Riemannian divergence theorem

Given: The stated oriented manifold, vector field, and countable choice.

Coordinate formula for riemannian divergence: In coordinates, ${\mathrm{div}}_{g}X=(\det G{)}^{-1\mathrm{/}2}{\sum }_{i=1}^n{\mathrm{\partial }}_i((\det G{)}^{1\mathrm{/}2}{X}^i)$.

The riemannian volume form is the unique positive unit top form: The Riemannian volume form is the unique positive unit section of ${\bigwedge }^n{T}^{\ast }M$ for the specified orientation and normalized exterior metric.

Pullback of a riemannian metric is riemannian exactly for immersions: $F^{\ast }h$ is Riemannian if and only if $F$ is an immersion. In general it is positive semidefinite, with radical $\ker d{F}_p$ at $p$.

The general Stokes theorem: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $M$ be an oriented smooth $n$-manifold with boundary, $n\ge 1$, and let $\eta \in {\mathrm{\Omega }}_c^{n-1}(M)$. With $j:\mathrm{\partial }M\hookrightarrow M$ and the outward-normal-first orientation, ${\int }_{M}d\eta ={\int }_{\mathrm{\partial }M}{j}^{\ast }\eta \mathrm{.}$ An empty boundary contributes zero; in dimension one its integral is a finite signed sum of point values.

Induced boundary orientation: For an oriented manifold with boundary, orient $T_p\mathrm{\partial }M$ by the outward-normal-first rule: an outward vector first, followed by a positive boundary determinant, is a positive determinant of $T_{p}M$.

The boundary tangent space is the boundary-tangent hyperplane: For $p\in \mathrm{\partial }M$ of an $n\ge 1$ dimensional manifold and the inclusion $i:\mathrm{\partial }M\hookrightarrow M$, the differential $d{i}_p$ identifies $T_p\mathrm{\partial }M$ with the hyperplane of boundary-tangent vectors in $T_{p}M$.

The gradient is characterized by inner products: The gradient is the unique smooth vector field $Y$ satisfying $g(Y,X)=Xf$ for every smooth vector field $X$.

Riemannian metrics induce metrics on dual tensor and exterior bundles: A Riemannian metric induces smooth metrics on dual, tensor and exterior bundles. On decomposable covectors, $⟨{\alpha }_1\wedge \cdots \wedge {\alpha }_k,{\beta }_1\wedge \cdots \wedge {\beta }_k⟩=\det (⟨{\alpha }_i,{\beta }_j⟩)$; increasing orthonormal wedge monomials have norm one.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): The Axiom of Countable Choice, written ${\mathrm{A}\mathrm{C}}_{\omega }$, is the following statement. > For every family $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets indexed by > $\mathbb{N}$ there is a function $f$ with domain $\mathbb{N}$ such that > $f(n)\in X_n$ for every $n\in \mathbb{N}$. Equivalently, in the vocabulary of def-choice-function: every at most countable family of nonempty sets (def-countable) has a choice function.