(STATEMENT, DERIVE AND PHYSICALLY INTER PRATE THE STOKE’S THEOREM)
(ASSIGNMENT #1 SEMESTER FALL 2018)
Submission Date (October 8, 2018)
FAZAL UL REHMAN
Course code (Course title)
Degree program Title
(BS PHYSICS 1 SECTION A)
DR. MUHAMMAD RIZWAN
Department of physics
UNIVERSITY OF GUJRAT
HAFIZ HAYAT CAMPUS
The line integral of a vector function around a closed curve is equal to the surface integral of the curl of that vector over a surface which is bounded by that closed curve is called Stoke’s Theorem.
Line integral of A = Surface integral of curl of AA . dr = ? (?×A) da
Consider a surface S bounded by a closed curve C placed in vector field A as shown in figure. Take a length element of this closed curve. It is tangent to curve C and makes angle ? with vector function.
Now divides curve C into N-loops C1, C2, C3 and so on to CN curve . Then
Line integral of A = i=1N?A . drMultiply and divide R.H.S by ?aiai
=i=1N(?A .dr)×?ai?aiIn limiting case i=1N?? and ?ai?daai?da limit ?Siaproaches to zero ?A. dr/?ai = ?×ASo play applying these conditions
Line integral of A= ?(?×A ).da (Hence proved)…………. (A)
IN TERMS OF RECTANGULAR COMPONENTS
A= Axi +Ayj+Azkdr= dxi +dyj + dzkn= cos?i +cos?j +cos?kOn solving ?× AA A , WE find
(?× A.) . n?Az?y-?Ay?zcos? -?Az?x-?Ax?zcos?+?Ay?x-?Ax?ycos?……….. (1)
A.dr= Axdx + Aydy + Azdz ………… (2)
So, by putting these values in (A)
(Axdx+Aydy+Azdz )=??Az?y-?Ay?zcos?-?Az?x-?Ax?zcos?+?Ay?x-?Ax?ycos?This is called a rectangular component of stoke’s theorem.
Stoke’s theorem provides insight into physical interpretation of the curl. In a vector field, the rotation of vector field is at a maximum when the curl of the vector field and normal vector have same direction.
In other words, while the tendency to rotate will vary from point to point on the surface.Stoke’s theorem says that collective measure of this rotational tendency to taken over the entire surface is equal to the tendency of a fluid to circulate around the boundary curve. By using Stoke’stheorem , we will find a total net flow in or out of the closed curve.
APPLICATIONS OF STOKES’THEOREM:
3. Sufficient conditions of a vector field to be conservative.