Page 147 - Intro to Tensor Calculus

142

α
But the vectors A i and B are parallel vector fields and must satisfy the relations given by equations (1.5.52).
This implies that equation (1.5.53) can be written in the form

" #
dΨ dT α i i k dx j γ i du β α
= + T α − T γ A i B . (1.5.54)
dt dt kj dt βα dt
g a
The quantity inside the brackets of equation (1.5.54) is defined as the intrinsic tensor derivative with respect
to the parameter t along the curve C. This intrinsic tensor derivative is written

δT i dT i i dx j γ du β
α = α + T k − T i . (1.5.55)
dt dt kj α dt βα γ dt
g a
The spatial representation of the curve C is related to the surface representation of the curve C through the
defining equations. Therefore, we can express the equation (1.5.55) in the form

" #
δT i ∂T i i ∂x j γ du β
α = α + T k − T i (1.5.56)
dt ∂u β kj α ∂u β βα γ dt
g a
The quantity inside the brackets is a mixed tensor which is defined as the tensor derivative of T i with
α
β
respect to the surface coordinates u . The tensor derivative of the mixed tensor T α i with respect to the
β
surface coordinates u is written
∂T α i i k ∂x j γ i
i
T = + T − T .
α,β β α β γ
∂u kj ∂u βα
g a
In general, given a mixed tensor T i...j which is contravariant with respect to transformations of the
α...β
space coordinates and covariant with respect to transformations of the surface coordinates, then we can
define the scalar field along the surface curve C as

i...j α β
Ψ(t)= T A i ··· A j B ··· B (1.5.57)
α...β
β
α
where A i ,...,A j and B ,...,B are parallel vector fields along the curve C. The intrinsic tensor derivative
is then derived by differentiating the equation (1.5.57) with respect to the parameter t.
Tensor derivatives of the metric tensors g ij ,a αβ and the alternating tensors ijk , αβ and their associated
tensors are all zero. Hence, they can be treated as constants during the tensor differentiation process.
Generalizations
i
In a Riemannian space V n with metric g ij and curvilinear coordinates x ,i =1, 2, 3, the equations of a
i
2
1
α
i
surface can be written in the parametric form x = x (u ,u )where u ,α =1, 2 are called the curvilinear
coordinates of the surface. Since
∂x i α
i
dx = du (1.5.58)
∂u α
i
α
then a small change du on the surface results in change dx in the space coordinates. Hence an element of
arc length on the surface can be represented in terms of the curvilinear coordinates of the surface. This same
element of arc length can also be represented in terms of the curvilinear coordinates of the space. Thus, an
element of arc length squared in terms of the surface coordinates is represented
2
α
ds = a αβ du du β (1.5.59)