Page 71 - Intro to Tensor Calculus

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EXAMPLE 1.3-1. Show the metric components g ij are covariant tensors of the second order.
i
Solution: In a coordinate system x ,i =1,... ,N the element of arc length squared is

2
i
ds = g ij dx dx j (1.3.6)
i
while in a coordinate system x ,i =1,... ,N the element of arc length squared is represented in the form

2
m
n
ds = g mn dx dx . (1.3.7)
The element of arc length squared is to be an invariant and so we require that
i
n
m
g dx dx = g ij dx dx j (1.3.8)
mn
Here it is assumed that there exists a coordinate transformation of the form defined by equation (1.2.30)
together with an inverse transformation, as in equation (1.2.32), which relates the barred and unbarred
i
i
coordinates. In general, if x = x (x), then for i =1,... ,N we have
∂x i m j ∂x j n
i
dx = m dx and dx = n dx (1.3.9)
∂x ∂x
Substituting these differentials in equation (1.3.8) gives us the result
i
i
∂x ∂x j m n ∂x ∂x j m n
m
n
g mn dx dx = g ij m n dx dx or g mn − g ij m n dx dx =0
∂x ∂x ∂x ∂x
i
∂x ∂x j
For arbitrary changes in dx m this equation implies that g and consequently g ij transforms
mn = g ij m n
∂x ∂x
as a second order absolute covariant tensor.
EXAMPLE 1.3-2. (Curvilinear coordinates) Consider a set of general transformation equations from
rectangular coordinates (x, y, z) to curvilinear coordinates (u, v, w). These transformation equations and the
corresponding inverse transformations are represented
x = x(u, v, w) u = u(x, y, z)

y = y(u, v, w) v = v(x, y, z) (1.3.10)
z = z(u, v, w). w = w(x, y, z)

3
3
1
2
1
2
Here y = x, y = y, y = z and x = u, x = v, x = w are the Cartesian and generalized coordinates
and N =3. The intersection of the coordinate surfaces u = c 1 ,v = c 2 and w = c 3 define coordinate curves
of the curvilinear coordinate system. The substitution of the given transformation equations (1.3.10) into
r
the position vector ~ = x b e 1 + y b e 2 + z b e 3 produces the position vector which is a function of the generalized
coordinates and
r ~ = ~(u, v, w)= x(u, v, w) b e 1 + y(u, v, w) b e 2 + z(u, v, w) b e 3
r