Mathematical Methods for Physics and Engineering : A Comprehensive Guide

LINE, SURFACE AND VOLUME INTEGRALS

R

Q

P

S

x

y

z

T

h 1 ∆u 1 eˆ 1

h 2 ∆u 2 eˆ 2

h 3 ∆u 3 eˆ 3

Figure 11.10 A general volume ∆Vin orthogonal curvilinear coordinates

u 1 ,u 2 ,u 3 .PTgives the vectorh 1 ∆u 1 ˆe 1 ,PSgivesh 2 ∆u 2 ˆe 2 andPQgives

h 3 ∆u 3 ˆe 3.

By considering the infinitesimal planar surface elementPQRSin figure 11.10, show that

(11.17) leads to the usual expression for∇×ain orthogonal curvilinear coordinates.

The planar surfacePQRSis defined by the orthogonal vectorsh 2 ∆u 2 eˆ 2 andh 3 ∆u 3 ˆe 3
at the pointP. If we traverse the loop in the directionPSRQthen, by the right-hand
convention, the unit normal to the plane isˆe 1 .Writinga=a 1 ˆe 1 +a 2 eˆ 2 +a 3 ˆe 3 , the line
integral around the loop in this direction is given by
∮

PSRQ

a·dr=a 2 h 2 ∆u 2 +

[

a 3 h 3 +

∂

∂u 2

(a 3 h 3 )∆u 2

]

∆u 3

−

[

a 2 h 2 +

∂

∂u 3

(a 2 h 2 )∆u 3

]

∆u 2 −a 3 h 3 ∆u 3

=

[

∂

∂u 2

(a 3 h 3 )−

∂

∂u 3

(a 2 h 2 )

]

∆u 2 ∆u 3.

Therefore from (11.17) the component of∇×ain the directionˆe 1 atPis given by

(∇×a) 1 = lim

∆u 2 ,∆u 3 → 0

[

1

h 2 h 3 ∆u 2 ∆u 3

∮

PSRQ

a·dr

]

=

1

h 2 h 3

[

∂

∂u 2

(h 3 a 3 )−

∂

∂u 3

(h 2 a 2 )

]

.

The other two components are found by cyclically permuting the subscripts 1, 2, 3.

Finally, we note that we can also write the∇^2 operator as a surface integral by

settinga=∇φin (11.15), to obtain

∇^2 φ=∇·∇φ= lim

V→ 0

(

1

V

∮

S

∇φ·dS

)

.