复变函数与电场线方程

青青子衿

复变函数的图像是如何与平面曲线搭建起关系的？

Infinity

因为电场有势，可以用势函数（电势）的梯度表示，而平面静电场是无源场，故而势函数满足拉普拉斯方程。而解析复变函数满足柯西黎曼条件，从而使得实数分量和虚数分量都满足拉普拉斯方程（称之为调和函数），且由于相互正交。因此，可以用一个复变函数来表示一个电场的势函数和流函数，流函数代表电通量，故而可以表示电场线。

因此，随意给定一个解析复变函数，就能表示一个特定分布的静电场。

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vc2.pdf page26

Theorem 8.3. A vector field $\mathbf{v}$ defined on a simply connected domain $\Omega \subset \mathbb{R}^2$ admits a potential, $\mathbf{v}=\nabla \varphi$ for some $\varphi: \Omega \rightarrow \mathbb{R}$ if and only if $\nabla \wedge \mathbf{v} \equiv 0$

We can also apply Green's Theorem 8.1 to flux line integrals of the form (6.17). Using the identification (6.19) followed by (8.1), we find that
\[
\oint_{\partial \Omega} \mathbf{v} \cdot \mathbf{n} d s=\oint_{\partial \Omega} \mathbf{v}^{\perp} \cdot d \mathbf{x}=\iint_{\Omega} \nabla \wedge \mathbf{v}^{\perp} d x d y .
\]
However, note that the curl of the orthogonal vector field (6.18), namely
\[\tag{8.3}
\nabla \wedge \mathbf{v}^{\perp}=\frac{\partial v_1}{\partial x}+\frac{\partial v_2}{\partial y}=\nabla \cdot \mathbf{v},
\]
coincides with the divergence of the original velocity field. Combining these together, we have proved the divergence or flux form of Green's Theorem:
\[\tag{8.4}
\iint_{\Omega} \nabla \cdot \mathbf{v} d x d y=\oint_{\partial \Omega} \mathbf{v} \cdot \mathbf{n} d s .
\]
As before, $\Omega$ is a bounded domain, and $\mathbf{n}$ is the unit outward normal to its boundary $\partial \Omega$.

In the fluid flow interpretation, the right hand side of (8.4) represents the net fluid flux out of the region $\Omega$. Thus, the double integral of the divergence of the flow vector must equal this net change in area. Thus, in the absence of sources or sinks, the divergence of the velocity vector field, $\nabla \cdot \mathbf{v}$ will represent the local change in area of the fluid at each point. In particular, if the fluid is incompressible if and only if $\nabla \cdot \mathbf{v} \equiv 0$ everywhere.

An ideal fluid flow is both incompressible, $\nabla \cdot \mathbf{v}=0$, and irrotational, $\nabla \wedge \mathbf{v}=\mathbf{0}$. Assuming its domain is simply connected, we introduce velocity potential $u(x, y)$, so that $\nabla u=\mathbf{v}$. Therefore
\[\tag{8.5}
0=\nabla \cdot \mathbf{v}=\nabla \cdot \nabla u=u_{x x}+u_{y y}
\]
Therefore, the velocity potential for an incompressible, irrotational fluid flow is a harmonic function, i.e., it satisfies the Laplace equation. Water waves are typically modeled in this manner, and so many problems in fluid mechanics rely on the solution to Laplace's equation.

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