Question

State and prove the Gauss's theorem of divergence.
State and prove the Stoke's theorem​

Answer 1

Answer:

' In Calculus, the most important theorem is the “Divergence Theorem”. This theorem is used to solve many tough integral problems. It compares the surface integral with the volume integral. It means that it gives the relation between the two. In this article, you will learn the divergence theorem statement, proof, Gauss divergence theorem, and examples in detail.

Divergence Theorem Statement

The divergence theorem states that the surface integral of the normal component of a vector point function “F” over a closed surface “S” is equal to the volume integral of the divergence of F⃗ taken over the volume “V” enclosed by the surface S. Thus, the divergence theorem is symbolically denoted as:

∬v∫▽F⃗ .dV=∬sF⃗ .n⃗ .dS

Divergence Theorem Proof

The divergence theorem-proof is given as follows:

Assume that “S” be a closed surface and any line drawn parallel to coordinate axes cut S in almost two points. Let S1 and S2 be the surface at the top and bottom of S. These are represented by z=f(x,y)and z=ϕ(x,y) respectively.

F⃗ =F1i⃗ +F2j⃗ +F3k⃗ , then we have

∫∫∫∂F3∂zdV=∫∫∫∂F3∂zdxdydz

∬R[∫z=f(x,y)z=Φ(x,y)∂F3∂z]dxdy

∬R[F3(x,y,z)]z=f(x,y)z=Φ(x,y)dxdy"

"The Stoke's theorem states that “the surface integral of the curl of a function over a surface bounded by a closed surface is equal to the line integral of the particular vector function around that surface.” Where, C = A closed curve. S = Any surface bounded by C."