Question

State and prove stoke's theorem.

Answer 1

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.”

stoke's law

∮CF⃗ .dr→=∬S(▽×F⃗ ).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.

stoke's law

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.

Answer 2

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STOKES LAW

• The law that the force that retards a sphere moving through a viscous fluid is directly proportional to the velocity of the sphere, the radius of the sphere, and the viscosity of the fluid. the law that the frequency of luminescence induced by radiation is usually less than the frequency of the radiation.

According to this law F = 6πrηv, where

• r is the radius of the ball
• v is its velocity
• η is the viscosity of the medium.