Page 47 - Intro to Tensor Calculus

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EXAMPLE 1.2.2. (Spherical Coordinates) (ρ, θ, φ)
Consider the transformation

x = x(ρ, θ, φ)= ρ sin θ cos φ y = y(ρ, θ, φ)= ρ sin θ sin φ z = z(ρ, θ, φ)= ρ cosθ

from rectangular coordinates (x, y, z) to spherical coordinates (ρ, θ, φ). By letting

3
2
1
3
2
1
y = x, y = y, y = z x = ρ, x = θ, x = φ
the above set of equations has the form found in equation (1.2.8) with u = ρ, v = θ, w = φ the generalized
coordinates. One could place bars over the x s in this example in order to distinguish these coordinates from
0
the x s of the previous example. The spherical coordinates (ρ, θ, φ) are illustrated in the figure 1.2-3.
0

Figure 1.2-3. Spherical coordinates.

Scalar Functions and Invariance

We are now at a point where we can begin to define what tensor quantities are. The first definition is
for a scalar invariant or tensor of order zero.