Mathematical Physics Vol 1

Chapter 4. Field theory

162

Thus, the integral A 12 depends on the start and end point of the path, but not on the path between these points itself. b) Let F = F 1 i + F 2 j + F 3 k . Then the integral along the curve c , between points P 1 ( x 1 , y 1 , z 1 ) and P ( x , y , z ) , is

( x , y , z ) Z ( x 1 , y 1 , z 1 )

( x , y , z ) Z ( x 1 , y 1 , z 1 )

φ ( x , y , z )=

F · d r =

F 1 d x + F 2 d y + F 3 d z .

From here it follows

( x + ∆ x , y , z ) Z ( x 1 , y 1 , z 1 )

( x , y , z ) Z ( x 1 , y 1 , z 1 )

φ ( x + ∆ x , y , z ) − φ ( x , y , z )=

F · d r −

F · d r =

( x + ∆ x , y , z ) Z ( x 1 , y 1 , z 1 )

( x 1 , y 1 , z 1 ) Z ( x , y , z ) ( x + ∆ x , y , z ) Z ( x , y , z )

F · d r +

F · d r =

=

( x + ∆ x , y , z ) Z ( x , y , z )

F · d r =

F 1 d x + F 2 d y + F 3 d z .

=

Since the last integral, according to the assumption, does not depend on the path between the points with coordinates ( x , y , z ) and ( x + ∆ x , y , z ) , we can choose a straight line that passes through these two points as the path, so that d y = d z = 0 (the line is parallel to the x axis, see Figure 4.29b). Then

(a) Work along the x axis.

(b) Work from point P 1 topoint P 2 .

Figure 4.29

( x + ∆ x , y , z ) Z ( x , y , z )

φ ( x + ∆ x , y , z ) − φ ( x , y , z )=

F 1 d x .

From here we obtain

( x + ∆ x , y , z ) Z ( x , y , z )

φ ( x + ∆ x , y , z ) − φ ( x , y , z ) ∆ x

1 ∆ x

F 1 d x .

(4.187)

=