Page 22 - Intro to Tensor Calculus

18

Transformation Equations

Consider two sets of N independent variables which are denoted by the barred and unbarred symbols
i
i
i
x and x with i =1,... ,N. The independent variables x ,i =1,... ,N can be thought of as defining
the coordinates of a point in a N−dimensional space. Similarly, the independent barred variables define a
point in some other N−dimensional space. These coordinates are assumed to be real quantities and are not
complex quantities. Further, we assume that these variables are related by a set of transformation equations.

1
2
i
i
N
x = x (x , x ,..., x ) i =1,... ,N. (1.1.7)
It is assumed that these transformation equations are independent. A necessary and sufficient condition that
these transformation equations be independent is that the Jacobian determinant be different from zero, that
is
1
∂x 1 ∂x 1 ∂x
N
∂x 1 ∂x 2 ··· ∂x
2 2
i ∂x ∂x ∂x 2
x ∂x 1 ∂x 2 ··· N
J( )= = ∂x ∂x 6=0.
.
x ∂¯x j . . . . . . . . .
.
.
N N N
∂x ∂x ∂x
∂x 1 ∂x 2 ··· ∂x N
This assumption allows us to obtain a set of inverse relations
1
2
N
i
i
x = x (x ,x ,...,x ) i =1,... ,N, (1.1.8)
0
0
where the x s are determined in terms of the x s. Throughout our discussions it is to be understood that the
given transformation equations are real and continuous. Further all derivatives that appear in our discussions
are assumed to exist and be continuous in the domain of the variables considered.
EXAMPLE 1.1-17. The following is an example of a set of transformation equations of the form
defined by equations (1.1.7) and (1.1.8) in the case N =3. Consider the transformation from cylindrical
coordinates (r, α, z) to spherical coordinates (ρ, β, α). From the geometry of the figure 1.1-5 we can find the
transformation equations
r = ρ sin β
α = α 0 <α < 2π
z = ρ cos β 0 <β <π

with inverse transformation
p
2
ρ = r + z 2
α = α
r
β = arctan( )
z
Now make the substitutions

3
1
2
3
1
2
(x ,x ,x )= (r, α, z) and (x , x , x )= (ρ, β, α).