082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

862 Chapter 26 Vector calculus
pronounced‘del-squared φ’,andoccursinLaplace’sequation ∇ 2 φ = 0andotherpartial
differential equations.
Example26.9 If φ = 2x 2 −y 2 −z 2 ,find ∇φ, ∇ · (∇φ)anddeducethat φ satisfiesLaplace’sequation.
Solution ∇φ = 4xi −2yj −2zk
∇·(∇φ)=4−2−2=0
that is,
∇ 2 φ = 0
Hence φ satisfies Laplace’s equation.
Example26.10 If φ(x,y,z)isanarbitrarydifferentiablescalarfield,showthatcurl(grad φ) = ∇×(∇φ)
isalways zero.
Solution Given φ = φ(x,y,z) we have, by definition,
Then
∇φ = ∂φ
∂x i + ∂φ
∂y j + ∂φ
∂z k
curl(grad φ) = ∇ × (∇φ)
i j k
∂ ∂ ∂
=
∂x ∂y ∂z
∂φ ∂φ ∂φ
∣
∂x ∂y ∂z
∣
( ( ∂ ∂φ
=
∂y ∂z
( ( ∂ ∂φ
+
∂x ∂y
)
withsimilarresultsfortheothermixedpartialderiva-
Now,since ∂ ∂x
tives, itfollows that
∇×(∇φ)=0
( ) ∂φ
= ∂ ∂y ∂y
( ∂φ
∂x
for any scalar field φ whatsoever.
)
− ∂ ∂z
( ∂φ
∂y
)
− ∂ ∂y
( ∂φ
∂x
)) ( ∂
i −
∂x
))
k
( ) ∂φ
− ∂ ∂z ∂z
( )) ∂φ
j
∂x
Foranarbitrarydifferentiable scalar field φ
∇×(∇φ)=0