Mathematical Tools for Physics

12—Tensors 376

All previous statements concerning the symmetry properties of tensors are unchanged because they were
made in a way independent of basis, though it’s easy to see that the symmetry properties of the tensor are
reflected in the symmetry of the covariant or the contravariant components (but not in the mixed components
usually).

Metric Tensor
Take as an example the metric tensor:
g(~u,~v) =~u.~v.

The linear function found by pulling off the~ufrom this is the identity operator.

g(~v) =~v

This tensor is symmetric, so this must be reflected in its covariant and contravariant components. Take as a basis
the vectors

e

e^2

e

2

e 1

1

Let|~e 2 |= 1and|e 1 |= 2; the angle between them being 45 ◦. A little geometry shows that

|~e^1 |=

1

√

2

and |~e^2 |=

√

2

Assume this problem is two dimensional in order to simplify things.
Compute the covariant components:

g 11 =g(~e 1 ,~e 1 ) = 4

g 12 =g(~e 1 ,~e 2 ) =

√

2

g 21 =g(~e 2 ,~e 1 ) =

√

2

g 22 =g(~e 2 ,~e 2 ) = 1

(

grc

)

=

(

4

√

√^2

2 1

)