Mathematical Tools for Physics

12—Tensors 382

In spherical coordinates

x^1 =r, x^2 =θ, x^3 =φ

and the radial coordinate axis is defined by the equationsθ=,constant andφ=constant.
The use ofx^1 , x^2 , andx^3 (xi) for the coordinate system is a matter of convenience. It will make the
notation somewhat more uniform. Despite the use of superscripts, these arenotthe components of any vectors.

Coordinate Basis
In order to discuss components of vectors, I’ll need basis vectors. These are defined by the following considerations:
Imagine a particle moving around in the manifold. At each instant of time the particle will have a definite position.
It will also have a definite velocity. So far no coordinate system is needed. The position of the particle is typically
described by specifying its coordinates as functions of time,xi(t). The basis vectors at a point are defined so
that the following equation holds:

~v=~ei

dxi
dt
The summation convention still holds. This is called a coordinate basis, and it willnot necessarily be a set of
unit vectors that you are accustomed to such as (ˆx,ˆy,ˆz) or (ˆr,θˆ,φˆ).
Look at some special cases. First, rectangular coordinates. If a particle is moving along thex^1 -axis (= the
x-axis) thendx^2 /dt= 0 =dx^3 /dt. Also, the velocity will be in thexdirection and of sizedx^1 /dt. This gives

~e 1 =xˆ

as you would expect. Similarly
~e 2 =y, ~eˆ 3 =z.ˆ
Second, examine plane polar coordinates, so there are onlyx^1 andx^2. Letx^1 =r, andx^2 =θ. A particle
moving on a straight line away from the origin will have a velocity

ˆr

dr

dt

, which gives ~e 1 =ˆr.

If the particle is moving around a circle of radiusrinstead, then its velocity is

θ rˆ dθ

dt

, which gives ~e 2 =rθˆ (38)