Page 127 - Intro to Tensor Calculus

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Parallel Vector Fields
i
i
i
Let y = y (t), i =1, 2, 3 denote a space curve C in a Cartesian coordinate system and let Y define a
i
constant vector in this system. Construct at each point of the curve C the vector Y . This produces a field
of parallel vectors along the curve C. What happens to the curve and the field of parallel vectors when we
transform to an arbitrary coordinate system using the transformation equations
1
3
2
i
i
y = y (x ,x ,x ), i =1, 2, 3
with inverse transformation
i i 1 2 3
x = x (y ,y ,y ), i =1, 2, 3?
The space curve C in the new coordinates is obtained directly from the transformation equations and can
be written
i
3
2
1
i
i
x = x (y (t),y (t),y (t)) = x (t), i =1, 2, 3.
i
i
The field of parallel vectors Y become X in the new coordinates where
∂y i
j
i
Y = X . (1.4.55)
∂x j
Since the components of Y i are constants, their derivatives will be zero and consequently we obtain by
differentiating the equation (1.4.55), with respect to the parameter t, that the field of parallel vectors X i
must satisfy the differential equation
j
2 i
dX ∂y i j ∂ y dx m dY i
+ X = =0. (1.4.56)
j
dt ∂x j ∂x ∂x m dt dt
Changing symbols in the equation (1.4.7) and setting the Christoffel symbol to zero in the Cartesian system
of coordinates, we represent equation (1.4.7) in the form
2 i
∂ y α ∂y i
=
j
∂x ∂x m jm ∂x α
and consequently, the equation (1.4.56) can be reduced to the form
δX j dX j j k dx m
= + X =0. (1.4.57)
δt dt km dt

The equation (1.4.57) is the differential equation which must be satisfied by a parallel field of vectors X i
i
along an arbitrary curve x (t).