β as component indices and α as a label giving the particular tensor referred to. There is one such tensor for each basis vector eα . However, this is not terribly useful, since under a change of coordinates the basis changes and the important quantities in the new system are the new tensors eβ which are obtained from the old ones eα in a complicated way: they are different tensors, not just different components of the same tensor. So the set

in one frame is not obtained by a simple tensor transformation from the set μ

of another frame. The easiest example of this is Cartesian coordinates, where α βμ ≡ 0, while they are not zero in other frames. So in many books it is said that μαβ are not components of tensors. As we have seen, this is not strictly true: μαβ are the (μ, β) com-

combination

Vβ ,α + Vμ β μα

isa component of a single tensor V.

5.5 N o n co o rd i n a t e b a s e s

In this whole discussion we have generally assumed that the non-Cartesian basis vectors were generated by a coordinate transformation from (x, y) to some (ξ , η). However, as we shall show below, not every field of basis vectors can be obtained in this way, and we shall have to look carefully at our results to see which need modification (few actually do). We will almost never use non-coordinate bases in our work in this course, but they are frequently encountered in the standard references on curved coordinates in flat space, so we should pause to take a brief look at them now.

Polar coordinate basis

The basis vectors for our polar coordinate system were defined by

eα = β α eβ ,

where primed indices refer to polar coordinates and unprimed to Cartesian. Moreover, we had

β α = ∂xβ /∂xα ,

where we regard the Cartesian coordinates {xβ } as functions of the polar coordinates {xα }. We found that

eα · eβ ≡ gα β =δα β ,

i.e. that these basis vectors are not unit vectors.

So if, for example, the books calculate the divergence of the vector, they obtain, instead of our Eq. (5.56),

The difference between Eqs. (5.56) and (5.87) is purely a matter of the basis for V.

General remarks on noncoordinate bases

The principal differences between coordinate and noncoordinate bases arise from the fol-

But if no coordinates exist for {eμ}, then Eq. (5.89) must fail. For example, if we let Eq. (5.88) define φ,μˆ , then we have

for any coordinate system {xβ } and noncoordinate basis {eαˆ }. It is thus convenient to continue with the notation, Eq. (5.88), and to make the rule that φ,μ = ∂φ/∂xμ only in a coordinate basis.