The divergence of a vector function F(x, y, z)is a scalar function defined by
div F =
∂F 1
∂x +
∂F 2
∂y +
∂F 3
∂z
If one uses the notation x = (x 1 , x 2 , x 3 )the divergence is expressed
div F(x ) =
∂F 1
∂x 1 +
∂F 2
∂x 2 +
∂F 3
∂x 3
The del operator can be used to represent the divergence using the dot product
operation
div F =∇·F=
(
∂
∂x ˆe^1 +
∂
∂y ˆe^2 +
∂
∂z ˆe^3
)
·
(
F 1 ˆe 1 +F 2 ˆe 2 +F 3 eˆ 3
)
=
∂F 1
∂x +
∂F 2
∂y +
∂F 3
∂z
The curl of a vector function F(x, y, z)is defined by the determinant operation^9
curlF =∇× F =
∣∣
∣∣
∣∣
ˆe 1 ˆe 2 eˆ 3
∂
∂x
∂
∂y
∂
∂z
F 1 F 2 F 3
∣∣
∣∣
∣∣
curlF =∇× F =
(
∂F 3
∂y −
∂F 2
∂z
)
ˆe 1 −
(
∂F 3
∂x −
∂F 1
∂z
)
ˆe 2 +
(
∂F 2
∂x −
∂F 1
∂y
)
ˆe 3
If the notation F =F(x 1 , x 2 , x 3 )is used, then the curl is sometimes represented in the
form
curlF=
(
∂F 3
∂x 2
−∂F^2
∂x 3
,∂F^1
∂x 3
−∂F^3
∂x 1
,∂F^2
∂x 1
−∂F^1
∂x 2
)
where the unit base vectors are to be understood. The operations of gradient,
divergence and curl will be investigated in more detail in the next chapter.
Taylor Series for Vector Functions
Consider a vector function
F=F(x ) = F(x 1 , x 2 ) = F 1 (x 1 , x 2 )ˆe 1 +F 2 (x 1 , x 2 )ˆe 2
which is continuous and possesses (n+ 1) partial derivatives. The Taylor series
expansion for this function is just applying the Taylor series expansion to each of
the scalar functions F 1 , F 2. Associated with the vector h = (h 1 , h 2 ) is the vector
operator
h·∇ =h 1 ∂
∂x 1
+h 2 ∂
∂x 2
(^9) See chapter 10 for properties of determinants.