Page 46 - Intro to Tensor Calculus

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Figure 1.2-2. Cylindrical coordinates.

In this symbolic notation, we let T θ denote the identity transformation. That is, using the parameter
value of θ in the given set of transformation equations produces the identity transformation. The inverse
transformation can then be expressed in the form of finding the parameter value β such that T α T β = T θ .

Cartesian Coordinates

At times it is convenient to introduce an orthogonal Cartesian coordinate system having coordinates
i
y , i =1, 2,...,N. This space is denoted E N and represents an N-dimensional Euclidean space. Whenever
i
the generalized independent coordinates x ,i =1,... ,N are functions of the y s, and these equations are
0
functionally independent, then there exists independent transformation equations
2
1
i
N
i
y = y (x ,x ,...,x ), i =1, 2,...,N, (1.2.34)
with Jacobian different from zero. Similarly, if there is some other set of generalized coordinates, say a barred
i
0
0
system x ,i =1,... ,N where the x s are independent functions of the y s, then there will exist another set
of independent transformation equations
1
i
N
2
i
y = y (x , x ,..., x ), i =1, 2,...,N, (1.2.35)
with Jacobian different from zero. The transformations found in the equations (1.2.34) and (1.2.35) imply
0
0
that there exists relations between the x s and x s of the form (1.2.30) with inverse transformations of the
form (1.2.32). It should be remembered that the concepts and ideas developed in this section can be applied
to a space V N of any finite dimension. Two dimensional surfaces (N = 2) and three dimensional spaces
(N = 3) will occupy most of our applications. In relativity, one must consider spaces where N =4.
EXAMPLE 1.2-1. (cylindrical coordinates (r, θ, z)) Consider the transformation
x = x(r, θ, z)= r cos θ y = y(r, θ, z)= r sin θ z = z(r, θ, z)= z
from rectangular coordinates (x, y, z) to cylindrical coordinates (r, θ, z), illustrated in the figure 1.2-2. By
letting
3
1
2
3
1
2
y = x, y = y, y = z x = r, x = θ, x = z
the above set of equations are examples of the transformation equations (1.2.8) with u = r, v = θ, w = z as
the generalized coordinates.