Page 146 - Intro to Tensor Calculus

141

Tensor Derivatives
α
α
Let u = u (t) denote the parametric equations of a curve on the surface defined by the parametric
2
1
i
i
equations x = x (u ,u ). We can then represent the surface curve in the spatial geometry since the surface
2
1
i
i
i
curve can be represented in the spatial coordinates through the representation x = x (u (t),u (t)) = x (t).
i
i
i
Recall that for x = x (t) a given curve C , the intrinsic derivative of a vector field A along C is defined as
the inner product of the covariant derivative of the vector field with the tangent vector to the curve. This
intrinsic derivative is written
" #
δA i i dx j ∂A i i k dx j
= A ,j = + A
δt dt ∂x j jk dt
g
or
δA i dA i i k dx j
= + A
δt dt jk dt
g
α
where the subscript g indicates that the Christoffel symbol is formed from the spatial metric g ij . If A is a
surface vector defined along the curve C, the intrinsic derivative is represented
δA α α du β ∂A α α γ du β
= A ,β = + A
δt dt ∂u β βγ dt
a
or
δA α dA α α γ du β
= + A
δt dt βγ dt
a
where the subscript a denotes that the Christoffel is formed from the surface metric a αβ .
Similarly, the formulas for the intrinsic derivative of a covariant spatial vector A i or covariant surface
vector A α are given by
j
δA i dA i k dx
= − A k
δt dt ij dt
g
and
β
δA α dA α γ du
= − A α .
δt dt αβ dt
a
i
Consider a mixed tensor T which is contravariant with respect to a transformation of space coordinates
α
i
α
i
x and covariant with respect to a transformation of surface coordinates u . For T defined over the surface
α
α
i
curve C, which can also be viewed as a space curve C, define the scalar invariant Ψ = Ψ(t)= T A i B where
α
α
A i is a parallel vector field along the curve C when it is viewed as a space curve and B is also a parallel
vector field along the curve C when it is viewed as a surface curve. Recall that these parallel vector fields
must satisfy the differential equations
j α α β
δA i dA i k dx δB dB α γ du
= − A k =0 and = + B =0. (1.5.52)
δt dt ij dt δt dt βγ dt
g a
The scalar invariant Ψ is a function of the parameter t of the space curve since both the tensor and the
parallel vector fields are to be evaluated along the curve C. By differentiating the function Ψ with respect
to the parameter t there results
dΨ dT α i α i dA i α i dB α
= A i B + T α B + T A i . (1.5.53)
α
dt dt dt dt