Mathematical Physics Vol 1

Chapter 4. Field theory

186

vector product, and the integral is calculated along a closed contour.

Solution If ◦ represents the scalar product. then ∇ ◦ A = lim ∆ V → 0 1

∆ V x ∆ S

d S ◦ A

or

1 ∆ V x ∆ S 1 ∆ V x ∆ S

div A = lim ∆ V → 0

d S · A =

= lim A · n d S . Similarly, if ◦ represents the vector product, then (see Example 123) ∇ ◦ A = ∇ × A = rot A = lim ∆ V → 0 1 ∆ V x ∆ S d S × A = = lim ∆ V → 0 1 ∆ V x ∆ S n × A d S . Finally, if ◦ represents the multiplication of a vector by a scalar φ , we obtain ∇ ◦ φ = lim ∆ V → 0 1 ∆ V x ∆ S d S ◦ φ , or ∇ φ = lim ∆ V → 0 1 ∆ V x ∆ S φ d S . (see Example 127a). ∆ V → 0

Exercise 129 Let S be a closed surface, V the space bounded by the surface S , and r the position vector of a point ( x , y , z ) with respect to the coordinate origin. Show that the integral I = x S n · r r 3 d S

a) I = 0, if O lies outside the surface S , b) I = 4 π if O lies inside the surface S .