classical Stokes’ theorem

classical Stokes’ theorem

Let

M

be a

compact

,

oriented

two-dimensional

differentiable manifold

(

surface

) with

boundary

in

ℝ

3

,
and

𝐅

be a

C

2

-smooth

vector field

defined on an

open set

in

ℝ

3

containing

M

.
Then

∬

M

(

∇

×

𝐅

)

⋅

𝑑

𝐀

=

∫

∂

⁡

M

𝐅

⋅

𝑑

𝐬

.

Here, the boundary of

M

,

∂

⁡

M

(which is a curve)
is given the induced orientation from

M

. The symbol

∇

×

𝐅

denotes the curl of

𝐅

.
The symbol

d

⁢

𝐬

denotes the

line element

d

⁢

s

with a

direction

parallel

to the

unit tangent

vector

𝐭

to

∂

⁡

M

, while

d

⁢

𝐀

denotes
the

area element

d

⁢

A

of the surface

M

with a direction parallel to the unit outward normal

𝐧

to

M

. In precise terms:

d

⁢

𝐀

=

𝐧

⁢

d

⁢

A

,

d

⁢

𝐬

=

𝐭

⁢

d

⁢

s

.

The classical Stokes’ theorem reduces to Green’s theorem on the plane if the surface

M

is taken to lie in the xy-plane.

The classical Stokes’ theorem, and
the other “Stokes’ type” theorems
are special cases of the general Stokes’ theorem involving

differential forms

.
In fact, in the proof we present below, we appeal to the general Stokes’ theorem.

Physical interpretation

(To be written.)

Proof using differential forms

The proof becomes a triviality once we express

(

∇

×

𝐅

)

⋅

d

⁢

𝐀

and

𝐅

⋅

d

⁢

𝐬

in terms of differential forms.

Proof.

Define the differential forms

η

and

ω

by

η

p

⁢

(

𝐮

,

𝐯

)

=

⟨

curl

⁡

𝐅

⁢

(

p

)

,

𝐮

×

𝐯

⟩

,

ω

p

⁢

(

𝐯

)

=

⟨

𝐅

⁢

(

p

)

,

𝐯

⟩

.

for points

p

∈

ℝ

3

, and

tangent vectors

𝐮

,

𝐯

∈

ℝ

3

.
The symbol

⟨

,

⟩

denotes the

dot product

in

ℝ

3

.
Clearly, the functions

η

p

and

ω

p

are linear and

alternating

in

𝐮

and

𝐯

.

We claim

η

=

∇

×

𝐅

⋅

d

⁢

𝐀

on

M

.

(1)

ω

=

𝐅

⋅

d

⁢

𝐬

on

∂

⁡

M

.

(2)

To prove (1), it suffices to check
it holds true when we evaluate the left- and right-hand sides
on an

orthonormal basis

𝐮

,

𝐯

for the

tangent space

of

M

corresponding to the orientation of

M

,
given by the unit outward normal

𝐧

.
We calculate

∇

×

𝐅

⋅

d

⁢

𝐀

⁢

(

𝐮

,

𝐯

)

=

⟨

curl

⁡

𝐅

,

𝐧

⟩

⁢

d

⁢

A

⁢

(

𝐮

,

𝐯

)

definition of

d

⁢

𝐀

=

𝐧

⁢

d

⁢

A

=

⟨

curl

⁡

𝐅

,

𝐧

⟩

definition of

volume form

d

⁢

A

=

⟨

curl

⁡

𝐅

,

𝐮

×

𝐯

⟩

since

𝐮

×

𝐯

=

𝐧

=

η

⁢

(

𝐮

,

𝐯

)

.

For equation (2), similarly, we only have to check that it holds
when both sides are evaluated at

𝐯

=

𝐭

,
the unit tangent vector of

∂

⁡

M

with the induced orientation of

∂

⁡

M

.
We calculate again,

𝐅

⋅

d

⁢

𝐬

⁢

(

𝐭

)

=

⟨

𝐅

,

𝐭

⟩

⁢

d

⁢

s

⁢

(

𝐭

)

definition of

d

⁢

𝐬

=

𝐭

⁢

d

⁢

s

=

⟨

𝐅

,

𝐭

⟩

definition of volume form

d

⁢

s

=

ω

⁢

(

𝐭

)

.

Furthermore,

d

⁢

ω

=

η

.
(This can be checked by a calculation
in

Cartesian coordinates

, but in fact this equation
is one of the coordinate-free

definitions

of the curl.)

The classical Stokes’ Theorem now follows
from the general Stokes’ Theorem,

∫

M

η

=

∫

M

𝑑

ω

=

∫

∂

⁡

M

ω

.

∎

References

1

Michael Spivak.

Calculus on Manifolds

. Perseus Books, 1998.

Title

classical Stokes’ theorem

Canonical name

ClassicalStokesTheorem

Date of creation

2013-03-22 15:27:52

Last modified on

2013-03-22 15:27:52

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

6

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 26B20

Related topic

GeneralStokesTheorem

Related topic

GaussGreenTheorem

Related topic

GreensTheorem

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