Coordinates transformation and Jacobian

In the Cartesian coordinates, a point is described by its coordinates $(x, y, z)$, which, in vector form, is written as

$$\mathbf{r}= x \hat{\mathbf{x}} + y \hat{\mathbf{y}} + z \hat{\mathbf{z}},$$ (100)

where $\mathbf{r}$ is the location vector of the point. The transformation between the Cartesian coordinates system, $(x, y, z)$, and a general coordinates system, $(x^1, x^2, x^3)$, can be written as

$$\mathbf{r}= x (x^1, x^2, x^3) \hat{\mathbf{x}} + y (x^1, x^2, x^3) \hat{\mathbf{y}} + z (x^1, x^2, x^3) \hat{\mathbf{z}} .$$ (101)

(For example, cylindrical coordinates $(R, \phi, Z)$ can be considered as a general coordinate systems, which are defined by $\mathbf{r}= R \cos \phi \hat{\mathbf{x}} + R \sin \phi \hat{\mathbf{y}} + Z \hat{\mathbf{z}}$.) The transformation function in Eq. (101) can be written as

$$\begin{array}{l} x = x (x^1, x^2, x^3)\\ y = y (x^1, x^2, x^3)\\ z = z (x^1, x^2, x^3) \end{array}$$ (102)

The Jacobian determinant (or simply called Jacobian) of the transformation in Eq. (102) is defined by

$$\mathcal{J}= \left\vert\begin{array}{ccc} \frac{\partial x}{\part... ...tial z}{\partial x_2} & \frac{\partial z}{\partial x_3} \end{array}\right\vert,$$ (103)

which can be written as

$$\mathcal{J}= \frac{\partial \mathbf{r}}{\partial x_1} \times \fra... ...tial \mathbf{r}}{\partial x_2} \cdot \frac{\partial \mathbf{r}}{\partial x_3} .$$ (104)

It is easy to prove that the Jacobian $\mathcal{J}$ in Eq. (104) can also be written (refer to another notes)

$$\mathcal{J}= (\nabla x_1 \times \nabla x_2 \cdot \nabla x_3)^{- 1} .$$ (105)

In most cases, the Jacobian of the transformation from the Cartesian coordinates to a particular coordinate system are simply called the Jacobian of that particular coordinate system without mentioning that this transformation is with respect to the Cartesian coordinates. Using this nomenclature, the Jacobian $\mathcal{J}$ of the Cartesian coordinates is obviously equal to one since the transformation is defined with respect to the Cartesian coordinates. If the Jacobian of a coordinate system is greater than zero, it is called a right-hand coordinate system. Otherwise it is called a left-hand system.