Stokes Theorem (also known as Generalized Stoke’s Theorem) is a declaration about the integration of differential forms on manifolds, which both generalizes and simplifies several theorems from vector calculus.

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.”

\(\oint _{C} \vec{F}.\vec{dr} = \iint_{S}(\bigtriangledown \times \vec{F}). \vec{dS}\)

Where,

C = A closed curve.

S = Any surface bounded by C.

F = A vector field whose components have continuous derivatives in an open region of R3 containing S.

This classical declaration, along with the classical divergence theorem, fundamental theorem of calculus , and Green’s theorem are basically special cases of the general formulation specified above.

This means that:

If you walk in the positive direction around C with your head pointing in the direction of n, the surface will always be on your left.

S is an oriented smooth surface bounded by a simple, closed smooth-boundary curve C with positive orientation.

### Gauss Divergence theorem:

The Gauss divergence theorem states that the vector’s outward flux through a closed surface is equal to the volume integral of the divergence over the area within the surface. Put differently, the sum of all sources subtracted by the sum of every sink results in the net flow of an area.

Gauss divergence theorem is a result that describes the flow of a vector field by a surface to the behavior of the vector field within the surface.

### Stokes’ Theorem Proof:

We assume that the equation of S is Z = g(x,y), (x,y)D

Where g has a continuous second order partial derivative.

D is a simple plain region whose boundary curve C1 corresponds to C.

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