Mathematical Methods for Physics and Engineering - Matematica.NET

TENSORSand so the covariant components of v are given by v i = ∂φ/∂u i . In (26.97),however, we require the contravariant components v i . These may be obtained byraising the index using the metric tensor, to givev j = g jk jk ∂φv k = g∂u k .Substituting this into (26.97) we obtain∇ 2 φ = √ 1 (∂ √gg jk ∂φ )g ∂u j ∂u k . (26.98)◮Use (26.98) to find the expression for ∇ 2 φ in an orthogonal coordinate system with scalefactors h i , i =1, 2, 3.For an orthogonal coordinate system √ g = h 1 h 2 h 3 ;further,g ij =1/h 2 i if i = j and g ij =0otherwise. Therefore, from (26.98) we have( )∇ 2 φ = 1 ∂ h 1 h 2 h 3 ∂φ,h 1 h 2 h 3 ∂u j h 2 j∂u jwhich 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 aswell. In a general space the operation curl v is defined by(curl v) ij = v i; j − v j; i ,which is an antisymmetric covariant tensor.In fact the difference of derivatives can be simplified, sincev i; j − v j; i = ∂v i∂u j − Γl ijv l − ∂v j∂u i +Γ l jiv l= ∂v i∂u j − ∂v j∂u i ,where the Christoffel symbols have cancelled because of their symmetry properties.Thus curl v can be written in terms of partial derivatives as(curl v) ij = ∂v i∂u j − ∂v j∂u i .Generalising slightly the discussion of section 26.17, in three dimensions we mayassociate with this antisymmetric second-order tensor a vector with contravariantcomponents,(∇×v) i = − 12 √ g ɛijk (curl v) jk= − 1 (2 √ ∂vjg ɛijk ∂u k − ∂v )k∂u j = √ 1 ɛ ijk ∂v kg ∂u j ;974