Mathematical Physics Vol 1

4.6 Examples

161

and the force is

Z c

F · d r = Z c

( 3 xy i − y 2 j ) · ( d x i + d y j )=

= Z c

3 xy d x − y 2 d y . Let us introduce the parameter t such that x = t . The parametric equation of the curve c is x = t and y = 2 t 2 . Parameter values t = 0 and t = 1 correspond to points O ( 0 , 0 ) and A ( 1 , 2 ) , respectively. Then d x = d t andd y = 4 t d t , and it follows that Z c F · d r = 1 Z 0 3 ( t )( 2 t 2 ) d t − ( 2 t 2 ) 2 4 t d t =

1 Z 0

( 6 t 3 − 16 t 5 ) d t = − 7 6 .

=

Exercise 99 Let F = ∇ φ , where φ is an unambiguous function with continuous second order partial derivatives. a) Show that the integral R c F · d r , calculated between points P 1 ( x 1 , y 1 , z 1 ) and P 2 ( x 2 , y 2 , z 2 ) , does not depend on the choice of the path between these two points. b) Conversely, if the integral R c F · d r is independent of the path c between any two points, then there exists a function φ such that F = ∇ φ . Prove this.

Solution a) Let us denote the integral R c

F · d r between points P 1 and P 2 by A 12 , that is A 12 = Z c F · d r .

According to the initial assumption F = ∇ φ , and it follows that Z c F · d r = Z c ∇ φ · d r = = Z c d φ , where d φ is a total differential (see Example 81, p.151), and if further follows that A 12 = φ ( P 2 ) − φ ( P 1 )= φ ( x 2 , y 2 , z 2 ) − φ ( x 1 , y 1 , z 1 ) .