Generalizing a surface integral to 4 dimensions

Question

I am trying to evaluate a surface integral, but instead of using a surface in $\mathbb{R}^3$, using a surface in $\mathbb{R}^4$.

That is to say,

$\oint_S f(x,y,z,w)\,dS$, where S is given by some $r(u,v,t) = \left( x(u,v,t) , y(u,v,t) , z(u,v,t) , w(u,v,t)\right)$

So like a line integral has a $|r'(t)|$, a surface integral has a factor of $|r_u \times r_v|$, I read up on a generalization of this using the square root of a Gramian matrix, which I had never heard of before researching it now, but I don't know how to calculate it exactly for a parametric function from $\mathbb{R}^3 \to \mathbb{R}^4$, like we have here for $r(u,v,t)$.

Can someone help me with this evaluation? Does it involve integrating differential forms and manifolds? I know a little bit about differential geometry, but not much.

How do I evaluate these integrals, and what is the $\mathbb{R}^3 \to \mathbb{R}^4$ analog of $|r_u \times r_v|$ ?

Answer 1

If you want to learn more about the general setting, take a look at this previous answer of mine on Integrating using surface and volume elements. The $|r'(t)|$ and $|r_u \times r_v|$ you mention for line and surface integrals (in $\Bbb{R}^3$) are simply the square root of the determinant Gramian matrix (I leave it to you to verify this).

In your particular case, since $S$ sits inside of some Euclidean space, we can give it the induced Riemannian metric (i.e we can take dot/inner products of vectors which are tangent to the surface $S$). So, here's what we do: first we're going to construct a $3\times 3$ matrix-valued function $G$ as follows: \begin{align} G &= \begin{pmatrix} \left\langle \frac{\partial r}{\partial u}, \frac{\partial r}{\partial u}\right\rangle & \left\langle \frac{\partial r}{\partial u}, \frac{\partial r}{\partial v} \right\rangle & \left\langle \frac{\partial r}{\partial u}, \frac{\partial r}{\partial t}\right\rangle \\ \left\langle \frac{\partial r}{\partial v}, \frac{\partial r}{\partial u} \right\rangle & \left\langle \frac{\partial r}{\partial v}, \frac{\partial r}{\partial v} \right\rangle & \left\langle \frac{\partial r}{\partial v}, \frac{\partial r}{\partial t} \right\rangle\\ \left\langle \frac{\partial r}{\partial t}, \frac{\partial r}{\partial u} \right\rangle & \left\langle \frac{\partial r}{\partial t}, \frac{\partial r}{\partial v} \right\rangle & \left\langle \frac{\partial r}{\partial t}, \frac{\partial r}{\partial t} \right\rangle \end{pmatrix} \end{align} Note that this is a matrix-valued function which means for each $(u,v,t)$, $G(u,v,t)$ is a $3\times 3$-symmetric matrix of numbers obtained by evaluating all the above partial derivatives at the point $(u,v,t)$.

Since the inner product is symmetric: $\langle v,w\rangle = \langle w,v\rangle$ (and in the case of this Euclidean inner product it is just $\sum_i v^iw^i$), it follows that $G$ is a symmetric matrix, so if you have to actually compute a specific example, you only have to compute the upper triangular portion. As a very explicit example, the $(1,3)$ entry of this matrix is \begin{align} \left\langle \frac{\partial r}{\partial u}, \frac{\partial r}{\partial t} \right\rangle &= \dfrac{\partial x}{\partial u}\dfrac{\partial x}{\partial t} + \dfrac{\partial y}{\partial u}\dfrac{\partial y}{\partial t} + \dfrac{\partial z}{\partial u}\dfrac{\partial z}{\partial t} + \dfrac{\partial w}{\partial u}\dfrac{\partial w}{\partial t}. \end{align} Now, suppose the parametrization is $r:A\subset \Bbb{R}^3\to r[A] = S\subset\Bbb{R}^4$. Then, \begin{align} \int_S f \, dS &= \int_A f\circ r \cdot \sqrt{\det G} \\ &\equiv \int_A f(r(u,v,t)) \cdot \sqrt{\det[G(u,v,t)]}\, du\,dv\,dt. \end{align} (where $\equiv$ means "same thing in different notation"). Now, this triple integral over $A\subset \Bbb{R}^3$ can be calculated for example using Fubini's theorem.