ABSTRACT
In the general case, the symmetric Jacobian matrix of the R3 → R3 transformation is102 defined as
1 0 00 r2 0 0 0 r2 sin(θ)2
whose associated Jacobian determinant reads98
|det(JT )| (3.286)98
= √
√ r4 sin(θ)2 = r2 sin(θ) (3.15)
Whenever the radial coordinate is constant though, the symmetric Jacobian matrix of the R2 → R3 transformation is instead defined as102
[ r2 0 0 r2 sin(θ)2
] (3.16)
whose associated Jacobian determinant reads98 √ det(JTT JT )
√ r4 sin(θ)2 = r2 sin(θ) (3.17)
of the R2 → R3 transformation is then defined as 102
[ 1 0 0 r2 sin(θ)2
] (3.18)
whose associated Jacobian determinant reads 98 √
= √ r2 sin(θ)2 = r sin(θ) (3.19)
In contrast, when the azimuthal coordinate is held constant instead, the symmetric Jacobian matrix of the R2 → R3 transformation is defined as 102
[ 1 0 0 r2
] (3.20)
whose associated Jacobian determinant reads 98 √
= √ r2 = r (3.21)
3.1.2 Cylindrical Coordinates
As illustrated in Figure 3.2, a three-dimensional cylindrical coordinate system is parameter-69 ized by the radial distance ρ ∈ [0,∞) of a point ~p to the longitudinal axis, the azimuthal angle φ ∈ (−pi,+pi] (or φ ∈ [0, 2pi)) between the azimuthal axis and the vector from the origin ~o to the projection of ~p onto the principal plane orthogonal to the longitudinal axis, and the longitudinal coordinate z ∈ (−∞,+∞) corresponding to the signed distance of ~p to the principal plane orthogonal to the longitudinal axis, or equivalently to the signed distance between ~o and the orthogonal projection of ~p onto the longitudinal axis.
