Page 120 - Intro to Tensor Calculus

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Rules for Covariant Differentiation

The rules for covariant differentiation are the same as for ordinary differentiation. That is:
(i) The covariant derivative of a sum is the sum of the covariant derivatives.
(ii) The covariant derivative of a product of tensors is the first times the covariant derivative of the second
plus the second times the covariant derivative of the first.
(iii) Higher derivatives are defined as derivatives of derivatives. Be careful in calculating higher order deriva-
tivesasin general
A i,jk 6= A i,kj .

EXAMPLE 1.4-6. (Covariant differentiation) Calculate the second covariant derivative A i,jk .
Solution: The covariant derivative of A i is

∂A i σ
A i,j = − A σ .
∂x j ij
By definition, the second covariant derivative is the covariant derivative of a covariant derivative and hence

∂ ∂A i σ m m
A i,jk =(A i,j ) ,k = − A σ − A m,j − A i,m .
∂x k ∂x j ij ik jk
Simplifying this expression one obtains

2
∂ A i ∂A σ σ ∂ σ
A i,jk = − − A σ
j
∂x ∂x k ∂x k ij ∂x k ij

∂A m σ m ∂A i σ m
− − A σ − − A σ .
∂x j mj ik ∂x m im jk
Rearranging terms, the second covariant derivative can be expressed in the form
2
∂ A i ∂A σ σ ∂A m m ∂A i m
A i,jk = − − −
j
∂x ∂x k ∂x k ij ∂x j ik ∂x m jk
(1.4.32)

∂ σ σ m m σ
− A σ k − − .
∂x ij im jk ik mj