Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
operate on V, and then we keep V ®xed and let r operate on f …r f is nonsense),
and as rf and V are vectors we complete their multiplication by taking their dot
product.
A vector V is said to be solenoidal if its divergence is zero: rV ˆ 0.
The operator r 2 , the Laplacian
The divergence of a vector ®eld is de®ned by the scalar product of the operator r
with the vector ®eld. What is the scalar product of r with itself ?
r 2 ˆrrˆ @ ^e
@x 1 ‡ @ ^e
1 @x 2 ‡ @
@
^e
2 @x 3 ^e
3 @x 1 ‡ @ ^e
1 @x 2 ‡ @
^e
2 @x 3
3
This important quantity
ˆ @2
@x 2 ‡ @2
1
@x 2 ‡ @2
2
@x 2 :
3
r 2 ˆ @2
@x 2 ‡ @2
1
@x 2 ‡ @2
2
@x 2 …1:49†
3
is a scalar di€erential operator which is called the Laplacian, after a French
mathematician of the eighteenth century named Laplace. Now, what is the divergence
of a gradient?
Since the Laplacian is a scalar di€erential operator, it does not change the
vector character of the ®eld on which it operates. Thus r 2 …r† is a scalar ®eld
if …r† is a scalar ®eld, and r 2 ‰r…r†Š is a vector ®eld because the gradient r…r†
is a vector ®eld.
The equation r 2 ˆ 0 is called Laplace's equation.
The curl of a vector
If V…x 1 ; x 2 ; x 3 † is a di€erentiable vector ®eld, then the curl or rotation of V,
written rV (or curl V or rot V), is de®ned by the vector product
^e 1 ^e 2 ^e 3
@ @ @
curl V ˆrV ˆ
@x 1 @x 2 @x 3
V 1 V 2 V 3
@V
ˆ ^e 3
1 @V
2 @V
‡ ^e 1
@x 2 @x 2 @V
3 @V
‡ ^e 2
3 @x 3 @x 3 @V
1
1 @x 1 @x 2
ˆ X
i;j;k
" ijk^e i
@V k
@x j
:
24
…1:50†