Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
Here we have used Einstein's summation convention: repeated indexes which
appear once in the lower and once in the upper position are automatically
summed over. Thus,
X N
ˆ1
A A ˆA A :
It is important to remember that indexes repeated in the lower part or upper part
alone are not summed over. An index which is repeated and over which summation
is implied is called a dummy index. Clearly, a dummy index can be replaced
by any other index that does not appear in the same term.
Contravariant and covariant vectors
A set of N quantities A … ˆ 1; 2; ...; N† which, under a coordinate change,
transform like the coordinate di€erentials, are called the components of a contravariant
vector or a contravariant tensor of the ®rst rank or ®rst order:
A ˆ @x
@x 0 A 0 :
…1:95†
This relation can easily be inverted to express A 0 in terms of A . We shall leave
this as homework for the reader (Problem 1.32).
If N quantities A … ˆ 1; 2; ...; N† in a coordinate system …x 1 ; x 2 ; ...; x N † are
related to N other quantities A 0 … ˆ 1; 2; ...; N† in another coordinate system
…x 01 ; x 02 ; ...; x 0N † by the transformation equations
A ˆ @x 0
@x A
…1:96†
they are called components of a covariant vector or covariant tensor of the ®rst
rank or ®rst order.
One can show easily that velocity and acceleration are contravariant vectors
and that the gradient of a scalar ®eld is a covariant vector (Problem 1.33).
Instead of speaking of a tensor whose components are A or A we shall simply
refer to the tensor A or A .
Tensors of second rank
From two contravariant vectors A and B we may form the N 2 quantities A B .
This is known as the outer product of tensors. These N 2 quantities form the
components of a contravariant tensor of the second rank: any aggregate of N 2
quantities T which, under a coordinate change, transform like the product of
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