3D Generalization of Surface Intregral

[maverick_starstrider]

Hi,

I'm pondering the relation for the surface integral of a membrane

$$\sqrt{1+\nabla \psi \cdot \nabla \psi}$$

My two questions are:

1) Does this expression have an identical form if the "membrane" is a scalar field in 3 dimensions (like Temperature)
2) How does one derive this result for 3 dimensions (I see how it is done in 2D but that requires cross products which don't exist if we add another dimension)

Thanks for the help.

 

[maverick_starstrider]

I know there is a similar results in spivak for higher dimensional manifolds I just really don't know enough about calculus on manifolds (or wedge products) to get an expression of gradients out of it. For physical reasons $$\sqrt{1+\nabla \phi \nabla \phi}$$ is just the solution I want I just don't know the derivation for 3 dimensional manifolds (i.e. volumes). Or if that is the correct results for that matter. Can the volume integral of a scalar function in any way be equated to something like

$$\sqrt{1+\left(\frac{df}{dx}\right)^2+\left(\frac{df}{dy}\right)^2+\left(\frac{df}{dz}\right)^2}$$

 

[maverick_starstrider]

"Surface area" of a 3 dimensional manifold in 4 dimensional space, anyone?

 

[maverick_starstrider]

If anyone's curious I answered this myself. It does indeed generalize. It is determined by an integral of a triple wedge product of the form (in cartesian coordinates)

$$\int_V \left| \frac{\partial \vec{r}}{\partial x} \wedge \frac{\partial \vec{r}}{\partial y} \wedge \frac{\partial \vec{r}}{\partial z} \right| dV$$

where $$r = (x,y,z, \phi(x,y,z))$$. This becomes (if you do the product)

$$\int_V \left| (-\frac{\partial \phi}{\partial x},-\frac{\partial \phi}{\partial y},-\frac{\partial \phi}{\partial z},1) \right|$$

which is

$$\sqrt{1 + \nabla \phi \cdot \nabla \phi}$$

I can then use this to generate a series expansion for the fluctuations in Landau's theory of phase transitions (without saying some hand-wavy nonsense like $$\frac{1}{2} \nabla \phi \cdot \nabla \phi$$ is the "simplest" gradient term one can think of that obeys symmetries). I just expand this guy (which will be directly proportional to the amount of fluctuations) to get the first and as many higher order terms as I want.