TENSORS
and so the covariant components ofvare given byvi=∂φ/∂ui. In (26.97),
however, we require the contravariant componentsvi. These may be obtained by
raising the index using the metric tensor, to give
vj=gjkvk=gjk∂φ
∂uk.Substituting this into (26.97) we obtain
∇^2 φ=1
√
g∂
∂uj(
√
ggjk∂φ
∂uk). (26.98)
Use (26.98) to find the expression for∇^2 φin an orthogonal coordinate system with scale
factorshi,i=1, 2 , 3.For an orthogonal coordinate system
√
g=h 1 h 2 h 3 ;further,gij=1/h^2 iifi=jandgij=0
otherwise. Therefore, from (26.98) we have
∇^2 φ=1
h 1 h 2 h 3∂
∂uj(
h 1 h 2 h 3
h^2 j∂φ
∂uj)
,
which agrees with the results of section 10.10.
CurlThe special vector form of the curl of a vector field exists only in three dimensions.
We therefore consider a more general form valid in higher-dimensional spaces as
well. In a general space the operation curlvis defined by
(curlv)ij=vi;j−vj;i,which is an antisymmetric covariant tensor.
In fact the difference of derivatives can be simplified, sincevi;j−vj;i=∂vi
∂uj−Γlijvl−∂vj
∂ui+Γljivl=∂vi
∂uj−∂vj
∂ui,where the Christoffel symbols have cancelled because of their symmetry properties.
Thus curlvcan be written in terms of partial derivatives as
(curlv)ij=∂vi
∂uj−∂vj
∂ui.Generalising slightly the discussion of section 26.17, in three dimensions we mayassociate with this antisymmetric second-order tensor a vector with contravariant
components,
(∇×v)i=−1
2√
gijk(curlv)jk=−1
2√
gijk(
∂vj
∂uk−∂vk
∂uj)
=1
√
gijk∂vk
∂uj;