Mathematical Physics Vol 1

4.6 Examples

175

Exercise 116

y i + x j

Let F = −

x 2 + y 2 . a) Calculate ∇ × F . b) Calculate

I F · d r for any closed path and explain the results. Solution a) For the given field F

i

j

k

∂ ∂ x

∂ ∂ y

∂ ∂ z

∇ × F =

= 0

− y x 2 + y 2

x x 2 + y 2

0

for all points ( x , y ) in the plane, except in O ( 0 , 0 ) . In point O the field F is not defined. b) Observe the integral along the closed curve C I C F · d r = I C − y d x + x d y x 2 + y 2 . Let x = ρ cos φ , y = ρ sin φ , where ( ρ , φ ) are polar coordinates. From here, by differentiating, we obtain

d x = − ρ sin φ d φ + d ρ cos φ , d y = ρ cos φ d φ + d ρ sin φ ,

pa je

= d φ = d arctan y x .

− y d x + x d y x 2 + y 2

We will consider two possible cases. The first, when the coordinate origin O is inside the curve ABCDA (see Figure 4.26a), and the second, when the point O is outside the region surrounding this curve (Figure 4.26b). In the first case 2 π Z 0 d φ = 2 π , and in the second φ 0 Z φ 0 d φ = 0 (see Example on p. 141).