Stokes theorem studies and problems and solutions

Stokes theorem

1. Stoke's theoremStoke's theorem

2. Topics of discussionTopics of discussion
•History
•Definition of Stoke’s Theorem
•Mathematical expression
•Proof of theorem
•Physical significance
•Practical applications of Stoke’s
Theorem.

3. STOKES’ THEOREMSTOKES’ THEOREM
•The theorem is named after the Irish
mathematical physicist Sir George
Stokes (1819–1903).
–What we call Stokes’ Theorem was actually
discovered by the Scottish physicist Sir William
Thomson (1824–1907, known as Lord Kelvin).
–Stokes learned of it in a letter from Thomson in
1850.

4. Statement of Stoke’s Statement of Stoke’s
TheoremTheorem
It states that line integral of a vector
field A round any close curve C is
equal to the surface integral of the
normal component of curl of vector
A over an unclosed surface ‘S’.

5. Mathematical Mathematical
expressionexpression




A . dr= (X A) .dS

Stoke’s theorem thus converts surface integral in to a
line integral over any path which constitutes the
boundary of the surface.
C S

6. where is known as
Del Operator
It is treated as a vector in Cartesian
coordinate system but it has no meaning
unless it is operated upon a scalar or vector.
It is given by
=î∂/∂x+ ĵ∂/∂y+ ^∂/∂zk

7. X
Y
Z
∆ s
1= k
∆s
1
^
^
A
dr
c
o

8. Proof of theoremProof of theorem
In order to prove this
theorem, we consider
that surface ‘S’ is
divided in to
infinitesimally small
surface elements
∆S1,∆S2,∆S3…..etc,
having boundaries
C1,C2,C3…etc.
Boundary of each
element is traced out
anti-clock wise.
x
y
z
c
A
∆S
1 =k∆S
1
^

9. The line integral of a vector field A round the boundary of a unit
area in x-y plane is equal to the component of curl A along
positive z-direction. Thus the line integral of a vector field A
along the boundary of ith surface is equal to the product of the
curl A and normal component of area ∆Si i.e.
∫A.dr =(Curl A). k∆Si =( XA).k∆Si
^
^
c
A similar process is applied to the surface element,tracing them all in
the same sense then above equation holds good for each surface
element and if we add all such equations ,we have
∑∫ A.dr=∑( X A).K∆Sii ...........(1)
i=1
i=1
n
n

10. Then all the integrals within the interior of surface
cancel, because the two integrals are in opposite
directions along the common side between two
adjacent area elements. The only portions of the line
integrals that are left are those along the sides which
lie on the boundary C.

11. Lt ∑( . A).k∆S
I = ∫∫( X A). kdS
N∞
S
i0
i=1
N
s
^
^
Then equation can be written as
Hence the equations(1)reduces to
∫A.dr=∫∫(
XA).kdS
^
c
This is the Stoke’s Theorem for a plane surface.

12. Physical significancePhysical significance
•If ∫ A.dl =0 for any closed path, then A is called
irrotational or conservative field.
•If A denotes the force F then ∫F.dr=0 means that
total work done by the force in taking a body
round a closed curve is zero i.e total energy
remains conserved throughout the motion.

13. Practical applicationsPractical applications
•It is used for determining whether a
vector field is conservative or not.
•It allows one to interpret the curl of
vector field as measure of swirling
about an axis.