Mathematical Physics Vol 1

4.6 Examples

129

Problem50 A scalar function φ = 2 x 3 y 2 z 4 is given. a) Calculate ∇ · ∇ φ . b) Show on this example that ∇ · ∇ φ = ∇ 2 φ = △ φ , where ∇ 2 ≡△≡ ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 is the so-called Laplace operator (expressed with respect to Cartesian coor dinates).

Solution a) Let us start from the definition of the gradient

∂ ∂ x

∂ ∂ y

∂ ∂ z

2 x 3 y 2 z 4 + j

( 2 x 3 y 2 z 4 )+ k

( 2 x 3 y 2 z 4 )=

∇ φ = i

= 6 x 2 y 2 z 4 i + 4 x 3 yz 4 j + 8 x 3 y 2 z 3 k .

Let us now find the divergence of this vector, i.e. ∇ · ∇ φ = i ∂ ∂ x + j ∂ ∂ y + k ∂ ∂ z ·

6 x 2 y 2 z 4 i + 4 x 3 yz 4 j + 8 x 3 y 2 z 3 k =

∂ ∂ x

∂ ∂ y

∂ ∂ z

6 x 2 y 2 z 4 +

4 x 3 yz 4 +

8 x 3 y 2 z 3 =

=

= 12 xy 2 z 4 + 4 x 3 z 4 + 24 x 3 y 2 z 2 .

b) Given that

∂ 2 φ ∂ x 2

∂ 2 φ ∂ y 2

∂ 2 φ ∂ z 2

= 12 xy 2 z 4 + 4 x 3 z 4 + 24 x 3 y 2 z 2 ,

△ φ =

+

+

by comparison with the result under a), we can see on this example that ∇ · ∇ φ = △ φ = ∇ 2 φ .

Problem51 Prove the following properties of divergence

a) ∇ · ( A + B )= ∇ · A + ∇ · B b) ∇ · ( φ A )=( ∇ φ ) · A + φ ∇ · A

Solution a) Given that (with respect to the Cartesian coordinate system) A = A 1 i + A 2 j + A 3 k and B = B 1 i + B 2 j + B 3 k ,