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Stoke’s Curl Theorem

Statement: “The surface integral of the curl of a vector field A taken over any surface S is equal to the line integral of A around the closed curve forming the periphery of the surface S.”

Or

” The flux of the curl of a vector field A over any closed surface S of any arbitrary shape is equal to the line integral of vector A, taken over the boundary of that surface.”

Mathematically it may be represented as

 

ʃʃ S (∆ x A ) Ds  =ʃ ʃS  curl A  .ds =ʃ A .dl

 

Hence this theorem is used to convert surface integral into line integral.

 

Proof: Let us consider a closed curve

 

 

C1C2C3C4C1 enclosing a surface area S in a vector field A as shown in Figure 7.14.

 

The line integral of A over the boundary of the closed curve C1C2C3C4C1 may be given as

ʃ A .dl

 

where dl is the small line element of the boundary of curve C1C2C3C4C1.

 

 

Now when the whole surface is divided into large number of infinitesimal rectangular elements each of surface area dS, then the vector area of each element will be n dS = dS, where n is unit vector along the normal on dS.

 

But the curl of a vector field A at any point in the field is equal to the line integral of that field per unit area taken along the boundary of an infinitesimal area drawn around that point. The  directiondf the line integral is normal to the surface.

 

Therefore, the line integral of A along the boundary of each of the rectangular element

dS may be written as

(curl F) . dS

 

But the line integrals taken along the common sides of the continuous element (dS) mutually cancel each other due to being equal in magnitude and opposite in direction as shown in Figure 7.15, therefore, if we want to find out total line integral for all dS elements, then only the sides of the elements lying on the boundary or periphery of the surface S will contribute to the line integral.

Thus the total line integral of the vector A along the boundary C1C2C3C4C1 of the

surface S

 

i.e.,                  ʃ ʃs(cur  F).dS will be equal to the ʃ F . dl,

i.e.,                  ʃ ʃS (curl F) .dS = ʃ ʃS/v x F) .dS = ʃF .dl.