Differential forms, integration and the generalized Stokes theorem (part II)

Today let's see how to integrate differential forms. We begin with $\mathbb{R}^n$ and then move on to manifolds. The $n$-form $dx^1 \wedge \cdots \wedge dx^n$ is a basis for the space of linear $n$-forms on $\mathbb{R}^n$; therefore any differential $n$-form on $\mathbb{R}^n$ is given by \[ \omega = f(x^1, \ldots, x^n) dx^1 \wedge \cdots \wedge dx^n \] To recover the function $f$ from $\omega$ we can just evaluate it on the standard basis: \[ f(p) = \omega_p(e_1, \ldots, e_n)\] Therefore $n$-forms in $\mathbb{R}^n$ are in direct correspondence to scalar functions. Integrating $\omega$ is easy: we set \[ \int_{\mathbb{R}^n} \omega = \int_{\mathbb{R}^n} f(x^1, \ldots, x^n) dx^1 \cdots dx^n \] where the integral on the right hand side is the ordinary Riemann integral of the function $f$[1]. The same definition holds for integrating $\omega$ over any domain $D \subseteq \mathbb{R}^n$ (some reasonably well-behaved subset, possibly with boundary).

Now let $M$ be an oriented $n$-dimensional manifold[2]. For simplicity we'll assume given a (properly oriented) parametrization[3] $\phi : D \to M$ of $M$. If $\omega$ is an $n$-form on $M$, we define \[ \int_M \omega = \int_D \phi^* \omega \] where $\phi^* \omega$ is the pullback form defined by, given tangent vectors $v_1, \ldots, v_n$ at the point $p \in D$[4], \[ (\phi^* \omega)_p(v_1, \ldots, v_n) = \omega_{\phi(p)}(D_p \phi \cdot v_1, \ldots, D_p \phi \cdot v_n) \] This integral can be proven to be independent of the parametrization, provided it induces the given orientation on $M$.

Example: Consider a vector field $X$ on $\mathbb{R}^n$. We can re-interpret it as a differential $1$-form $X^\flat$ defined by \[ X^\flat_p(v) = X(p) \cdot v \] Exercise: Show that, if $X = (X^1(x^1, \ldots, x^n), \ldots, X^n(x^1, \ldots, x^n))$, then \[ X^\flat = X^1 dx^1 + \cdots + X^n dx^n \] Now suppose $\gamma : [0, 1] \to \mathbb{R}^n$ parametrizes a curve. We have \[ \int_{\gamma} X^\flat = \int_{[0,1]} \gamma^* X^\flat \] By definition, \[ (\gamma^* X^\flat)_t(v) = X^\flat(\gamma'(t)v) = X(\gamma(t)) \cdot \gamma'(t)v \] where $v$ is a tangent vector at $t \in [0, 1]$. Since this is just a number, $dt(v) = v$. The expression in coordinates is then \[ \gamma^* X^\flat = X(\gamma(t)) \cdot \gamma'(t)\ dt \] Therefore \[ \int_\gamma X^\flat = \int_{0}^{1} X(\gamma(t)) \cdot \gamma'(t)\ dt\] is the line integral of the vector field $X$ along the curve $\gamma$. Observe that, indeed, such line integrals get a flipped sign if you switch the direction of the parametrization of $\gamma$.

Example: Consider a vector field $X$ on $\mathbb{R}^3$. We can re-interpret it as a $2$-form $\omega$[5] given by \[ \omega_p(v, w) = \det(X(p), v, w) \] Exercise: Show that, for $X = (X^1(x,y,z),X^2(x,y,z),X^3(x,y,z))$, \[ \omega = X^1 dy \wedge dz - X^2 dx \wedge dz + X^3 dx \wedge dy \] Suppose $\phi : D \to \mathbb{R}^3$, where $\phi = \phi(u,v)$ and $D \subseteq \mathbb{R}^2$, parametrizes a surface $\Sigma$ oriented by the normal $\hat{n}$, so that \[ \frac{\partial \phi}{\partial u} \times \frac{\partial \phi}{\partial v} = \left| \frac{\partial \phi}{\partial u} \times \frac{\partial \phi}{\partial v} \right| \hat{n} \] Taking $e_1 = (1, 0)$ and $e_2 = (0, 1)$ we find \begin{equation*} \begin{split} (\phi^* \omega)_p(e_1, e_2) & = \omega_{\phi(p)} \left( D_p \phi \cdot e_1, D_p \phi \cdot e_2 \right) \\ & = \omega_{\phi(p)} \left( \frac{\partial \phi}{\partial u}, \frac{\partial \phi}{\partial v} \right) \\ & = \det \left( X(\phi(p)), \frac{\partial \phi}{\partial u}, \frac{\partial \phi}{\partial v} \right) \\ & = X(\phi(p)) \cdot \left( \frac{\partial \phi}{\partial u} \times \frac{\partial \phi}{\partial v} \right) \\ & = X(\phi(p)) \cdot \hat{n} \left| \frac{\partial \phi}{\partial u} \times \frac{\partial \phi}{\partial v} \right| \end{split} \end{equation*} Therefore \[ \phi^* \omega = \left( X(\phi(p)) \cdot \hat{n} \right) \left| \frac{\partial \phi}{\partial u} \times \frac{\partial \phi}{\partial v} \right| du \wedge dv \] and thus \[ \int_\Sigma \omega = \int_D \left( X(\phi(p)) \cdot \hat{n} \right) \left| \frac{\partial \phi}{\partial u} \times \frac{\partial \phi}{\partial v} \right| du dv \] is the flux of $X$ through $\Sigma$. Again, we would've gotten a flipped sign by choosing the opposite orientation $-\hat{n}$.

This previous example can be generalized to arbitrary dimensions: a vector field $X$ on $\mathbb{R}^n$ can be re-interpreted as the $(n-1)$-form $\omega$ given by \[ \omega_p(v_1, \ldots, v_{n-1}) = \det(X(p), v_1, \ldots, v_{n-1}) \] In coordinates, \[ \omega = X^1 dx^2 \wedge \cdots \wedge dx^n - X^2 dx^1 \wedge dx^3 \wedge \cdots \wedge dx^n + \cdots + (-1)^{n-1} X^n dx^1 \wedge dx^2 \wedge \cdots \wedge dx^{n-1} \] Then the integral of $\omega$ over an oriented hypersurface $\Sigma \subset \mathbb{R}^n$ is equal to the flux of $X$ through $\Sigma$.

Next time we'll be ready to see how it all fits together into the generalized Stokes theorem.

Notes

[1] Of course this may or may not converge -- we typically integrate compactly supported forms only, i.e. those where $f$ is zero outside a compact set ↩

[2] If you don't know the precise definitions, don't worry -- it'll be clear in the examples. ↩

[3] More specifically, $\phi$ is a surjective map $D \to M$ that is injective and $C^1$ with invertible derivative within the interior of the domain $D \subseteq \mathbb{R}^n$, such that the boundary of $D$ maps to a set of $n$-dimensional volume equal to zero. (As an example, think of $D = [0, 2\pi] \times [0, \pi]$ and $\phi : D \to S^2$ the spherical coordinates parametrization). ↩

[4] Here $D_p \phi \cdot v_1$ is the directional derivative of $\phi$ at $p$ acting on the vector $v_1$ and so forth. ↩

[5] This is also known as $*X^b$, where $*$ is the Hodge star. ↩

Comments

Post a Comment

Popular posts from this blog

The joy of quaternions

Image
Quaternions are a lovely topic that usually don't get much attention in the standard undergraduate and graduate textbooks on mathematics. In this post I'll try to go over the basic ideas and motivations of the theory in a hopefully illuminating way; let's get started. The complex numbers $\mathbb{C}$ form a wonderful tool to describe geometry in a plane, since (among other things) complex multiplication describes at once both rotations and scalings in a neatly concise way. Hamilton's plan, which eventually led him to the discovery of quaternions, was to develop an analogous number system that would help describe geometry in three-dimensional space. So how might you do that? Naively, you might try to just extend the complex numbers and take numbers of the form \[ a + bi + cj \] where $a,b,c \in \mathbb{R}$, $i$ is the usual imaginary unit and $j$ is a new quantity which also satisfies $j^2 = -1$. To obtain a multiplication of these new numbers, essentially we just nee
Read more

Classical field theory and the Euler-Lagrange equations

In this post I'll try to briefly give a more-precise-than-you-usually-find explanation of the Euler-Lagrange equations for classical fields. This is a point where textbooks tend to be sloppy so hopefully this post can clear things up somewhat, even though I will ignore the analytical details of convergence and smoothness. In a follow-up post I plan to explain Noether's theorem for fields. Let's begin with the simplest example: a real scalar field. That is simply a function $\phi : \mathbb{R}^{n+1} \to \mathbb{R}$ (assumed to be sufficiently smooth and to decay sufficiently fast at infinity). Here $\mathbb{R}^{n+1}$ stands for one dimension of time and $n$ dimensions of space. We write the coordinates on $\mathbb{R}^{n+1}$ as $x^\mu$, where $\mu$ ranges from $0$ to $n$, with $\mu = 0$ being the time dimension. One obtains the equations of motion for $\phi$ from a Lagrangian density $\mathcal{L}$. In this case, the Lagrangian density is a function [1] $\mathcal{L} : \mat
Read more

Green, Gauss, Stokes: the classical theorems of integral calculus (part I)

Image
In this series of posts I will present the classical theorems of integral calculus in a unified, geometric perspective; later I will show how this generalizes to differential forms, the exterior derivative and the general Stokes theorem. The objective here is not to be rigorous, but to emphasize the geometric intuition in these topics. Let's start off easy, with the ordinary derivative of a function $f : \mathbb{R} \to \mathbb{R}$. Here's an unusual way to think about it: given an interval $[a, b]$, oriented positively (from $a$ to $b$), we can think of its boundary $\partial [a, b]$ as the discrete collection of points $\{a, b\}$, with a twist: $b$ is positively oriented, and $a$ is negatively oriented (it'll become clear what this means). If we integrate the function $f$ along the boundary of $[a, b]$, since an integral over a discrete set is really just a sum we get (keeping in mind the orientations) \[ \int_{\partial [a, b]} f = f(b) - f(a) \] The 'volume' of
Read more