Using the del operator ∇the divergence can be represented
div v =∇·v =(
∂
∂x
eˆ 1 + ∂
∂yˆe 2 + ∂
∂zˆe 3 )·(v 1 ˆe 1 +v 2 ˆe 2 +v 3 ˆe 3 )div v =∇·v =
∂v 1
∂x +∂v 2
∂y +∂v 3
∂z(7 .64)Again make note of the fact that the del operator is not commutative and ∇·v =v ·∇.
If the divergence of a vector field is zero, ∇· v = 0 , then the vector field is called
solenoidal.
If v =v (x, y, z ) = v 1 (x, y, z )ˆe 1 +v 2 (x, y, z )ˆe 2 +v 3 (x, y, z)ˆe 3 denotes a vector field with
components which are well defined, continuous and everywhere differential, then the
curl or rotation of v is written^4 curlv =∇× v =curlv =∇× v =
∣∣
∣∣
∣∣ˆe 1 eˆ 2 ˆe 3
∂
∂x∂
∂y∂
∂z
v 1 v 2 v 3∣∣
∣∣∣∣which
can be expressed in the expanded determinant form^5
curlv =∇× v =curlv =∇× v =
∣∣
∣∣
∣∣ˆe 1 ˆe 2 eˆ 3
∂
∂x∂
∂y∂
∂z
v 1 v 2 v 3∣∣
∣∣
∣∣=ˆe 1∣∣
∣∣∂
∂y∂
∂z
v 2 v 3∣∣
∣∣−ˆe 2∣∣
∣∣∂
∂x∂
∂z
v 1 v 3∣∣
∣∣+ˆe 3∣∣
∣∣∂
∂x∂
∂y
v 1 v 2∣∣
∣∣curlv =∇× v =
(
∂v 3
∂y −∂v 2
∂z)
ˆe 1 −(
∂v 3
∂x −∂v 1
∂z)
eˆ 2 +(
∂v 2
∂x −∂v 1
∂y)
ˆe 3(7 .65)If the curl of a vector field is zero, curlv =∇× v = 0 , then the vector field is said to
be irrotational.
Example 7-10. Find the gradient of the scalar field φ=x^2 y+zxy^2 at the
point (1, 1 ,2).
Solution Using the above definition show that
grad φ=∇φ= (2xy +zy^2 )ˆe 1 + (x^2 + 2 xyz )ˆe 2 +xy^2 eˆ 3and grad φ
(1, 1 ,2)= 4 ˆe 1 + 5 ˆe 2 +eˆ 3.(^4) The curl of v is sometimes referred to as the rotation of v and written rotv.
(^5) See chapter 10 for properties of determinants.