Mathematical Tools for Physics

12—Tensors 383

this is no longer just a unit vector. Not only does it have dimensions of length but its magnitude varies from
point to point. Note thatˆθisnotone of the basis vectors in this notation.
Third example: a coordinate system again in a plane, but where the axes are not orthogonal to each other,
but are rectilinear anyway.

a

x

x

1

2

0 1 2

0

1

2

Still keep the definition

~v=~ei

dxi

dt

=~e 1

dx^1

dt

+~e 2

dx^2

dt

. (39)

If the particle moves along thex^1 -axis (or parallel to it) then by the definition of the axes,x^2 is a constant
anddx^2 /dt= 0. Suppose that the coordinates measure centimeters, so that theperpendiculardistance between
the lines is one centimeter. The distance between the points(0,0)and(1,0)is then 1 cm/sinα= cscαcm. If
in∆t=one second, particle moves from the first to the second of these points,∆x^1 =one cm, sodx^1 /dt=
1 cm/sec. The speed however, iscscαcm/sec because the distance moved is greater by that factor. This means
that
|~e 1 |= cscα

and this is greater than one; it is not a unit vector. The magnitudes of~e 2 is the same. The dot product of these
two vectors is~e 1 .~e 2 = cosα/sin^2 α.

Reciprocal Coordinate Basis
The reciprocal basis vectors are constructed from the direct basis by the equation

~ei.~ej=δij

In rectangular coordinates the direct and reciprocal bases coincide because the basis is orthonormal. For the tilted
basis of Eq. ( 39 ),
~e 2 .~e^2 = 1 =|~e 2 ||~e^2 |cos

(

90 ◦−α

)

= (cscα)|~e^2 |sinα=|~e^2 |