Question

what is the difference between gauss's divergence theorem and stoke's theorem

Answer 1

Answer:

Down vote

In a sense, Stokes', Green's, and Divergence theorems are all special cases of the generalized Stokes theorem for differential forms

∫∂Ωω=∫Ωdω

but I don't think that's what you're asking about.

The usual (3-dimensional) Stokes' and Divergence theorems both involve a surface integral, but they are in rather different circumstances.

In the Divergence theorem, the surface S is the boundary of a bounded region R of space, and you're taking the flux through this surface of a vector field F defined in R and on its boundary:

∬SF⋅dS=∭RdivFdV

In Stokes' theorem, the surface is generally not the boundary of a region: instead it has a boundary which is a curve C; you're taking the flux, not of an arbitrary vector field, but of the curl of some other field:

∬ScurlG⋅dS=∮CG⋅dr

There is one situation where both apply: suppose your surface S is the boundary of a bounded region R, and your vector field F happens to be the curl of some other field G. Since the divergence of the curl is 0, the Divergence theorem says the result is 0. On the other hand, for Stokes the surface has no boundary (it's a closed surface), so Stokes integrates G around an empty curve and gets 0 as well.

Answer 2

Answer:

• Gauss's divergence theorem : is a mathematical statement of the physical fact.
• Stoke's theorem : it is a line integral of vector field over a loop is equal to fux of its curl through the enclosed surface.
• Gauss's theorem enables integral taken over a volume to be replaced by one over the surface bounding that volume & vice versa.
• Stoke's theorem : it enables an integral taken around a closed curved to be replaced by one over any surface bounded by that curve.
• Gauss's theorem is a result of flow of vector.
• Stoke's theorem provides a relationship between line integral and surface integral.

Explanation: