Page 148 - Vector Analysis

144 CHAPTER 4. Vector Calculus

thus

lim 1¿ u¨T ds = 1 3 [ B (u ¨ pe2) (a) ´ e2j B (u ¨ pe1) ]
2πr Cr r 2 e1j B xj B xj (a)
rÑ0 ÿ

j=1

1 3 ( ) B uk (a) 1 3 B uk (a)
2 B xj 2 B xj
ÿ ÿ
(δj r δks
= e1j e2k ´ e2j e1k = ´ δj s δkr )e1r e2s ,
j,k,r,s=1
j,k=1

where 䨨’s are the Kronecker deltas. Using (4.9), we further conclude that

lim 1 ¿ u¨T ds = 1 3 εijk εirse1r e2s B uk (a) .
2πr Cr r 2 B xj
rÑ0 ÿ

i,j,k,r,s=1

3

Since pe1 ˆ pe2 = pe3, we have e3i = ř εirse1re2s; thus the identity above shows that

r,s=1

lim 1 ¿ u¨T ds = 1 3 εijk e3i B uk (a) = 1 3 (3 εijk B uk ) .
2πr Cr r 2 B xj 2 ÿ B xj (a) e3i
rÑ0 ÿ ÿ
j,k=1
i,j,k=1 i=1

(The blue expression of) (4.21) and the identity above motivate the following

Definition 4.85 (The curl operator). Let u : Ω Ď Rn Ñ Rn, n = 2 or n = 3, be a vector
field.

1. For n = 2, the curl of u is a scalar function defined by

2

curlu = ÿ ε3ijuj,i .

i,j=1

2. For n = 3, the curl of u is a vector-valued function defined by

3

(curlu)i = ÿ εijkuk,j .

j,k=1

The function curlu is also called the vorticity of u, and is usually denoted by one single
Greek letter ω.

Having the curl operator defined, for the three-dimensional case the circulation of a
vector field u on the plane with normal N is given by curlu ¨ N .

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