Stoke's Theorem

 

                                 Stoke’s theorem :

Statement : The closed line integral of any vector function is equal to the surface integral of the “curl of vector function”, considering the surface is bounded by the closed curve.

stoke's theorem 

 

In short :  Stoke's theorem enable us to convert a surface integral into a line integral.

Proof of Stoke's Theorem

B

stoke's theorem figure 1

 

 

 

 

 

 

 

 

 

 

 

Consider an arbitrary surface(S) area bounded by a closed curve (L).

 Assume the surface is made up of large no. of elementary surface area  (small elements  ∆s_i).

Here large no. means it should be countable.

Considering one such element  ∆s_i   ( ABCD )  shown in figure 1

Curl of any vector,

We have  ∆s_i  element (s),

so,

Above equations holds good for each surface elements

So, for the whole surface

Now,

stoke's theorem  fig.2

All the common boundary or shared boundary get cancelled out any line or closed curved inside the surface have common boundary and hence after integration of all these closed curve will be equivalent to closed line integral of the outer boundary of vector  ( Explained in fig. 2)               

Quick Explanation (fig.2 ABCEFDA)

Let for ABCDA,

A is in upward direction    ↑ and

A.dl cos0⁰ =  A.dl

For CEFDA, dl is ↓ downward

A.dl cos 180⁰ =  -A.dl  

Therefore, 

curl along CD is 

A.dl – A.dl = 0 ( the boundary CD get cancelled out and similarly for the whole surface) 

If  N  →  ∞  then ∆s_i →  0, in this case Σ  can be replaced by integration for continuous addition

Therefore,

This is the stoke’s theorem.

Q. State and prove stoke’s theorem. ( LNMU,2019, paper 1, part 1 )

 

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