Notation
The position vector r =xˆe 1 +yˆe 2 +zˆe 3 is sometimes represented in matrix no-
tation as a row vector r = (x, y, z)or a column vector r =col(x, y, z). Sometimes the
substitution x=x 1 ,y=x 2 and z=x 3 is made and these vectors are represented as
x = (x 1 , x 2 , x 3 )or x =col(x 1 , x 2 , x 3 )and a vector function
F(x, y, z) = F 1 (x, y, z )ˆe 1 +F 2 (x, y, z)ˆe 2 +F 3 (x, y, z)ˆe 3
is represented in the form of either a row vector or column vector
F(x ) = (F 1 (x ), F 2 (x ), F 3 (x )) or F(x ) = col(F 1 (x ), F 2 (x ), F 3 (x ))
where the representation of the basis vectors ˆe 1 ,ˆe 2 ,ˆe 3 is to be understood and col
is used to denote a column vector.
This change in notation is made in order that scalar and vector concepts can
be extended to represent scalars and vectors in higher dimensions. For example,
the representation x = (x 1 , x 2 , x 2 ,... , x n) would represent an n−dimensional vector,
The scalar function φ=φ(x ) = φ(x 1 , x 2 ,... , x n) would represent a scalar function
of n-variables and the vector F(x ) = (F 1 (x ), F 2 (x ),... , F n(x )) would represent an n-
dimensional vector function of position.
Gradient, Divergence and Curl
The gradient of a scalar function φ=φ(x, y, z)is the vector function
grad φ=∂φ
∂x
ˆe 1 +∂φ
∂y
eˆ 2 +∂φ
∂z
ˆe 3
If the scalar function is represented in the form φ=φ(x 1 , x 2 , x 3 ), then the gradient
vector is sometimes expressed in the form
grad φ=
(
∂φ
∂x 1
, ∂φ
∂x 2
, ∂φ
∂x 3
)
where it is to be understood that the partial derivatives are to be evaluated at the
point (x 1 , x 2 , x 3 ) = (x, y, z ). The vector operator
∇=
∂
∂x
ˆe 1 + ∂
∂y
ˆe 2 + ∂
∂z
ˆe 3
called the “del operator”or “nabla”, is sometimes used to represent the gradient as
an operator operating upon a scalar function to produce
grad φ=∇φ=
(
∂
∂x ˆe^1 +
∂
∂y ˆe^2 +
∂
∂z ˆe^3
)
φ=
∂φ
∂x ˆe^1 +
∂φ
∂y ˆe^2 +
∂φ
∂z ˆe^3
