[Math] Green’s formula in n-dimensions

estimationfinite element methodgreen's theorempartial differential equationsstability-theory

I'll try to get straight to the point. The professor teaching my FEM course is a bit of a stereotype professor in that he is quite hard to understand. Green's formula is used quite extensively in proving various stability estimates. The version used in the lecture notes looks like this,

$\int\limits_{\Omega} u\Delta\varphi=\int\limits_{\partial\Omega}un^T\nabla\varphi-\int\limits_{\Omega}\nabla u^T\nabla\varphi$, where $n$ is the unit normal to $\partial\Omega$.

What is confusing me is that the formula is used by the lecture notes in proving stability estimates for functions $u,\varphi\in\Omega\subset\mathbf{R}^2$ but also for functions $u,\varphi\in\Omega\subset\mathbf{R}^n$, and I strongly recall Green's formula only being applicable for functions $u,\varphi\in\Omega\subset\mathbf{R}^2$. Searching the internet also gives me this impression, but not in a fully convincing way – hence I find myself writing this.

Additionally, upon asking him after class, I got the contrary impression that the formula is in fact applicable if $\Omega\subset\mathbf{R}^n,\ n>2$. So I am confused by this and am wondering if someone could help sort this out.

This formula is quite useful so if it were the case that it is applicable in the n-dimensional case then that would be quite handy.

Best Answer

The Green formulas are most widely known in 2d, but they can easily be derived from the Gauss theorem (aka. divergence theorem) in $\mathbb R^n$.

In Wikipedia you can find them as Green identities. (also MathWorld which even provides the derivation using the Gauss theorem.)

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