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§ 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
Define yi,i=1,...,N as independent coordinates in an N dimensional orthogonal Cartesian coordinate
system. The distance squared between two points yi and yi + dyi,i=1,...,N is defined by the
expression
ds2 = dymdym =(dy1)2 +(dy2)2 + ···+(dyN )2. ()
Assume that the coordinates yi are related to a set of independent generalized coordinates xi,i=1,...,N
by a set of transformation equations
yi = yi(x1,x2,...,xN ),i=1,...,N. ()
To emphasize that each yi depends upon the x coordinates we sometimes use the notation yi = yi(x), for
i =1,...,N. The differential of each coordinate can be written as
∂ym
dym = dxj ,m=1,...,N, ()
∂xj
and consequently in the x-generalized coordinates the distance squared, found from the equation (),
es a quadratic form. Substituting equation () into equation () we find
∂ym ∂ym
ds2 = dxidxj = g dxidxj ()
∂xi ∂xj ij
where
∂ym ∂ym
g = ,i,j=1,...,N ()
ij ∂xi ∂xj
i
are called the metrices of the space defined by the coordinates x ,i=1,...,N. Here the gij are functions of
the x coordinates and is sometimes written as gij = gij (x). Further, the metrices gij are symmetric in the
indices i and j so that gij = gji for all values of i and j over the range of the indices. If we transform to
another coordinate system, say xi,i=1,...,N, then the element of arc length squared is expressed in terms
2 i j
of the barred coordinates and ds = gij dx dx , where gij = gij (x) is a function of the barred coordinates.
The following example demonstr