Consider some smooth curve $C \subset \mathbb{R^n}$ and $\gamma:[a,b] \subset\mathbb{R}\rightarrow C$ a parametrisation of $C$ and a continuous vector field $K:\mathbb{R^n} \rightarrow \mathbb{R^n}$. Let $\omega = K_{1}dx^{1}+…+K_{n} dx^{n}$ where $K_{1},…,K_{n}$ are the components of $K$ with respect to the standard basis of $\mathbb{R^n}$. Now the following holds:
$$\int_{c}\vec{K}\cdot\vec{ds} = \int_{c}\vec{K}\cdot\hat{n}\space ds:=\int_{a}^{b}\langle K(\gamma(t)),\dot{\gamma}(t)\rangle \space dt =\int_{a}^{b}\sum_{i=1}^{n}K_{i}(\gamma(t))\space\dot{\gamma_{i}}(t)\space dt$$ where $\langle.,.\rangle$ is the standard inner product.
One can also write the same integral using a differential form:
$$\int_{\gamma}\omega:=\int_{a}^{b}\gamma^{*}\omega=\int_{a}^{b}\omega(\gamma(t))\space \dot{\gamma}(t)\space dt =\int_{a}^{b}\sum_{i=1}^{n}K_{i}(\gamma(t))\space\dot{\gamma_{i}}(t)\space dt$$
Similarly let $S \subset \mathbb{R^3}$ be a smooth surface (2-dim submanifold) and $\phi:U\subset\mathbb{R^2}\rightarrow S$ a parametrisation of $S$. $\space F:\mathbb{R^3}\rightarrow\mathbb{R^3}$ a continuous vectorfield. Let $\eta = F_{1}\space dx\wedge dy -F_{2}\space dx \wedge dz +F_{3}\space dy \wedge dz$ where $F_{1},F_{2},F_{3}$ are the components of F with respect to the standard basis of $\mathbb{R^3}$.
Now the following holds:
$$\int_{S}\vec{F}\cdot \vec{dA} = \int_{S}\vec{F}\cdot\hat{n}\space dA :=\int_{U}\langle F,\frac{\partial\phi}{\partial u}\times\frac{\partial\phi}{\partial v}\rangle\space d\mu(u,v)$$
And the same Integral using the differential form:
$$\int_{S}\eta:=\int_{U}\phi^{*}\eta= \int_{U}\langle F,\frac{\partial\phi}{\partial u}\times\frac{\partial\phi}{\partial v}\rangle\space d\mu(u,v)$$
My Question is:
How do I express the following integrals using differential forms?
Let $\space f:\mathbb{R^n}\rightarrow\mathbb{R}$ and $g:\mathbb{R^3}\rightarrow \mathbb{R}$ be continuous functions.
$$\int_{c}f\space ds := \int_{a}^{b}f(\gamma(t))\space\lVert\dot{\gamma}(t)\rVert\space dt$$
$$\int_{S}g\space dA := \int_{U} g(\phi(u,v)) \space\lVert\frac{\partial\phi}{\partial u}\times\frac{\partial\phi}{\partial v}\rVert\space d\mu(u,v)$$
Help is greatly appreciated.
Vincent Pfenninger
Best Answer
For an oriented m-dimensional Riemannian manifold $(M,g)$ there is a unique m-form $\omega$ such that $\omega_{p}(e_{1},\ldots,e_{m})=1$ for $\lbrace e_{i} \rbrace_{i=1}^{m}\subset T_{p}M$ a g-orthonormal basis ordered according to the orientation. For a function $f$ on $M$ which is sufficiently nice one defines $\int\limits_{M} f:= \int\limits_{M}f \cdot \omega$. For a chart $\phi: U \to \phi(U)=:O$ it is easy to check that $\phi^{*}\omega=\sqrt{\det(g(x))} dx^{1}\wedge \cdots \wedge dx^{m}$ where $g$ is the matrix of $\phi^{*}g = \sum\limits_{i,j}g_{ij}(x) dx^{i} \otimes dx^{j}$. Thus $$ \begin{aligned}\int\limits_{O}f &= \int\limits_{O}f \cdot \omega = \int\limits_{U} \phi^{*}(f \cdot \omega) \\ &=\int\limits_{U}f(\phi(x))\sqrt{\det(g(x))} dx^{1}\wedge \cdots \wedge dx^{m} \equiv \int\limits_{U}f(\phi(x))\sqrt{\det(g(x))} dx. \end{aligned}$$
You are dealing with submanifolds $M$ of Euclidean space $(\mathbb{R}^{n},\delta)$, for which you naturally use the Riemannian metric induced by restricting the Euclidean metric to your submanifold, $g=\delta\big\vert_{M}$. As $\delta=\sum\limits_{i=1}^{n}dy^{i} \otimes dy^{i}$ for the usual coordinates you have for a chart $\det(g(x))=\det((D\phi(x))^{t}D\phi(x))$ where $D\phi(x)$ is the matrix of the differential of $\phi$, leaving us with $$ \int\limits_{O} f = \int\limits_{U}f(\phi(x))\sqrt{\det((D\phi(x))^{t}D\phi(x))} dx.$$ For curves $\gamma: I \to \mathbb{R}^{n}$ it obviously reduces to $\det(g(t))=\Vert \dot{\gamma}(t) \Vert_{2}^{2}$, giving the formula you wrote down. For a surface in $\mathbb{R}^{3}$ it also gives the formula you want, but I leave it to you to check that.
Remark. You probably are not familiar with certain notions I have used here, since no one actually introduces these notions in Analysis I/II in the same generality. You might want to look those things up if you are really interested into certain details or wait for the course in Differential Geometry.