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Let $\phi$ and $\psi$ be two smooth scalar fields:

$$\phi:\mathbb{R}^2\rightarrow\mathbb{R}$$

$$\psi:\mathbb{R}^2\rightarrow\mathbb{R}$$

Consider a closed surface $S$ in $\mathbb{R}^2$ bounded by a contour $C$.

Let the following be the Green's second identity applied to the fields $\phi$ and $\psi$:

$$\intop_{S}\left(\phi\nabla^{2} \psi-\psi\nabla^{2}\phi\right)dS=\oint_{C}\left(\phi\boldsymbol{\nabla}\psi-\psi\boldsymbol{\nabla}\phi\right)\cdot\pmb{n}\;dl$$

  1. This relation holds if one of the functions maps to complex space $\mathbb{C}$, instead of $\mathbb{R}$? Say, $\psi:\mathbb{R}^2\rightarrow \mathbb{C}$?
  2. (Stronger) the same relation holds if the two functions map to complex space ($\phi:\mathbb{R}^2\rightarrow \mathbb{C}$ and $\psi:\mathbb{R}^2\rightarrow \mathbb{C}$), instead of $\mathbb{R}$?
Marcos Henriques
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Oct 11 '22 at 22:19
  • Yes, it holds for the complex case because both sides are $\Bbb{C}$-bilinear with respect to $(phi,\psi)$. If that’s confusing, then just write out $\phi=\phi_1+i\phi_2$, likewise for $\psi$ and expand everything out. – peek-a-boo Oct 12 '22 at 18:59

0 Answers0