Is there a divergence theorem for higher dimensions and what is it called?

[Jonny_trigonometry]

The fundamental theorem of calculus is basically the divergence theorem but dealing with a ball in R^1 instead of a ball in R^3. The fundamental theorem of Calculus relates the stuff inside the ball to its boundary, just like how the divergence theorem relates the stuff inside a volume with its surface. I was just wondering if there is such a thing as a divergence theorem which relates the stuff inside a hyper volume (a volume in R^4) to a volume--its boundary (a volume in R^3)--and so on for higher R^n. Is there such a theorem and if so, what is it called?

 

[genneth]

It's called Stoke's Theorem, and is one of the most beautiful theorems about. If only more people knew differential geometry, I'm sure it could beat out that old crap by Euler

$$\int_M d\omega = \int_{\partial M} \omega$$

Where M is an m-dimensional oriented manifold and omega is an (m+1)-form. Unfortunately, it's beyond my ability to really state it formally without going into the precise definitions.

This theorem also sits at the focus point of various topics in topology. It's also possible to extend this theorem in such way that a discrete version of differential geometry can be made, where the definition of exterior derivative can be defined through requiring this theorem as an axiom.

 

[tacman]

Yes, there is such a theorem and it is called the generalized Stokes' theorem. It extends the divergence theorem to higher dimensional spaces, such as R^4 and beyond. It also relates the "stuff" inside a hyper volume to its boundary, just like how the divergence theorem relates the "stuff" inside a volume to its surface. The generalized Stokes' theorem is an important tool in vector calculus and has many applications in physics and engineering. It is a fundamental theorem that helps us understand the relationships between different dimensions and volumes in higher dimensional spaces.