74 5. STOKES™ THEOREM

(6) Let S be any curve that connects the point (0, 0) to the point (3, 0). What is

ω? WHY???

S

Exercise 5.14. Calculate the volume of a ball of radius 1, {(ρ, θ, φ)|ρ ¤ 1}, by inte-

grating some 2-form over the sphere of radius 1, {(ρ, θ, φ)|ρ = 1}.

Exercise 5.15. Calculate

13

x3 dx + x + xy 2 dy

3

C

where C is the circle of radius 2, centered about the origin. Answer: 8π

Exercise 5.16. Suppose ω = x dx + x dy is a 1-form on R2 . Let C be the ellipse

y2

x2

+ = 1. Determine the value of ω by integrating some 2-form over the region

4 9

C

bounded by the ellipse. (Hint: the region bounded by the ellipse can be parameterized

by φ(r, θ) = (2r cos(θ), 3r sin(θ)), where 0 ¤ r ¤ 1 and 0 ¤ θ ¤ 2π.) Answer: 6π

4. Vector calculus and the many faces of Stokes™ Theorem

Although the language and notation may be new, you have already seen Stokes™

Theorem in many guises. For example, let f (x) be a 0-form on R. Then df = f ′ (x)dx.

Let [a, b] be a 1-cell in R. Then Stokes™ Theorem tells us

b

f ′ (x)dx = f ′ (x)dx = f (x) = f (b) ’ f (a)

f (x) =

a b’a

[a,b] ‚[a,b]

Which is, of course, the Fundamental Theorem of Calculus. If we let R be some

2-chain in R2 then Stokes™ Theorem implies

‚Q ‚P

’

P dx + Q dy = d(P dx + Q dy) = dx dy

‚x ‚y

‚R R R

This is what we call “Green™s Theorem” in Calculus. To proceed further, we

restrict ourselves to R3 . In this dimension there is a nice correspondence between

vector ¬elds and both 1- and 2-forms.

4. VECTOR CALCULUS AND THE MANY FACES OF STOKES™ THEOREM 75

1

” ωF = Fx dx + Fy dy + Fz dz

F = Fx , Fy , Fz

2

” ωF = Fx dy § dz ’ Fy dx § dz + Fz dx § dy

On R3 there is also a useful correspondence between 0-forms (functions) and

3-forms.

3

f (x, y, z) ” ωf = f dx § dy § dz

We can use these correspondences to de¬ne various operations involving functions

and vector ¬elds. For example, suppose f : R3 ’ R is a 0-form. Then df is the

1-form, ‚f dx + ‚f dy + ‚f dz. The vector ¬eld associated to this 1-form is then

‚x ‚y ‚z

‚f ‚f ‚f

. In calculus we call this vector ¬eld grad f , or ∇f . In other words, ∇f

,,

‚x ‚y ‚z

is the vector ¬eld associated with the 1-form, df . This can be summarized by the

equation

1

df = ω∇f

It will be useful to think of this diagrammatically as well.

grad

f ’ ’ ∇f

’’

¦

¦

f ’ ’ df

’’

d

Example 5.9. Suppose f = x2 y 3z. Then df = 2xy 3z dx+3x2 y 2z dy +x3 y 3 dz.

The associated vector ¬eld, grad f , is then ∇f = 2xy 3 z, 3x2 y 2 z, x3 y 3 .

1

Similarly, if we start with a vector ¬eld, F, form the associated 1-form, ωF ,

di¬erentiate it, and look at the corresponding vector ¬eld, then the result is called

1

curl F, or ∇—F. So, ∇—F is the vector ¬eld associated with the 2-form, dωF. This

can be summarized by the equation

1 2

dωF = ω∇—F

This can also be illustrated by the following diagram.

76 5. STOKES™ THEOREM

curl

F ’’ ∇—F

’’

¦

¦ ¦

¦

1 1

ωF ’ ’

’’ dωF

d

Example 5.10. Let F = xy, yz, x2 . The associated 1-form is then

ωF = xy dx + yz dy + x2 dz.

1

The derivative of this 1-form is the 2-form

1

dωF = ’y dy § dz + 2x dx § dz ’ x dx § dy.

The vector ¬eld associated to this 2-form is curl F, which is

∇ — F = ’y, ’2x, ’x .

Lastly, we can start with a vector ¬eld, F = Fx , Fy , Fz , and then look at the

3-form, dωF = ( ‚Fx + ‚Fy + ‚Fz )dx § dy § dz (See Exercise 4.12). The function,

2

‚x ‚y ‚z

+ ‚Fy + ‚Fz is called div F, or ∇ · F. This is summarized in the following

‚Fx

‚x ‚y ‚z

equation and diagram.

2 3

dωF = ω∇·F

div

F ’’ ∇·F

’’

¦

¦ ¦

¦

2 2

ωF ’ ’ dωF

’’

d

Example 5.11. Let F = xy, yz, x2 . The associated 2-form is then

ωF = xy dy § dz ’ yz dx § dz + x2 dx § dy.

2

The derivative is the 3-form

2

dωF = (y + z) dx § dy § dz.

So div F is the function ∇ · F = y + z.

4. VECTOR CALCULUS AND THE MANY FACES OF STOKES™ THEOREM 77

Two important vector identities follow from the fact that for a di¬erential form,

ω, calculating d(dω) always yields zero (see Exercise 4.8 of Chapter 4). For the ¬rst,

consider the following diagram.

grad curl

f ’ ’ ∇f ’ ’ ∇ — (∇f )

’’ ’’

¦ ¦

¦ ¦

f ’ ’ df ’ ’

’’ ’’ ddf

d d

This shows that if f is a 0-form then the vector ¬eld corresponding to ddf is

∇ — (∇f ). But ddf = 0, so we conclude

∇ — (∇f ) = 0

For the second identity, consider this diagram.

curl div

F ’ ’ ∇ — F ’ ’ ∇ · (∇ — F)

’’ ’’

¦ ¦

¦ ¦ ¦

¦

1 1 1

ωF ’ ’

’’ ’’

’’

dωF ddωF

d d

1

This shows that if ddωF is written as g dx § dy § dz then the function g is equal

1

to ∇ · (∇ — F). But ddωF = 0, so we conclude

∇ · (∇ — F) = 0

In vector calculus we also learn how to integrate vector ¬elds over parameterized

curves (1-chains) and surfaces (2-chains). Suppose ¬rst that σ is some parameterized

curve. Then we can integrate the component of F which points in the direction of

the tangent vectors to σ. This integral is usually denoted F · ds, and its de¬nition

σ

1

is precisely the same as the de¬nition we learned here for ωF . A special case of this

σ

1

integral arises when F = ∇f , for some function, f . In this case, ωF is just df , so the

∇f · ds is the same as

de¬nition of df .

σ σ

We also learn to integrate vector ¬elds over parameterized surfaces. In this case,

the quantity we integrate is the component of the vector ¬eld which is normal to the

surface. This integral is often denoted F · dS. Its de¬nition is precisely the same as

S

2

(see Exercises 2.20 and 2.21). A special case of this is when F = ∇—G,

that of ωF

S

78 5. STOKES™ THEOREM

2 1

(∇ — G) · dS must

for some vector ¬eld, G. Then ωG is just dωG , so we see that

S

1

be the same as dωG .

S

The most basic thing to integrate over a 3-dimensional region (i.e. a 3-chain),

„¦, in R3 is a function f (x, y, x). In calculus we denote this integral as f dV . Note

„¦

3

A special case is when f = ∇ · F, for some

that this is precisely the same as ωf .

„¦

(∇ · F)dV . But we can write this integral

vector ¬eld F. In this case f dV =

„¦ „¦

2

with di¬erential forms as dωF.

„¦

We summarize the equivalence between the integrals developed in vector calculus

and various integrals of di¬erential forms in the following table:

Vector Calculus Di¬erential Forms

1

F · ds ωF

σ σ

∇f · ds df

σ σ

2