# Surface integral with vector integrand

1. Jun 15, 2010

### thojrie

If we integrate a vector field over a surface, $$\int_S \vec{F} \cdot \vec{dS}$$, we get the flux through that surface. What does it mean if the integrad were a vector instead, $$\int_S \vec{F} dS$$? I can't picture the Riemann sum.

2. Jun 15, 2010

### thrillhouse86

But the surface is a vector because its an orientated surface right (surface + direction) ?
I mean when you do your surface integrals, you don't just define the surface, you define direction of the normal vector of the differential elements of the surface surface

Supposing you have some surface dS
you parameterise it with two variables, say s and t
then we can trace out our surface with some function x(s,t)
and then we take the dot product of the surface normal with our vector field:

$$\int_S {\mathbf v}\cdot \,d{\mathbf {S}} = \int_S ({\mathbf v}\cdot {\mathbf n})\,dS=\iint_T {\mathbf v}(\mathbf{x}(s, t))\cdot \left({\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right) ds\, dt.$$

The key part here is that
$$d\mathbf{S} = \mathbf{n}dS$$

With regards to visualising it as a Riemann sum, this will need to be checked by some one with more mathematical knowledge, but treat the whole dot product
$${\mathbf v}(\mathbf{x}(s, t))\cdot \left({\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right)$$
As the Jacobian which maps your complicated surface integral of a vector field to a simple two dimensional integral which you can then use your standard Riemann sums on

Last edited: Jun 15, 2010
3. Jun 16, 2010

### thojrie

Hi thrillhouse86, thanks for the detailed reply! I don't think I phrased my question well enough though.

Because the dot product of two vectors is a scalar, when we find the flux through a surface the integrand and answer are both scalars. If instead, the integrand were a vector, the integral becomes,

$$\int_S \vec{F} dS$$

which will give a vector answer. I assume this is analogous to flux (but with a direction), however I'm having trouble breaking it down and sussing out exactly what it means.

4. Jun 16, 2010

### elibj123

I don't think it has a specific meaning like flux, it will simply "sum up" all of the different vectors that the field creates on the surface.

If the field would represent something like "Surface current density" then it will give you the total/effective current on the surface.

5. Jun 16, 2010

### HallsofIvy

Writing $\vec{F}= f\vec{i}+ g\vec{j}+ h\vec{k}$, that integral would be
[tex]\int_S f dS\vec{i}+ \int_S g dS\vec{j}+ \int h dS\vec{k}[/itex].

6. Jun 19, 2010

### thojrie

Hey thanks guys. That helped.