Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
We ®rst write
^e 1 ^e 2 ^e 3
@ @ @
r…V† ˆ
@x 1 @x 2 @x 3
;
V 1 V 2 V 3
then notice that
@
@x 1
…V 2 †ˆ @V 2
@x 1
‡ @
@x 1
V 2 ;
so we can expand the determinant in the above equation as a sum of two determinants:
^e 1 ^e 2 ^e 3
^e 1 ^e 2 ^e 3
@ @ @
@ @ @
r…V† ˆ
‡
@x 1 @x 2 @x 3
@x 1 @x 2 @x 3
V 1 V 2 V 3 V 1 V 2 V 3
ˆ …r V†‡…r†V:
Alternatively, we can simplify the proof with the help of the permutation symbols
" ijk :
r…V† ˆX
i; j;k
ˆ X i; j;k
" ijk^e i
@
@x j
…V k †
@V
" k
ijk^e i ‡ X @
"
@x ijk^e i V
j @x k
i; j;k j
ˆ …r V†‡…r†V:
A vector ®eld that has non-vanishing curl is called a vortex ®eld, and the curl of
the ®eld vector is a measure of the vorticity of the vector ®eld.
The physical signi®cance of the curl of a vector is not quite as transparent as
that of the divergence. The following example from ¯uid ¯ow will help us to
develop a better feeling. Fig. 1.15 shows that as the component v 2 of the velocity
v of the ¯uid increases with x 3 , the ¯uid curls about the x 1 -axis in a negative sense
(rule of the right-hand screw), where @v 2 =@x 3 is considered positive. Similarly, a
positive curling about the x 1 -axis would result from v 3 if @v 3 =@x 2 were positive.
Therefore, the total x 1 component of the curl of v is
‰curl vŠ 1 ˆ @v 3 =…@x 2 @v 2 =@x 3 ;
which is the same as the x 1 component of Eq. (1.50).
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