Mathematical Physics Vol 1

Chapter 4. Field theory

152

Exercise 83 If v is a constant vector, prove that ∇ ( r · v )= v .

Exercise 84 If v = v 1 i + v 2 j + v 3 k , prove that

d v =( ∇ v 1 · d r ) i +( ∇ v 2 · d r ) j +( ∇ v 3 · d r ) k .

Exercise 85 Find the increment of the function φ = 4 xz 3 − 3 x 2 y 2 z in point A ( 2 , − 1 , 2 ) in the direction of vector n = 2 i − 3 j + 6 k .

Solution

376 / 7.

Exercise 86 Find the increment of the function ϕ = 4 e 2 x − y + z inpoint A ( 1 , 1 , − 1 ) in the direction of point B ( − 3 , 5 , 6 ) .

Solution

− 20 / 9.

Exercise 87 In the direction of which vector, from point A ( 1 , 3 , 2 ) , is the increment of the function φ = 2 xz − y 2 the largest? What is that increment?

Solution v = 4 i − 6 j + 2 k ,

√ 14.

2

Exercise 88 Find the angle between the surfaces xy 2 z = 3 x + z 2 and 3 x 2 − y 2 + 2 z = 1 in point A ( 1 , − 2 , 1 ) .