathbf{n} d S=\iiint_D \operatorname{curl}\mathbf P d V .$$以上证明和MathWorld相同

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对于标量场$p$使用21题:$$\iint_S p\mathbf{n} d S=\iiint_D \nabla p d V .$$使用$\nabla p=\rho \mathrm{g}$$$\iint_S p\mathbf{n} d S=\iiint_D \rho \mathrm{g} d V .$$即$$-\mathbf B=\bf W$$又见Archimedes Principle and Gauss's Divergence Theorem