Stokes Theorem

Stokes Theorem

Stokes Theorem is also referred to as the generalized Stokes Theorem. It is a declaration about the integration of differential forms on different manifolds. It generalizes and simplifies the several theorems from vector calculus. According to this theorem, a line integral is related to the surface integral of vector fields. Students will learn a stokes theorem with explanations and examples. Let us begin it!

 

Source: en.wikipedia.org

Stokes Theorem Meaning:

Stokes’ theorem relates the surface integral of the curl of the vector field to a line integral of the vector field around some boundary of a surface. It is named after George Gabriel Stokes.  Although the first known statement of the theorem is by William Thomson and it appears in a letter of his to Stokes.

Stoke’s theorem statement is “the surface integral of the curl of a function over the surface bounded by a closed surface will be equal to the line integral of the particular vector function around it.” Stokes theorem gives a relation between line integrals and surface integrals. Depending upon the convenience, one integral can be computed in terms of the other.

Stokes Theorem Formula:

It is,

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

Where,

C = A closed curve.

S = Any surface bounded by C.

F = A vector field whose components are continuous derivatives in S.

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

Thus, it means that if we walk in the positive direction around C with our head pointing in the direction of n, then the surface will always be on our left. S is oriented smooth surface bounded by a simple, closed smooth-boundary curve C with positive orientation. It can be noted that the surface S can actually be any surface so long as its boundary curve is given by C. This is something that can be used to our advantage to simplify the surface integral on the occasion.

Stokes theorem physics and stokes theorem mathematics are very popular. We can find many applications of stokes theorem in Physics. It helps to derive many useful formulae and equations. For example, stokes theorem in electromagnetic theory is very popular in Physics.

Gauss Divergence theorem:

In vector calculus, divergence theorem is also known as Gauss’s theorem. It relates the flux of a vector field through the closed surface to the divergence of the field in the volume enclosed.

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. The sum of all sources subtracted by the sum of every sink will result in the net flow of an area. Gauss divergence theorem is the result that describes the flow of a vector field by a surface to the behaviour of the vector field within it.

Stokes’ Theorem Proof:

We can assume that the equation of S is Z and it is g(x,y), (x,y)D.

Where g has a continuous second-order partial derivative.

D is a simple plain region whose boundary curve \(C_{1}\) corresponds to C.

We can easily explain this with a 3D air projection.

Some Solved Examples for You

Example-1: Give some applications of stokes theorem.

Solution: Stokes’ theorem has a wide area of applications. It is also useful for the interpretation of the curl of the vector field of any kind. This theorem is often useful in physics, especially in the area of electromagnetism. Stokes’ theorem and its customized form are very important for finding line integral of some particular curve as well as in determining the curl of a bounded surface.