The Unapologetic Mathematician

(Hyper-)Surface Integrals

The flip side of the line integral is the surface integral.

Given an $n$-manifold $M$ we let $S$ be an oriented $n-1$-dimensional “hypersurface”, which term we use because in the usual case of $M=\mathbb{R}^3$ a hypersurface has two dimensions — it’s a regular surface. The orientation on a hypersurface consists of $n-1$ tangent vectors which are all in the image of the derivative of the local parameterization map, which is a singular cube.

Now we want another way of viewing this orientation. Given a metric on $M$ we can use the inverse of the Hodge star from $M$ on the orientation $n-1$-form of $S$, which gives us a covector $\nu(p)\in\mathcal{T}^*_pM$ defined at each point $p$ of $S$. Roughly speaking, the orientation form of $S$ defines an $n-1$-dimensional subspace of $\mathcal{T}^*_pM$: the covectors “tangent to $S$“. The covector $\nu(p)$ is “perpendicular to $S"$, and since $S$ has only one fewer dimension than $M$ there are only two choices for the direction of such a vector. The choice of one or the other defines a distinction between the two sides of $S$.

If we flip around our covectors into vectors — again using the metric — we get something like a vector field defined on $S$. I say “something like” because it’s only defined on $S$, which is not an open subspace of $M$, and strictly speaking a vector field must be defined on an open subspace. It’s not hard to see that we could “thicken up” $S$ into an open subspace and extend our vector field smoothly there, so I’m not really going to make a big deal of it, but I want to be careful with my language; it’s also why I didn’t say we get a $1$-form from the Hodge star. Anyway, we will call this vector-valued function $dS$, for reasons that will become apparent shortly.

Now what if we have another vector-valued function $F$ defined on $S$ — for example, it could be a vector field defined on an open subspace containing $S$. We can define the “surface integral of $F$ through $S$“, which measures how much the vector field flows through the surface in the direction our orientation picks out as “positive”. And we measure this amount at any given point $p$ by taking the covector at $p$ provided by the Hodge star and evaluating it at the vector $F(p)$. This gives us a value that we can integrate over the surface. This evaluation can be flipped around into our vector field notation and language, allowing us to write down the integral as

$\displaystyle\int\limits_SF\cdot dS$

because the “dot product” $\left[F\cdot dS\right](p)$ is exactly what it means to evaluate the covector dual to $dS(p)$ at the vector $F(p)$. This should look familiar from multivariable calculus, but I’ve been saying that a lot lately, haven’t I?

We can also go the other direction and make things look more abstract. We could write our vector field as a $1$-form $\alpha$, which lets us write our surface integral as

$\displaystyle\int\limits_S\langle\alpha,\nu\rangle\omega_M=\int\limits_S\alpha\wedge*\nu$

where $\omega_M$ is the volume form on $M$. But then we know that $*\nu=\omega_S$ the volume form on $S$. That is, we can define a hypersurface integral of any $1$-form $\alpha$ across any hypersurface $S$ equipped with a volume form $\omega_S$ by

$\displaystyle\int\limits_S\alpha\wedge\omega_S$

whether or not we have a metric on $M$.

October 27, 2011 - Posted by | Differential Geometry, Geometry