Mathematical Physics Vol 1

4.6 Examples

153

Solution

√ 6 14

arccos

.

Exercise 89 Find the constants a and b such that the surface ax 2 − byz =( a + 2 ) x is normal to the surface 4 x 2 y + z 3 = 4 at point M ( 1 , − 1 , 2 ) .

Solution a = 5 / 2, b = 1.

Exercise 90

a) Prove that the functions u ( x , y , z ) , v ( x , y , z ) and w ( x , y , z ) are functionally depen dent, that is, ( F ( u , v , w )= 0) iff ∇ u · ∇ v × ∇ w = 0. b) Express ∇ u · ∇ v × ∇ w = 0 in the form of a determinant. c) Are the functions u = x + y + z , v = x 2 + y 2 + z 2 and w = xy + yz + zx dependent?

Solution

∂ u ∂ x ∂ v ∂ x ∂ w ∂ x

∂ u ∂ y ∂ v ∂ y ∂ w ∂ y

∂ u ∂ z ∂ v ∂ z ∂ w ∂ z

b)

,

1 1 1 2 x 2 y 2 z y + z x + z y + z

c) Yes, as in this case

= 0 . Their functional dependence is of

the from u 2 − v − 2 w = 0.

Exercise 91

If A = 3 xyz 2 i + 2 xy 3 j − x 2 yz k i φ = 3 x 2 − yz find a) ∇ · A ,

b) A · ( ∇ φ ) , c) ∇ · ( φ A ) , d) ∇ · ( ∇ φ ) , at point (1,-1,1).