Mathematical Physics Vol 1

4.1 Scalar field

87

b)

grad ( U + V )=

=

k ( U + V )=

∂ ∂ x

∂ ∂ y

∂ ∂ z

i +

j +

∂ ( U + V ) ∂ x

∂ ( U + V ) ∂ y

∂ ( U + V ) ∂ z

i +

j +

k =

=

∂ U ∂ x

∂ U ∂ y

∂ U ∂ z

∂ V ∂ x

∂ V ∂ y

∂ V ∂ z

i +

j +

k +

i +

j +

k =

=

=

k U +

k V =

∂ ∂ x

∂ ∂ y

∂ ∂ z

∂ ∂ x

∂ ∂ y

∂ ∂ z

i +

j +

i +

j +

= grad U + grad V .

c)

grad ( U · V )=

=

k ( U · V )=

∂ ∂ x

∂ ∂ y

∂ ∂ z

i +

j +

∂ ( U · V ) ∂ x

∂ ( U · V ) ∂ y

∂ ( U · V ) ∂ z j · V + U ·

i +

j +

k =

=

∂ U ∂ x

∂ V ∂ x

∂ U ∂ y

∂ V ∂ y

∂ U ∂ z

∂ V ∂ z

i · V + U ·

i +

j +

k · V + U ·

k =

=

=

k U · V + U ·

k V =

∂ ∂ x

∂ ∂ y

∂ ∂ z

∂ ∂ x

∂ ∂ y

∂ ∂ z

i +

j +

i +

j +

= grad U · V + U · grad V .

As an exercise, prove that the remaining three gradient properties are also valid.

4.1.4 Nabla operator or Hamilton operator

Introducing the differential operator, called "nabla" 4 or Hamiltonian 5 , defined by

∂ ∂ x

∂ ∂ y

∂ ∂ z

∇ =

i +

j +

k ,

(4.30)

the gradient of the function f can be expressed in the following form

∂ f ∂ x

∂ f ∂ y

∂ f ∂ z

grad f = ∇ f =

i +

j +

k .

(4.31)

The symbol ∇ f is very often used in technics to denote the gradient. Properties of the nabla operator ∇

Let f and g be two scalar functions, and a and b two vector functions. The properties of the nabla operator can then be expressed as follows 4 According to the Hebrew letter ∇ used to denote this operator. 5 William Rowan Hamilton (1805-1865), Irish mathematician, known for his work in dynamics.