Mathematical Physics Vol 1

4.6 Examples

151

Thus, we have proved that the initial relation is valid for m = 1. The validity of the relation for m = 2 , 3 and n = 1 , 2 , 3 can be proved analogously. Using the Kronecker symbol δ mn = 1 , za m = n , 0 , za m̸ = n , we can rewrite the previous result in the following form

3 ∑ p = 1

ℓ pm ℓ pn = δ mn .

Exercise 79 If v = 2 x 2 i − 3 yz j + xz 2 k i φ = 2 z − x 3 y , find

a) v · ∇ φ , b) v × ∇ φ inpoint A ( 1 , − 1 , 1 ) .

Solution

b) 7 i − j − 11 k .

a) 5,

Exercise 80

f ′ ( r ) r r

Prove that ∇ f ( r )=

.

Exercise 81 If U is a differentiable function of variables x , y , and z prove that ∇ U · d r = d U .

Exercise 82 Let F be a differentiable scalar function of variables x , y , z , and t , where x , y , and z are differentiable functions of t . Prove that d F d t = ∂ F ∂ t + ∇ F · d r d t .