4.5 Closed Loops and Conservatism

Back

Definition 4.16. c : I −! Rn, a (piecewise) differentiable and continuous

function is called a loop iff

c(0) = c(1)

Remark 4.5.1. If V : Rn ! Rn is conservative and c is any loop in Rn,

Z

c

V = 0

This is obvious since

R

cV = '(c(0)) − '(c(1)) and c(0) = c(1).

4.5. CLOSED LOOPS AND CONSERVATISM 43

Proposition 4.5.1. If V is a continuous vector field on Rn and for every

loop ` in Rn, Z

`

V = 0

Then for every path c,

R

cV depends only on the endpoints of c and is independent

of the path.

Proof: If there were two paths, c1 and c2 between the same end points and

Z

c1

V 6=

Z

c2

V

then we could construct a loop by going out by c1 and back by c2 and

R

c1?c2

woud be nonzero, contradiction. _

Remark 4.5.2. This uses the fact that the path integral along any path in

one direction is the negative of the reversed path. This is easy to prove. Try

it. (Change of variable formula again)

Proposition 4.5.2. If V : Rn ! Rn is continuous on a connected open

set D _ Rn

And if

R

cV is independent of the path

Then V is conservative on D

Proof: Let 0 be any point. I shall keep it fixed in what follows and define

'(0) , 0

For any other point P ,

0

BBB@

x1

x2

...

xn

1

CCCA

2 D, we take a path from 0 to P which, in

some ball centred on P, comes in to P changing only xi, the ith component.

In the diagram in R2, figure 4.5, I come in along the x-axis.

In fact we choose P0 =

0

BBB@

x1 − a

x2

...

xn

1

CCCA

for some positive real number a, and the

path goes from 0 to P0, then in the straight line from P0 to P

44 CHAPTER 4. FIELDS AND FORMS

Figure 4.5: Sneaking Home Along the X axis

We have V

0

B@

x1

...

xn

1

CA

=

0

BBBBBBBBBBB@

V1

0

B@

x1

...

xn

1

CA

...

Vn

0

B@

x1

...

xn

1

CA

1

CCCCCCCCCCCA

for each Vi a continuous function

Rn to R, and

Z

c

V =

Z P0

0

V +

Z P

P0

V

Where I have specified the endpoints only since V has the independence of

path properly.

For every point P 2 D I define ' ( P) to be

R P

0 V, and I can rewrite the

above equation as

'(P) = '(P0) +

Z P

P0

V

= '(P0) +

Z 1

t=0

0

B@

V1(c(t))

...

Vn(c(t))

1

CA

q

0

BBBBB@

1

0

0...

U

1

CCCCCA

dt

4.5. CLOSED LOOPS AND CONSERVATISM 45

where c(t) =

0

BBB@

x1 − a + t

x2

...

xn

1

CCCA

for 0 _ t _ a.

Since the integration is just along the x1 line we can write

'(P) = '(P0) +

Z x=x1

x=x0

V1

0

BBBBB@

x1

x2

x3

...

xn

1

CCCCCA

dx

Differentiating with respect to x1

@'

@x1

= 0 +

@

@x1

Z x=x1

x=x0

V1

0

BBB@

x

x2

...

xn

1

CCCA

dx

Recall the Fundamental theorem of calculus here:

_

d

dx

Z t=x

t=0

f(t)dt = f(x)

_

to conclude that

@'

@x1

= V1

0

BBB@

x1

x2

...

xn

1

CCCA

Similarly for

@'

@xi

for all i 2 [1 . . . n]

In other words, V = r' as claimed. _

Remark 4.5.3. We need D(the “domain”) to be open so we could guarantee

the existence of a little ball around it so we could get to each point from all

the n-directions.

46 CHAPTER 4. FIELDS AND FORMS

Figure 4.6: A Hole and a non-hole

Remark 4.5.4. So far I have cheerfully assumed that

V : Rn ! Rn

is a continuous vector field and

c : I ! Rn

is a piecewise differentiable curve.

There was one place where I was sneaky and defined V on R2 by

V

_

x

y

_

, −1

(x2 + y2)3/2

_

x

y

_

This is not defined at the origin. Lots of vector fields in Physics are like this.

You might think that one point missing is of no consequence. Wrong! One of

the problems is that we can have the integral along a loop is zero provided

the loop does not circle the origin, but loops around the origin have

non-zero integrals.

For this reason we often want to restrict the vector field to be continuous

(and defined) over some region which has no holes in it.

It is intuitively easy enough to see what this means:

in figure 4.6, the left region has a big hole in it, the right hand one does not.

Saying this is algebra is a little bit trickier.

Definition 4.17. For any sets X, Y, f : X ! Y is 1-1.

8a, b 2 X, f(a) = f(b) ) a = b.

4.5. CLOSED LOOPS AND CONSERVATISM 47

Definition 4.18. For any subset U _ Rn, @U is the boundary of U and is

defined to be the subset of Rn of points p having the property that every

open ball containing p contains points of U and points not in U.

Definition 4.19. The unit square is

__

x

y

_

2 R2 : 0 _ x _ 1, 0 _ y _ 1

_

Remark 4.5.5. @I2 is the four edges of the square. It is easy to prove that

there is a 1-1 continuous map from @I2 to S1, the unit circle, which has a

continuous inverse.

Exercise 4.5.1. Prove the above claim

Definition 4.20. D is simply connected iff every continuous map f : @I2 −!

D extends to a continuous 1-1 map ˜ f : I2 −! D, i.e. ˜ f|@I2 = f

Remark 4.5.6. You should be able to see that it looks very unlikely that

if we have a hole in D and the map from @ to D circles the hole, that we

could have a continuous extension to I2. This sort of thing requires proof

but is too hard for this course. It is usually done by algebraic topology in

the Honours year.

Proposition 4.5.3. If F= Pi + Qj is a vector field on D 2 R2 and D is

open, connected and simply connected, then if

@Q

@x

−

@P

@y

= 0

on D, there is a “potential function” f : D −! R such that F = rf, that is,

F is conservative.

Proof No Proof. Too hard for you at present.

48 CHAPTER 4. FIELDS AND FORMS

Навигация