Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR DIFFERENTIATION OF A VECTOR FIELD
The result is a vector ®eld. In the expansion of the determinant the operators
@=@x i must precede V i ; P ijk stands for P P P
i j k ; and " ijk are the permutation
symbols: an even permutation of ijk will not change the value of the resulting
permutation symbol, but an odd permutation gives an opposite sign. That is,
" ijk ˆ " jki ˆ " kij ˆ" jik ˆ" kji ˆ" ikj ; and
" ijk ˆ 0 if two or more indices are equal:
A vector V is said to be irrotational if its curl is zero: rV…r† ˆ0. From this
de®nition we see that the gradient of any scalar ®eld …r† is irrotational. The proof
is simple:
^e 1 ^e 2 ^e 3
@ @ @
r…r† ˆ
@x 1 @x 2 @x 3
…x 1 ; x 2 ; x 3 †ˆ0 …1:51†
@ @ @
@x 1 @x 2 @x 3
because there are two identical rows in the determinant. Or, in terms of the
permutation symbols, we can write r…r† as
@ @
r…r† ˆX
" ijk^e i …x
@x j @x 1 ; x 2 ; x 3 †:
k
ijk
Now " ijk is antisymmetric in j, k, but @ 2 =@x j @x k is symmetric, hence each term in
the sum is always cancelled by another term:
@ @ @ @
" ijk ‡ "
@x j @x ikj ˆ 0;
k @x k @x j
and consequently r…r† ˆ0. Thus, for a conservative vector ®eld F, wehave
curl F ˆ curl (grad † ˆ0.
We learned above that a vector V is solenoidal (or divergence-free) if its divergence
is zero. From this we see that the curl of any vector ®eld V(r) must be
solenoidal:
@
r …r V† ˆX
…r V†
@x i ˆ X
!
@ X @
"
i @x ijk V
i @x k ˆ 0; …1:52†
j
i
because " ijk is antisymmetric in i, j.
If …r† is a scalar ®eld and V(r) is a vector ®eld, then
i
j;k
r…V† ˆ…r V†‡…r†V:
…1:53†
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