Mathematical Physics Vol 1

Chapter 4. Field theory

166

Exercise 102 Show that if the integral

P 2 R

P 1 F · d r is independent of the path between any two points P 1

and P 2 of a given region, then H F · d r = 0 for a closed curve, and vice versa.

Solution Let P 1 AP 2 BP 1 be a closed curve. Then I F · d r = Z P 1 AP 2 BP 1

F · d r =

= Z = Z

F · d r + Z F · d r − Z

F · d r =

P 2 BP 1

P 1 AP 2

F · d r = 0 ,

P 1 BP 2

P 1 AP 2

because R

= R

P 1 BP 2 based on the assumption that the value of the integral does not

P 1 AP 2

depend on the path between points P 1 and P 2 .

Figure 4.30

Conversely, if H F · d r = 0 then 0 = I F · d r = Z P 1 AP 2 BP 1

F · d r = Z

F · d r + Z F · d r − Z

F · d r =

P 2 BP 1

P 1 AP 2

= Z

F · d r .

P 1 BP 2

P 1 AP 2

From there it follows that

Z P 1 AP 2

F · d r = Z

F · d r .

P 1 BP 2