divergence | lim Practice= Perfect

Tag Archives: divergence

Divergence on manifold

October 31, 2013 – 12:38 am

Divergence on manifold Suppose ${(M,g)}$ is a Riemannian manifold. ${X}$ is smooth vector field on it. ${\{e_i\}}$ is a frame at ${p}$ . Suppose ${X=X^ie_i}$ , then define

$\displaystyle \text{div} X=\sum_iX^i_{,i}$

Under a local coordinates near ${p}$ , choose ${\{\frac{\partial}{\partial u^i}\}}$ as the basis of tangent space, then

$\displaystyle \text{div} X=\frac{1}{\sqrt{G}}\sum_i\frac{\partial(\sqrt{G}X^i)}{\partial u^i}$

where ${G=\det(g_{ij})}$ . One way to prove is straightforward verification. Let us see a more intuitive way, from the divergence theorem

$\displaystyle \int_{M}\text{div } XdV=\int_{\partial M}X\cdot \vec{n}\,dS$

Choose ${h\in C_c^\infty(U(p))}$ , where ${U(p)}$ is a neighborhood of ${p}$ , apply the divergence theorem to ${hX}$

$\displaystyle \int_{U}\text{div } (hX)dV=0$

which means

$\displaystyle \int_U h\text{div}\,XdV=-\int_U X\cdot \nabla h \,dV=-\int_U X^i\frac{\partial h}{\partial u^i}\sqrt{G}du^1\wedge\cdots\wedge du^n$

Denote ${\phi:U\rightarrow\mathbb{R}^n}$ is the diffeomorphism and ${\tilde{X}=X\circ\phi^{-1}}$ , ${\tilde{h}=h\circ \phi^{-1}}$ . From Green’s formula

$\displaystyle \int_U X^i\frac{\partial h}{\partial u^i}\sqrt{G}du^1\wedge\cdots\wedge du^n=\int_{\phi(U)}\tilde{X}^i\frac{\partial \tilde{h}}{\partial u^i}\sqrt{G}du^1\cdots du^n$

$\displaystyle =-\int_{\phi(U)}\tilde{h}\frac{\partial (\tilde{X}^i\sqrt{G})}{\partial u^i}du^1\cdots du^n=-\int_{U}\frac{1}{\sqrt{G}}\sum_i\frac{\partial(\sqrt{G}X^i)}{\partial u^i}hdV$