Page 70 - Intro to Tensor Calculus

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§1.3 SPECIAL TENSORS

Knowing how tensors are defined and recognizing a tensor when it pops up in front of you are two
different things. Some quantities, which are tensors, frequently arise in applied problems and you should
learn to recognize these special tensors when they occur. In this section some important tensor quantities
are defined. We also consider how these special tensors can in turn be used to define other tensors.

Metric Tensor

i
Define y ,i =1,... ,N as independent coordinates in an N dimensional orthogonal Cartesian coordinate
i i i
system. The distance squared between two points y and y + dy , i =1,... ,N is defined by the
expression
N 2
2 2
1 2
m
2
ds = dy dy m =(dy ) +(dy ) + ··· +(dy ) . (1.3.1)
i
i
Assume that the coordinates y are related to a set of independent generalized coordinates x ,i =1,... ,N
by a set of transformation equations
2
i
1
i
N
y = y (x ,x ,... ,x ), i =1,... ,N. (1.3.2)
i
i
i
To emphasize that each y depends upon the x coordinates we sometimes use the notation y = y (x), for
i =1,... ,N. The differential of each coordinate can be written as
∂y m j
m
dy = dx , m =1,...,N, (1.3.3)
∂x j
and consequently in the x-generalized coordinates the distance squared, found from the equation (1.3.1),
becomes a quadratic form. Substituting equation (1.3.3) into equation (1.3.1) we find
∂y m ∂y m i j i j
2
ds = dx dx = g ij dx dx (1.3.4)
i
∂x ∂x j
where
∂y m ∂y m
g ij = , i, j =1,... ,N (1.3.5)
i
∂x ∂x j
i
are called the metrices of the space defined by the coordinates x ,i =1,... ,N. Here the g ij are functions of
the x coordinates and is sometimes written as g ij = g ij (x). Further, the metrices g ij are symmetric in the
indices i and j so that g ij = g ji for all values of i and j over the range of the indices. If we transform to
i
another coordinate system, say x ,i =1,... ,N, then the element of arc length squared is expressed in terms
2
i
j
of the barred coordinates and ds = g dx dx , where g = g (x) is a function of the barred coordinates.
ij ij ij
The following example demonstrates that these metrices are second order covariant tensors.