Stoke's Theorem COEP
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Sujoy ShivdeStoke's Theorem COEP
Uploaded by
Sujoy ShivdeStoke’s theorem
It gives the relationship between two completely different quantities- line integral and the surface
integral‼! Yet another fascinating theorem- a marvel of vector calculus. We can think it as a 3-D
version of the Green’s theorem.
1 Statement
Let S be a smooth surface with a smooth bounding curve C. Then for any continuously differentiable
vector function, F (x, y, z) = [P, Q, R], where P, Q, R are functions of x, y, z.
I ZZ
F · dr = curl(F ) · dS
C S
We assume there is an orientation on both the surface and the curve that are related by the right
hand rule. That is, if you were to walk around the curve in its preferred direction with your head
pointing in the same direction as the normal vector ~n to the surface, then the surface would always
be on your left (see the following figure for reference).
Thus, Stoke’s theorem says that the line integral around the boundary curve of S of the tangential
component of F is equal to the surface integral of the normal component of the curl of F .
Equivalent form of the stokes theorem:
I ZZ
∂R ∂Q ∂P ∂R ∂Q ∂P
P dx + Qdy + Rdz = ( − )dydz + ( − )dzdx + ( − )dxdy
∂y ∂z ∂z ∂x ∂x ∂y
C S
*Note:
There are three types of questions possible‼!
1. Verify the Stoke’ss theorem → evaluate both line and surface integrals(We will accept your
answers to the plus-minus sign error)
1
2. Evaluate the line integral using Stoke’s theorem → evaluate surface integral
3. Evaluate the surface integral integral using Stoke’s theorem → evaluate line integral
Make sure to find appropriate parametrizations for both boundary C and the surface S if required!
2 Application of the Stoke’s theorem
This gives glimpses on how we can apply Stoke’s theorem to prove certain things.
• Show that the following line integral is zero along any closed curve C.
I
yzdx + xzdy + xydz
C
Ans. The vector field is F (x, y, z) = [yz, xz, xy], observe that curl(F ) = 0. Therefore,
I ZZ
yzdx + xzdy + xydz = 0 · dS = 0
C S
where S is any good surface (which we can create on our own) having the boundary C.
• Prove the Green’s theorem using Stokes theorem
• Draw an analogy between Stoke’s theorem, Green’s theorem and the Fundamental theorem
of calculus
3 Examples
1. Verify the Stoke’s theorem:
Consider F = [3y, 4z, −6x] and the surface S described by the paraboloid z = 16 − x2 − y 2
for z ≥ 0, as shown in the figure below.
Solution:
Line integral: Observe that boundary of the paraboloid is the circle x2 + y 2 = 16. Thus, we
2
can parametrize it as r(t) = [4cos t, 4sin t, 0]; 0 ≤ t ≤ 2π.
Therefore,
I Z2π
F · dr = F (x(t), y(t), z(t)] · r0 (t)dt
C 0
Z2π
= [12sin t; 0, −24cos t] · [−4sin t, 4cos t, 0]dt
0
Z2π
= (−48sin2 t + 0 + 0)dt
0
Z2π
= −48 sin2 tdt
0
= −48π
Surface integral: We can either project the paraboloid on the x-y plane or parametrize it to
find the surface integral! Give it a try, you must get −48π as the final answer.
2. Evaluate the line integral using Stoke’s theorem
Let F (x, y, z) = [−y 2 , x, z 2 ], C is the curve of intersection of the plane y + z = 2 and the
cylinder x2 + y 2 = 1. (Orient C to be counterclockwise when viewed from above.)
Ans:- π
3. Evaluate the surface integral using Stoke’s theorem
ZZ
curl(F ) · dS
S
where F = [y, −x, yx3 ] and S is the upper hemisphere of radius 4 and the upwards orientation.
For any doubt, please reach to me using Moodle platform → Dr. Pralhad M. Shinde