Mathematical Physics Vol 1

4.6 Examples

185

Analogously, we obtain

s ∆ S

φ n · j d S ∆ V

∇ φ · j = lim ∆ V → 0

(4.213)

s ∆ S

φ n · k d S ∆ V

∇ φ · k = lim ∆ V → 0

(4.214)

.

If we now multiply the equations (4.212), (4.213) and (4.214), by i , j , k , respec tively, and then add them, using ∇ φ =( ∇ φ · i ) i +( ∇ φ · j ) j +( ∇ φ · k ) k n =( n · i ) i +( n · j ) j +( n · k ) k , we obtain

s ∆ S

φ n d S ∆ V

∇ φ = lim ∆ V → 0

,

which was to be proved. b) As (Example 126, p. 183) y V

∇ × A d V = x ∆ S

n × A d S ,

similarly to the first part of this Example, by a scalar product with i we obtain

s ∆ S

( ∇ × A ) · i = lim ∆ V → 0 ( n × A ) · i d S ∆ V . Analogously for j and k . Multiplying by i , j and k , and then adding, yields

s ∆ S

n × A d S ∆ V

∇ × A = lim ∆ V → 0

.

R Note that these results can be taken as starting points for defining the gradient, divergence and rotor. The expressions for gradient, divergence and rotor defined in this way are expressed independently of the coordinate system, and are thus valid for any coordinate system, i.e. they are invariant with respect to the coordinate system.

Exercise 128 Define the equivalency operator

1 ∆ V x ∆ S

∇ ◦≡ lim ∆ V → 0 d S ◦ , where ◦ represents: multiplication of a vector by a scalar, the scalar product or the