벡터 미적분 전체 문제

Curl.

Green's Theorem.

(a) True. Let $\mathbf{F}$ be the vector field. We are assuming that $\nabla\times\mathbf{F}=0$ on all of $\mathbb{R}^3$. As a result, $\mathbf{F}=\nabla\phi$ for some potential function $\phi$. We are also assuming that $0=\nabla\cdot\mathbf{F} =\nabla\cdot\nabla\phi =\big(\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2}\big)\phi$. This is the definition of “$\phi$ is harmonic”.

(b) False. Let $\mathbf{F}$ be the vector field. We are assuming that $\mathbf{F}=\nabla\phi$ for some potential function $\phi$. If $S$ is any smooth closed surface, with $S$ being the boundary of the solid $V$, then, by the divergence theorem, the outward flux of $\mathbf{F}$ through $S$ is

$$\iint _S \mathbf{F}\cdot\hat{\mathbf{n}}\ \mathrm{d}S =\iiint _V \nabla\cdot\mathbf{F}\ \mathrm{d}V =\iiint _V \nabla\cdot\nabla \phi\ \mathrm{d}V =\iiint _V \big(\tfrac{\partial^2}{\partial x^2} +\tfrac{\partial^2}{\partial y^2} +\tfrac{\partial^2}{\partial z^2}\big)\phi\ \mathrm{d}V$$

If, for example, $\phi=x^2$, then $\big(\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2}\big)\phi=2$ and the flux of $\mathbf{F}$ through $S$ is twice the volume of $V$, which is not zero.