Mathematical Physics Vol 1

4.6 Examples

119

4.6 Examples 4.6.1 Gradient

Problem33 Find the gradient for the following functions a) φ = r , b) φ = ln r , c) φ = 1 r , where r is the position vector, and r its magnitude.

For the physical interpretation see Chapter 4.3.

R

Solution a) The position vector, expressed with respect to the Cartesian coordinate sys tem, r = x i + y j + z k , has the magnitude r = p x 2 + y 2 + z 2 , and thus the scalar function φ , expressed with respect to the Cartesian coordinate system, has the following form φ = r = p x 2 + y 2 + z 2 . Its gradient is

∂ p x 2 + y 2 + z 2 ∂ x x p x 2 + y 2 + z 2 = r 0 .

∂ p x 2 + y 2 + z 2 ∂ y y p x 2 + y 2 + z 2 j +

∂ p x 2 + y 2 + z 2 ∂ z

∇ φ =

i +

j +

k ⇒

z p x 2 + y 2 + z 2

∇ φ =

i +

k =

r r

=

b) The scalar function φ , expressed with respect to the Cartesian coordinate system, has the following form

1 2

ln ( x 2 + y 2 + z 2 ) ,

φ = ln r =

and thus its gradient is

1 2 ∇ ln ( x 2 + y 2 + z 2 ) ⇒

∇ φ =

1 2 1 2

∂ ∂ x

∂ ∂ y

ln ( x 2 + y 2 + z 2 )+ j

ln ( x 2 + y 2 + z 2 )+

∇ φ =

i

ln ( x 2 + y 2 + z 2 ) ⇒

∂ ∂ z

+ k

2 z x 2 + y 2 + z 2 ⇒

2 x x 2 + y 2 + z 2

2 y x 2 + y 2 + z 2

∇ φ =

i

+ j

+ k

x i + y j + z k x 2 + y 2 + z 2

r r 2

∇ φ =

=

.