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4.1 Definitions Galore

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You may have noticed that I have been careful to write vectors in Rn as

vertical columns 2

6664

...

7775

and in particular

2 R2 and not (x, y) 2 R2.

You will have been, I hope, curious to know why. The reason is that I want

to distinguish between elements of two very similar vector spaces, R2 and

R2_.

Definition 4.1. Dual Space

8n 2 Z+,

Rn_ , L(Rn,R)

is the (real) vector space of linear maps fom Rn to R, under the usual addition

and scaling of functions.

Remark 4.1.1. You need to do some checking of axioms here, which is

soothing and mostly mechanical.

Remark 4.1.2. Recall from Linear Algebra the idea of an isomorphism of

vector spaces. Intuitively, if spaces are isomorphic, they are pretty much the

same space, but the names have been changed to protect the guilty.

26 CHAPTER 4. FIELDS AND FORMS

Remark 4.1.3. The space Rn_ is isomorphic to Rn and hence the dimension

of Rn_ is n. I shall show that this works for R2_ and leave you to verify the

general result.

Proposition 4.1.1. R2_ is isomorphic to R2

Proof For any f 2 R2_, that is any linear map from R2 to R, put

a , f

and

b , f

Then f is completely specified by these two numbers since

2 R2, f

= f

+ y

Since f is linear this is:

+ yf

= ax + by

This gives a map

_ : L(R2,R) −! R2

where a and b are as defined above.

This map is easily confirmed to be linear. It is also one-one and onto and

has inverse the map

_ : R2 −! L(R2,R)

ax + by

Consequently _ is an isomorphism and the dimension of R2_ must be the

same as the dimension of R2 which is 2. _

Remark 4.1.4. There is not a whole lot of difference between Rn and Rn_,

but your life will be a little cleaner if we agree that they are in fact different

things.

4.1. DEFINITIONS GALORE 27

I shall write [a, b] 2 R2_ for the linear map

[a, b] : R2 _ −! R

ax + by

Then it is natural to look for a basis for R2_ and the obvious candidate is

([1, 0], [0, 1])

The first of these sends _

which is often called the projection on x. The other is the projection on y.

It is obvious that

[a, b] = a[1, 0] + b[0, 1]

and so these two maps span the space L(R2,R). Since they are easily shown

to be linearly independent they are a basis. Later on I shall use dx for the

map [1, 0] and dy for the map [0, 1]. The reason for this strange notation will

also be explained.

Remark 4.1.5. The conclusion is that although different, the two spaces

are almost the same, and we use the convention of writing the linear maps as

row matrices and the vectors as column matrices to remind us that (a) they

are different and (b) not very different.

Remark 4.1.6. Some modern books on calculus insist on writing (x, y)T for

a vector in R2. T is just the matrix transpose operation. This is quite intelligent

of them. They do it because just occasionally, distinguishing between

(x, y) and (x, y)T matters.

Definition 4.2. An element of Rn_ is called a covector. The space Rn_ is

called the dual space to Rn, as was hinted at in Definition 4.1.

Remark 4.1.7. Generally, if V is any real vector space, V_ makes sense.

If V is finite dimensional, V and V_ are isomorphic (and if V is not finite

dimensional, V and V_ are NOT isomorphic. Which is one reason for caring

about which one we are in.)

Definition 4.3. Vector Field A vector field on Rn is a map

V : Rn ! Rn.

28 CHAPTER 4. FIELDS AND FORMS

A continuous vector field is one where the map is continuous, a differentiable

vector field one where it is differentiable, and a smooth vector field is

one where the map is infinitely differentiable, that is, where it has partial

derivatives of all orders.

Remark 4.1.8. We are being sloppy here because it is traditional. If we

were going to be as lucid and clear as we ought to, we would define a space

of tangents to Rn at each point, and a vector field would be something more

than a map which seems to be going to itself. It is important to at least

grasp intuitively that the domain and range of V are different spaces, even if

they are isomorphic and given the same name. The domain of V is a space

of places and the codomain of V (sometimes called the range) is a space of

“arrows”.

Definition 4.4. Differential 1-Form A differential 1-form on Rn or covector

field on Rn is a map

! : Rn ! R_n

It is smooth when the map is infinitely differentiable.

Remark 4.1.9. Unless otherwise stated, we assume that all vector fields and

forms are infinitely differentiable.

Remark 4.1.10. We think of a vector field on R2 as a whole stack of little

arrows, stuck on the space. By taking the transpose, we can think of a

differential 1-form in the same way.

If V : R2 ! R2 is a vector field, we think of V

as a little arrow going

from

x + a

y + b

Example 4.1.1. Sketch the vector field on R2 given by

−y

Because the arrows tend to get in each others way, we often scale them down

in length. This gives a better picture, figure 4.1. You might reasonably look

at this and think that it looks like what you would get if you rotated R2

anticlockwise about the origin, froze it instantaneously and put the velocity

vector at each point of the space. This is 100% correct.

Remark 4.1.11. We can think of a differential 1-form on R2 in exactly the

same way: we just represent the covector by attaching its transpose. In fact

4.1. DEFINITIONS GALORE 29

Figure 4.1: The vector field [−y, x]T

covector fields or 1-forms are not usually distinguished from vector fields as

long as we stay on Rn (which we will mostly do in this course). Actually, the

algebra is simpler if we stick to 1-forms.

So the above is equally a handy way to think of the 1-form

! : R2 −! R2_ _

(−y, x)

Definition 4.5. i ,

and j ,

when they are used in R2.

Definition 4.6. i ,

5 and j ,

5 and k ,

5 when they are

used in R3.

This means we can write a vector field in R2 as

P(x, y)i + Q(x, y)j

and similarly in R3:

P(x, y, z)i + Q(x, y, z)j + R(x, y, z)k

30 CHAPTER 4. FIELDS AND FORMS

Definition 4.7. dx denotes the projection map R2 −! R which sends

to x. dy denotes the projection map

y The same symbols dx, dy

are used for the projections from R3 along with dz. In general we have dxi

from Rn to R sends

6664

...

7775

Remark 4.1.12. It now makes sense to write the above vector field as

−y i + x j

or the corresponding differential 1-form as

−y dx + x dy

This is more or less the classical notation.

Why do we bother with having two things that are barely distinguishable?

It is clear that if we have a physical entity such as a force field, we could

cheerfully use either a vector field or a differential 1-form to represent it. One

part of the answer is given next:

Definition 4.8. A smooth 0-form on Rn is any infinitely differentiable map

(function)

f : Rn ! R.

Remark 4.1.13. This is, of course, just jargon, but it is convenient. The

reason is that we are used to differentiating f and if we do we get

Df : Rn −! L(Rn,R)

x Df(x) =

@x1

@x2

, . . .

@xn

(x)

This I shall write as

df : Rn ! Rn_

4.1. DEFINITIONS GALORE 31

and, lo and behold, when f is a 0-form on Rn df is a differential 1-form on

Rn. When n = 2 we can take f : R2 −! R as the 0-form and write

df =

dx +

which was something the classical mathematicians felt happy about, the dx

and the dy being “infinitesimal quantities”. Some modern mathematicians

feel that this is immoral, but it can be made intellectually respectable.

Remark 4.1.14. The old-timers used to write, and come to think of it still

do,

rf(x) ,

@x1 ...

@xn

and call the resulting vector field the gradient field of f.

This is just the transpose of the derivative of the 0-form of course.

Remark 4.1.15. For much of what goes on here we can use either notation,

and it won’t matter whether we use vector fields or 1-forms. There will be

a few places where life is much easier with 1-forms. In particular we shall

repeat the differentiating process to get 2-forms, 3-forms and so on.

Anyway, if you think of 1-forms as just vector fields, certainly as far as

visualising them is concerned, no harm will come.

Remark 4.1.16. A question which might cross your mind is, are all 1-forms

obtained by differentiating 0-forms, or in other words, are all vector fields

gradient fields? Obviously it would be nice if they were, but they are not. In

particular,

−y

is not the gradient field rf for any f at all. If you ran around the origin in

a circle in this vector field, you would have the force against you all the way.

If I ran out of my front door, along Broadway, left up Elizabeth Street, and

kept going, the force field of gravity is the vector field I am working with. Or

against. The force is the negative of the gradient of the hill I am running up.

You would not however, believe that, after completing a circuit and arriving

home out of breath, I had been running uphill all the way. Although it might

feel like it.

Definition 4.9. A vector field that is the gradient field of a function (scalar

field) is called conservative.

32 CHAPTER 4. FIELDS AND FORMS

Definition 4.10. A 1-form which is the derivative of a 0-form is said to be

exact.

Remark 4.1.17. Two bits of jargon for what is almost the same thing is a

pain and I apologise for it. Unfortunately, if you read modern books on, say

theoretical physics, they use the terminology of exact 1-forms, while the old

fashioned books talk about conservative vector fields, and there is no solution

except to know both lots of jargon. Technically they are different, but they

are often confused.

Definition 4.11. Any 1-form on R2 will be written

! , P (x, y) dx + Q(x, y) dy.

The functions P(x, y),Q(x, y) will be smooth when ! is, since this is what it

means for ! to be smooth. So they will have partial derivatives of all orders.

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