Mathematical Methods for Physics and Engineering : A Comprehensive Guide

VECTOR CALCULUS

Prove the expression for∇·ain orthogonal curvilinear coordinates.

Let us consider the sub-expression∇·(a 1 ˆe 1 ). Noweˆ 1 =eˆ 2 ׈e 3 =h 2 ∇u 2 ×h 3 ∇u 3. Therefore

∇·(a 1 ˆe 1 )=∇·(a 1 h 2 h 3 ∇u 2 ×∇u 3 ),

=∇(a 1 h 2 h 3 )·(∇u 2 ×∇u 3 )+a 1 h 2 h 3 ∇·(∇u 2 ×∇u 3 ).

However,∇·(∇u 2 ×∇u 3 ) = 0, from (10.43), so we obtain

∇·(a 1 eˆ 1 )=∇(a 1 h 2 h 3 )·

(

eˆ 2

h 2

×

ˆe 3

h 3

)

=∇(a 1 h 2 h 3 )·

ˆe 1

h 2 h 3

;

letting Φ =a 1 h 2 h 3 in (10.60) and substituting into the above equation, we find

∇·(a 1 ˆe 1 )=

1

h 1 h 2 h 3

∂

∂u 1

(a 1 h 2 h 3 ).

Repeating the analysis for∇·(a 2 ˆe 2 )and∇·(a 3 ˆe 3 ), and adding the results we obtain (10.61),
as required.

Laplacian

In the expression for the divergence (10.61), let

a=∇Φ=

1

h 1

∂Φ

∂u 1

ˆe 1 +

1

h 2

∂Φ

∂u 2

eˆ 2 +

1

h 3

∂Φ

∂u 3

ˆe 3 ,

where we have used (10.60). We then obtain

∇^2 Φ=

1

h 1 h 2 h 3

[

∂

∂u 1

(

h 2 h 3

h 1

∂Φ

∂u 1

)

+

∂

∂u 2

(

h 3 h 1

h 2

∂Φ

∂u 2

)

+

∂

∂u 3

(

h 1 h 2

h 3

∂Φ

∂u 3

)]

,

which is the expression for the Laplacian in orthogonal curvilinear coordinates.

Curl

The curl of a vector fielda=a 1 eˆ 1 +a 2 ˆe 2 +a 3 ˆe 3 in orthogonal curvilinear

coordinates is given by

∇×a=

1

h 1 h 2 h 3

∣

∣

∣

∣

∣

∣

∣

∣

∣

h 1 eˆ 1 h 2 eˆ 2 h 3 ˆe 3

∂

∂u 1

∂

∂u 2

∂

∂u 3

h 1 a 1 h 2 a 2 h 3 a 3

∣

∣

∣

∣

∣

∣

∣

∣

∣

. (10.62)

Prove the expression for∇×ain orthogonal curvilinear coordinates.

Let us consider the sub-expression∇×(a 1 eˆ 1 ). Sinceˆe 1 =h 1 ∇u 1 we have

∇×(a 1 ˆe 1 )=∇×(a 1 h 1 ∇u 1 ),

=∇(a 1 h 1 )×∇u 1 + a 1 h 1 ∇×∇u 1.

But∇×∇u 1 =0,soweobtain

∇×(a 1 eˆ 1 )=∇(a 1 h 1 )×

ˆe 1

h 1

.