7.4 Position, Velocity, and Acceleration

7.4.1 Cartesian Coordinates

The Cartesian coordinate system uses the coordinate axis vectors ${\overrightarrow{e}}_x$, ${\overrightarrow{e}}_y$, and ${\overrightarrow{e}}_z$ which are independent of time and independent of the position vector $\overrightarrow{s}$ (see Fig. 7.4a). In the Cartesian coordinate systems, the position vector $\overrightarrow{s}$, the velocity vector $\overrightarrow{v}$, and the acceleration vector $\overrightarrow{a}$ are defined as

$\begin{array}{ll}\overrightarrow{s}=\left(\begin{array}{c}\hfill s_x\hfill \\ \hfill s_y\hfill \\ \hfill s_z\hfill \end{array}\right)\hfill & =s_x{\overrightarrow{e}}_x+s_y{\overrightarrow{e}}_y+s_z{\overrightarrow{e}}_z\hfill \\ \overrightarrow{v}=\frac{d\overrightarrow{v}}{dt}=\left(\begin{array}{c}\hfill \frac{d{s}_x}{dt}\hfill \\ \hfill \frac{d{s}_y}{dt}\hfill \\ \hfill \frac{d{s}_z}{d{t}^2}\hfill \end{array}\right)\hfill & =v_x{\overrightarrow{e}}_x+v_y{\overrightarrow{e}}_y+v_z{\overrightarrow{e}}_z\hfill \\ \overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}=\frac{d^2\overrightarrow{s}}{d{t}^2}\left(\begin{array}{c}\hfill \frac{d^2{s}_x}{d{t}^2}\hfill \\ \hfill \frac{d^2{s}_y}{d{t}^2}\hfill \\ \hfill \frac{d^2{s}_z}{d{t}^2}\hfill \end{array}\right)\hfill & =a_x{\overrightarrow{e}}_x+a_y{\overrightarrow{e}}_y+a_z{\overrightarrow{e}}_z\hfill \end{array}$

7.4.2 Cylindrical Coordinates

The coordinate axis of the cylindrical coordinate system is dependent on the position vector $\overrightarrow{s}$ and is thus also dependent on time if the position vector is a function of time (see Fig. 7.4b). This means that the coordinate system changes over time and whenever $\overrightarrow{s}$ changes. This is a bit confusing as we are used to time-and space-invariant coordinate system. However, this is not a significant problem, as it only involves a bit more calculus in order to establish the velocity $\overrightarrow{v}$ and the acceleration vector $\overrightarrow{a}$ both of which are derived from the position vector as temporal derivatives.

We have already derived the unit vectors of the cylindrical coordinate systems in Eq. 7.23 and Eq. 7.24. We have also learned that the z-coordinate is identical to the coordinate of the Cartesian coordinate system. We found

$\begin{array}{lll}{\overrightarrow{e}}_r\hfill & =& \left(\begin{array}{c}\hfill cos\phi \hfill \\ \hfill sin\phi \hfill \\ \hfill 0\hfill \end{array}\right)\\ {\overrightarrow{e}}_{\phi }\hfill & =& \left(\begin{array}{c}\hfill -sin\phi \hfill \\ \hfill cos\phi \hfill \\ \hfill 0\hfill \end{array}\right)\\ {\overrightarrow{e}}_z\hfill & =& \left(\begin{array}{l}0\\ 0\\ 1\end{array}\right)\end{array}$

As stated, the unit vectors depend on the unit vectors of the Cartesian coordinate system. In order to derive the velocity vector $\overrightarrow{v}$ and the acceleration vector $\overrightarrow{a}$ we need the temporal derivatives given by

$\frac{d{\overrightarrow{e}}_r}{dt}=\frac{d\left(\begin{array}{c}\hfill cos\phi \hfill \\ \hfill sin\phi \hfill \\ \hfill 0\hfill \end{array}\right)}{dt}=\left(\begin{array}{c}\hfill -\frac{d\phi }{dt}sin\phi \hfill \\ \hfill \frac{d\phi }{dt}cos\phi \hfill \\ \hfill 0\hfill \end{array}\right)=\frac{d\phi }{dt}{\overrightarrow{e}}_{\phi }$

$\frac{d{\overrightarrow{e}}_{\phi }}{dt}=\frac{d\left(\begin{array}{c}\hfill -sin\phi \hfill \\ \hfill cos\phi \hfill \\ \hfill 0\hfill \end{array}\right)}{dt}=\left(\begin{array}{c}\hfill -\frac{d\phi }{dt}cos\phi \hfill \\ \hfill \frac{d\phi }{dt}sin\phi \hfill \\ \hfill 0\hfill \end{array}\right)=\frac{d\phi }{dt}{\overrightarrow{e}}_r$

$\frac{d{\overrightarrow{e}}_z}{dt}=\frac{d\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 1\hfill \end{array}\right)}{dt}=\left(\begin{array}{c}\hfill 0\hfill \\ \hfill 0\hfill \\ \hfill 0\hfill \end{array}\right)$

Please note that we have to use the chain rule of differentiation here (see Eq. 3.14). As you can see, the derivatives can be written as functions of the coordinate axes. Using these we can now formulate the position vector $\overrightarrow{s}$, the velocity vector $\overrightarrow{v}$, and the acceleration vector $\overrightarrow{a}$ as

$\overrightarrow{s}=r{\overrightarrow{e}}_r+z{\overrightarrow{e}}_z=\left(\begin{array}{c}\hfill r\hfill \\ \hfill 0\hfill \\ \hfill z\hfill \end{array}\right)$

$\begin{array}{ll}\overrightarrow{v}\hfill & =\frac{d\overrightarrow{s}}{dt}=\frac{dr}{dt}{\overrightarrow{e}}_r+r\frac{d{\overrightarrow{e}}_r}{dt}+\frac{dz}{dt}{\overrightarrow{e}}_z+z\frac{d{\overrightarrow{e}}_z}{dt}\hfill \\ \hfill & =\frac{dr}{dt}{\overrightarrow{e}}_r+r\frac{d\phi }{dt}{\overrightarrow{e}}_{\phi }+\frac{dz}{dt}{\overrightarrow{e}}_z\hfill \\ \hfill & =\left(\begin{array}{c}\hfill \frac{dr}{dt}\hfill \\ \hfill r\frac{d\phi }{dt}\hfill \\ \hfill \frac{dz}{dt}\hfill \end{array}\right)\hfill \end{array}$

$\begin{array}{ll}\overrightarrow{a}\hfill & =\frac{d\overrightarrow{v}}{dt}=\frac{d^{2}r}{d{t}^2}{\overrightarrow{e}}_r+\frac{dr}{dt}\frac{d{\overrightarrow{e}}_r}{dt}+\frac{dr}{dt}\frac{d\phi }{dt}{\overrightarrow{e}}_{\phi }+r\left(\frac{d^2\phi }{d{t}^2}{\overrightarrow{e}}_{\phi }+\frac{d\phi }{dt}\frac{d{\overrightarrow{e}}_{\phi }}{dt}\right)+\frac{d^{2}z}{d{t}^2}{\overrightarrow{e}}_z\hfill \\ \hfill & =\left(\frac{d^{2}r}{d{t}^2}-r{\left(\frac{d\phi }{dt}\right)}^2\right){\overrightarrow{e}}_r+\left(2\frac{dr}{dt}\frac{d\phi }{dt}+r\frac{d^2\phi }{d{t}^2}\right){\overrightarrow{e}}_{\phi }+\frac{d^{2}z}{d{t}^2}{\overrightarrow{e}}_z\hfill \\ \hfill & =\left(\begin{array}{c}\hfill \frac{d^{2}r}{d{t}^2}-r{\left(\frac{d\phi }{dt}\right)}^2\hfill \\ \hfill 2\frac{dr}{dt}\frac{d\phi }{dt}+r\frac{d^2\phi }{d{t}^2}\hfill \\ \hfill \frac{d^{2}z}{d{t}^2}\hfill \end{array}\right)\hfill \end{array}$

where we have used Eq. 7.54, Eq. 7.55, and Eq. 7.56 to convert the time-derivative of the coordinate axes into functions of the axes only. Eq. 7.57, Eq. 7.58, and Eq. 7.59 are the sought position, velocity, and acceleration vectors in the cylindrical coordinate system.

7.4.3 Polar Coordinates

Given the fact that polar coordinates (see Fig. 7.4c) are a special case of cylindrical coordinates, we can note down the position, velocity, and acceleration vectors by copying the vectors from the cylindrical coordinates while ignoring the z-component. We therefore find from Eq. 7.57, Eq. 7.58, and Eq. 7.59

$\overrightarrow{s}=\left(\begin{array}{l}r\hfill \\ 0\hfill \end{array}\right)$

$\overrightarrow{v}=\left(\begin{array}{l}\frac{dr}{dt}\hfill \\ r\frac{d\phi }{dt}\hfill \end{array}\right)$

$\overrightarrow{a}=\left(\begin{array}{l}\frac{d^{2}r}{d{t}^2}-r{\left(\frac{d\phi }{dt}\right)}^2\hfill \\ 2\frac{dr}{dt}\frac{d\phi }{dt}+r\frac{d^2\phi }{d{t}^2}\hfill \end{array}\right)$

7.4.4 Spherical Coordinates

We have already derived the unit vectors of the spherical coordinate system (see Fig. 7.4d) given by Eq. 7.40, Eq. 7.42, and Eq. 7.41 as

$\begin{array}{ll}{\overrightarrow{e}}_{\rho }\hfill & =sin\theta cos\phi {\overrightarrow{e}}_x+sin\theta \text{sin}\phi {\overrightarrow{e}}_y+cos\theta {\overrightarrow{e}}_z\hfill \\ {\overrightarrow{e}}_{\theta }\hfill & =cos\theta cos\phi {\overrightarrow{e}}_x+cos\theta sin\phi {\overrightarrow{e}}_y-sin\theta {\overrightarrow{e}}_z\hfill \\ {\overrightarrow{e}}_{\phi }\hfill & =-sin\phi {\overrightarrow{e}}_x+cos\phi {\overrightarrow{e}}_y\hfill \end{array}$

where we again formulate the time-derivatives that we will require for finding the position, velocity, and acceleration vectors

$\begin{array}{ll}\frac{d{\overrightarrow{e}}_{\rho }}{dt}\hfill & \begin{array}{l}=\left(\frac{d\theta }{dt}cos\theta cos\phi -\frac{d\phi }{dt}sin\theta sin\phi \right){\overrightarrow{e}}_x+\left(\frac{d\theta }{dt}cos\theta sin\phi +\frac{d\phi }{dt}sin\theta cos\phi \right){\overrightarrow{e}}_y\hfill \\ -\frac{d\theta }{dt}sin\theta {\overrightarrow{e}}_z\hfill \end{array}\hfill \\ \hfill & \frac{d\phi }{dt}sin\theta \left(-sin\phi {\overrightarrow{e}}_x+\mathit{cos\phi }{\overrightarrow{e}}_y\right)+\frac{d\theta }{dt}\left(cos\theta cos\phi {\overrightarrow{e}}_x+cos\theta sin\phi {\overrightarrow{e}}_y\right)\hfill \\ \hfill & \frac{d\phi }{dt}sin\theta {\overrightarrow{e}}_{\phi }+\frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }\hfill \end{array}$

$\begin{array}{ll}\frac{d{\overrightarrow{e}}_{\phi }}{dt}\hfill & =-\frac{d\phi }{dt}cos\phi {\overrightarrow{e}}_x-\frac{d\phi }{dt}sin\phi {\overrightarrow{e}}_y\hfill \\ \hfill & =\frac{d\phi }{dt}\left(sin\theta {\overrightarrow{e}}_p+cos\theta {\overrightarrow{e}}_{\theta }\right)\hfill \end{array}$

$\begin{array}{ll}\frac{d{\overrightarrow{e}}_{\theta }}{dt}\hfill & \begin{array}{l}=\left(-\frac{d\theta }{dt}sin\theta cos\phi -\frac{d\phi }{dt}\mathit{cos\theta }sin\phi \right){\overrightarrow{e}}_x+\left(-\frac{d\theta }{dt}sin\theta sin\phi +\frac{d\phi }{dt}cos\theta cos\phi \right){\overrightarrow{e}}_y\\ -\frac{d\theta }{dt}cos\theta {\overrightarrow{e}}_z\end{array}\hfill \\ \hfill & -\frac{d\theta }{dt}\left(sin\theta cos\phi {\overrightarrow{e}}_x+sin\theta sin\phi {\overrightarrow{e}}_y+cos\theta {\overrightarrow{e}}_z\right)+\frac{d\phi }{dt}cos\theta \left(-sin\phi {\overrightarrow{e}}_x+\mathit{cos\phi }{\overrightarrow{e}}_y\right)\hfill \\ \hfill & -\frac{d\theta }{dt}{\overrightarrow{e}}_{\rho }+\frac{d\phi }{dt}cos\theta {\overrightarrow{e}}_{\phi }\hfill \end{array}$

The last expression that results in Eq. 7.65 is not intuitive and the replacing of ${\overrightarrow{e}}_x$ and ${\overrightarrow{e}}_y$ by ${\overrightarrow{e}}_{\rho }$ and ${\overrightarrow{e}}_{\theta }$ requires a bit of rewriting. However, if you check the expression, you will see that it is actually correct. Using Eq. 7.63, Eq. 7.64, and Eq. 7.65 we can now derive the position, velocity, and acceleration vectors in the spherical coordinate systems as

$\overrightarrow{s}=\rho {\overrightarrow{e}}_{\rho }$

$\begin{array}{ll}\overrightarrow{u}\hfill & =\frac{d\overrightarrow{s}}{dt}=\frac{dp}{dt}{\overrightarrow{e}}_{\rho }+\rho \frac{d{\overrightarrow{e}}_{\rho }}{dt}\hfill \\ \hfill & =\frac{d\rho }{dt}{\overrightarrow{e}}_{\rho }+\rho \frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }+\rho \frac{d\phi }{dt}sin\theta {\overrightarrow{e}}_{\phi }\hfill \end{array}$

$\begin{array}{ll}\overrightarrow{a}\hfill & =\frac{d\overrightarrow{v}}{dt}\hfill \\ \hfill & \begin{array}{l}=\frac{d^2\rho }{d{t}^2}{\overrightarrow{e}}_{\rho }+\frac{dp}{dt}\frac{d{\overrightarrow{e}}_{\rho }}{dt}+\frac{dp}{dt}\frac{d\phi }{dt}sin\theta {\overrightarrow{e}}_{\phi }+\rho \left(\frac{d^2\phi }{d{t}^2}sin\theta {\overrightarrow{e}}_{\phi }+\frac{d\phi }{dt}\left(\frac{d\theta }{dt}cos\theta {\overrightarrow{e}}_{\phi }+sin\theta \frac{d{\overrightarrow{e}}_{\phi }}{dt}\right)\right)\\ +\frac{d\rho }{dt}\frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }+\rho \left(\frac{d^2\theta }{d{t}^2}{\overrightarrow{e}}_{\theta }+\frac{d\theta }{dt}\frac{d{\overrightarrow{e}}_{\theta }}{dt}\right)\end{array}\hfill \\ \hfill & \begin{array}{l}=\frac{d^2\rho }{d{t}^2}{\overrightarrow{e}}_{\rho }+\frac{d\rho }{dt}\left(\frac{d\phi }{dt}sin\theta {\overrightarrow{e}}_{\phi }+\frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }\right)+\frac{d\rho }{dt}\frac{d\phi }{dt}sin\theta {\overrightarrow{e}}_{\phi }+\\ \rho \left(\frac{d^2\phi }{d{t}^2}sin\theta {\overrightarrow{e}}_{\phi }+\frac{d\phi }{dt}\left(\frac{d\theta }{dt}cos\theta {\overrightarrow{e}}_{\phi }+sin\theta \left(-\frac{d\phi }{dt}\left(sin\theta {\overrightarrow{e}}_{\rho }+cos\theta {\overrightarrow{e}}_{\theta }\right)\right)\right)\right)+\\ \frac{d\rho }{dt}\frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }+\rho \left(\frac{d^2\theta }{d{t}^2}{\overrightarrow{e}}_{\theta }+\frac{d\theta }{dt}\left(-\frac{d\theta }{dt}{\overrightarrow{e}}_{\rho }+\frac{d\phi }{dt}cos\theta {\overrightarrow{e}}_{\phi }\right)\right)\end{array}\hfill \\ \hfill & \begin{array}{l}=\frac{d^2\rho }{d{t}^2}{\overrightarrow{e}}_{\rho }+\frac{d\rho }{dt}\frac{d\phi }{dt}sin\theta {\overrightarrow{e}}_{\phi }+\frac{d\rho }{dt}\frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }+\frac{dp}{dt}\frac{d\phi }{dt}sin\theta {\overrightarrow{e}}_{\phi }+\rho \frac{d^2\phi }{d{t}^2}sin\theta {\overrightarrow{e}}_{\phi }+\rho \frac{d\phi }{dt}\frac{d\theta }{dt}cos\theta {\overrightarrow{e}}_{\phi }\\ -\rho \frac{d\phi }{dt}sin\theta \frac{d\phi }{dt}sin\theta {\overrightarrow{e}}_{\rho }-\rho \frac{d\phi }{dt}sin\theta \frac{d\phi }{dt}cos\theta {\overrightarrow{e}}_{\theta }+\frac{d\rho }{dt}\frac{d\theta }{dt}{\overrightarrow{e}}_{\theta }+\rho \frac{d^2\theta }{d{t}^2}{\overrightarrow{e}}_{\theta }\\ -\rho \frac{d\theta }{dt}\frac{d\theta }{dt}{\overrightarrow{e}}_{\rho }+\rho \frac{d\theta }{dt}\frac{d\phi }{dt}cos\theta {\overrightarrow{e}}_{\phi }\\ =\left(\frac{d^2\rho }{d{t}^2}-\rho {\left(\frac{d\phi }{dt}\right)}^2{sin}^2\theta -\rho {\left(\frac{d\theta }{dt}\right)}^2\right){\overrightarrow{e}}_{\rho }+\left(2\frac{d\rho }{dt}\frac{d\theta }{dt}-\rho {\left(\frac{d\phi }{dt}\right)}^{2}sin\theta cos\theta +\rho \frac{d^2\theta }{d{t}^2}\right){\overrightarrow{e}}_{\theta }\\ +\left(\left(2\frac{d\rho }{dt}\frac{d\phi }{dt}+\rho \frac{d^2\phi }{d{t}^2}\right)sin\theta +2r\frac{d\phi }{dt}\frac{d\theta }{dt}cos\theta \right){\overrightarrow{e}}_{\phi }\end{array}\hfill \end{array}$

Eq. 7.66, Eq. 7.67, and Eq. 7.68 are the sought vectors for position $\overrightarrow{s}$, velocity $\overrightarrow{u}$, and acceleration $\overrightarrow{a}$. The derivation is not too complicated given Eq. 7.63, Eq. 7.64, and Eq. 7.65 and careful application of the chain rule of differentiation (see Eq. 3.14).

7.5 Jacobian Matrix

7.5.1 Definition and Usage

The Jacobian matrix or simply Jacobian is a matrix which is required for the conversion of surface and volume integrals from one coordinate system to another. For a vector $\overrightarrow{F}=F_1\left(x_1,\dots ,x_n\right){\overrightarrow{e}}_1+F_2\left(x_1,\dots ,x_n\right){\overrightarrow{e}}_2+\dots +F_n\left(x_1,\dots ,x_n\right){\overrightarrow{e}}_n$

$J_{\overrightarrow{F}}=\left(\begin{array}{l}\begin{array}{llll}\frac{\partial F_1\left(x_1,\dots ,x_n\right)}{\partial x_1}\hfill & \frac{\partial F_1\left(x_1,\dots ,x_n\right)}{\partial x_2}\hfill & \dots \hfill & \frac{\partial F_1\left(x_1,\dots ,x_n\right)}{\partial x_n}\hfill \\ \frac{\partial F_2\left(x_1,\dots ,x_n\right)}{\partial x_1}\hfill & \frac{\partial F_2\left(x_1,\dots ,x_n\right)}{\partial x_2}\hfill & \dots \hfill & \frac{\partial F_2\left(x_1,\dots ,x_n\right)}{\partial x_n}\hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \end{array}\\ \begin{array}{llll}\frac{\partial F_n\left(x_1,\dots ,x_n\right)}{\partial x_1}\hfill & \frac{\partial F_n\left(x_1,\dots ,x_n\right)}{\partial x_2}\hfill & \cdots \hfill & \frac{\partial F_n\left(x_1,\dots ,x_n\right)}{\partial x_n}\hfill \end{array}\end{array}\right)$

As you can see, the Jacobian matrix sums up all the changes of each component of the vector $\overrightarrow{F}$ along each coordinate axis, respectively. Jacobian matrices are used to transform the infinitesimal vectors from one coordinate system to another. We will mostly be interested in the Jacobian matrices that allow transformation from the Cartesian to a different coordinate system.

7.5.2 Cartesian/Cylindrical Coordinates Conversion

We will begin with the Jacobian matrix for the conversion of Cartesian to cylindrical coordinates. We have already determined the vector (see Eq. 7.22) in Cartesian coordinates that is

$\overrightarrow{F}\left(r,\phi ,z\right)=rcos\phi {\overrightarrow{e}}_x+rsin\phi {\overrightarrow{e}}_y+z{\overrightarrow{e}}_z$

Now we build the Jacobian for this vector given according to Eq. 7.69 by

$\begin{array}{ll}J\hfill & =\left(\begin{array}{lll}\frac{\partial x}{\partial r}\hfill & \frac{\partial x}{\partial \phi }\hfill & \frac{\partial x}{\partial z}\hfill \\ \frac{\partial y}{\partial r}\hfill & \frac{\partial y}{\partial \phi }\hfill & \frac{\partial y}{\partial z}\hfill \\ \frac{\partial z}{\partial r}\hfill & \frac{\partial z}{\partial \phi }\hfill & \frac{\partial z}{\partial z}\hfill \end{array}\right)=\left(\begin{array}{lll}\frac{\partial \left(rcos\phi \right)}{\partial r}\hfill & \frac{\partial \left(rcos\phi \right)}{\partial \phi }\hfill & \frac{\partial \left(rcos\phi \right)}{\partial z}\hfill \\ \frac{\partial \left(rsin\phi \right)}{\partial r}\hfill & \frac{\partial \left(rsin\phi \right)}{\partial \phi }\hfill & \frac{\partial \left(rsin\phi \right)}{\partial z}\hfill \\ \frac{\partial z}{\partial r}\hfill & \frac{\partial z}{\partial \phi }\hfill & \frac{\partial z}{\partial z}\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{lll}cos\phi \hfill & -rsin\phi \hfill & 0\hfill \\ sin\phi \hfill & rcos\phi \hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right)\hfill \end{array}$

As stated, the Jacobian matrix allows us to express a relationship between the infinitesimal vectors $\left(\begin{array}{l}dx\hfill \\ \mathit{dy}\hfill \\ dz\hfill \end{array}\right)$ and $\left(\begin{array}{l}dr\hfill \\ d\phi \hfill \\ dz\hfill \end{array}\right)$ by using

$\begin{array}{ll}\left(\begin{array}{l}dx\hfill \\ \mathit{dy}\hfill \\ dz\hfill \end{array}\right)\hfill & =J\left(\begin{array}{l}dr\hfill \\ d\phi \hfill \\ dz\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{lll}\frac{\partial x}{\partial r}\hfill & \frac{\partial x}{\partial \phi }\hfill & \frac{\partial x}{\partial z}\hfill \\ \frac{\partial y}{\partial r}\hfill & \frac{\partial y}{\partial \phi }\hfill & \frac{\partial y}{\partial z}\hfill \\ \frac{\partial z}{\partial r}\hfill & \frac{\partial z}{\partial \phi }\hfill & \frac{\partial z}{\partial z}\hfill \end{array}\right)\left(\begin{array}{l}dr\hfill \\ d\phi \hfill \\ dz\hfill \end{array}\right)=\left(\begin{array}{l}\frac{\partial x}{\partial r}dr+\frac{\partial x}{\partial \phi }d\phi +\frac{\partial x}{\partial z}dz\hfill \\ \frac{\partial y}{\partial r}dr+\frac{\partial y}{\partial \phi }d\phi +\frac{\partial y}{\partial z}dz\hfill \\ \frac{\partial z}{\partial r}dr+\frac{\partial z}{\partial \phi }d\phi +\frac{\partial z}{\partial z}dz\hfill \end{array}\right)\hfill \end{array}$

One can easily confirm that this calculus is correct by performing the matrix/vector multiplication which yields the total differentials according to Eq. 3.12. If the inverse transformation is to be made, the inverse Jacobian matrix J^− 1 must be used. Without actually deriving the matrix (which is straightforward, but a little tedious) the inverse Jacobi matrix J^− 1 is given by

${\mathrm{J}}^{-1}=\left(\begin{array}{lll}\frac{\partial r}{\partial x}\hfill & \frac{\partial r}{\partial y}\hfill & \frac{\partial r}{\partial z}\hfill \\ \frac{\partial \phi }{\partial x}\hfill & \frac{\partial \phi }{\partial y}\hfill & \frac{\partial \phi }{\partial z}\hfill \\ \frac{\partial z}{\partial x}\hfill & \frac{\partial z}{\partial y}\hfill & \frac{\partial z}{\partial z}\hfill \end{array}\right)=\left(\begin{array}{lll}cos\phi \hfill & sin\phi \hfill & 0\hfill \\ -\frac{sin\phi }{r}\hfill & \frac{cos\phi }{r}\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right)$

We will discuss examples of matrix inversion in section 25.2.2. The inverse Jacobian is used to transfer the infinitesimal vector $\left(\begin{array}{l}dr\hfill \\ d\phi \hfill \\ dz\hfill \end{array}\right)$ in cylindrical coordinates to the infinitesimal vector $\left(\begin{array}{l}dx\hfill \\ \mathit{dy}\hfill \\ dz\hfill \end{array}\right)$ in Cartesian coordinates by

$\begin{array}{ll}\left(\begin{array}{l}dr\hfill \\ d\phi \hfill \\ dz\hfill \end{array}\right)\hfill & ={\mathrm{J}}^{-1}\left(\begin{array}{l}dx\hfill \\ \mathit{dy}\hfill \\ dz\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{lll}\frac{\partial r}{\partial x}\hfill & \frac{\partial r}{\partial y}\hfill & \frac{\partial r}{\partial z}\hfill \\ \frac{\partial \phi }{\partial x}\hfill & \frac{\partial \phi }{\partial y}\hfill & \frac{\partial \phi }{\partial z}\hfill \\ \frac{\partial z}{\partial x}\hfill & \frac{\partial z}{\partial y}\hfill & \frac{\partial z}{\partial z}\hfill \end{array}\right)\left(\begin{array}{l}dx\hfill \\ \mathit{dy}\hfill \\ dz\hfill \end{array}\right)=\left(\begin{array}{l}\frac{\partial r}{\partial x}dx+\frac{\partial r}{\partial y}\mathit{dy}+\frac{\partial r}{\partial z}dz\hfill \\ \frac{\partial \phi }{\partial x}dx+\frac{\partial \phi }{\partial y}\mathit{dy}+\frac{\partial \phi }{\partial z}dz\hfill \\ \frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}\mathit{dy}+\frac{\partial z}{\partial z}dz\hfill \end{array}\right)\hfill \end{array}$

which is, again, simply the total differential (see Eq. 3.12). Inverse matrices can only be constructed if the determinant of the matrix is nonzero (see the following section on the Jacobian determinant for a quick review of what exactly the determinant of a matrix is). There exist several methods for constructing the inverse matrix of a mix with nonzero determinant, e.g., the Gauß-Jordan¹ elimination (see section 25.2.2). In case of the Jacobian constructed for our matrix, the inverse matrix is given by Eq. 7.72. As stated, Jacobian and inverse Jacobian are used to transfer the infinitesimal vectors between the two coordinate systems according to

$\begin{array}{ll}\left(\begin{array}{l}dx\hfill \\ \mathit{dy}\hfill \\ dz\hfill \end{array}\right)\hfill & =\mathrm{J}\left(\begin{array}{l}dr\hfill \\ d\phi \hfill \\ dz\hfill \end{array}\right)\hfill \\ \left(\begin{array}{l}dr\hfill \\ d\phi \hfill \\ dz\hfill \end{array}\right)\hfill & ={\mathrm{J}}^{-1}\left(\begin{array}{l}dx\hfill \\ \mathit{dy}\hfill \\ dz\hfill \end{array}\right)\hfill \end{array}$

If you perform the matrix multiplication J·J^− 1 you would find the unit matrix.

Jacobian Determinant. Besides the Jacobian matrix, we will also require the Jacobian determinant which is given by

$detJ=\left|\begin{array}{llll}\frac{\partial F_1\left(x_1,\dots ,x_n\right)}{\partial x_1}\hfill & \frac{\partial F_1\left(x_1,\dots ,x_n\right)}{\partial x_2}\hfill & \cdots \hfill & \frac{\partial F_1\left(x_1,\dots ,x_n\right)}{\partial x_n}\hfill \\ \frac{\partial F_2\left(x_1,\dots ,x_n\right)}{\partial x_1}\hfill & \frac{\partial F_2\left(x_1,\dots ,x_n\right)}{\partial x_2}\hfill & \cdots \hfill & \frac{\partial F_2\left(x_1,\dots ,x_n\right)}{\partial x_n}\hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \frac{\partial F_n\left(x_1,\dots ,x_n\right)}{\partial x_1}\hfill & \frac{\partial F_n\left(x_1,\dots ,x_n\right)}{\partial x_2}\hfill & \cdots \hfill & \frac{\partial F_n\left(x_1,\dots ,x_n\right)}{\partial x_n}\hfill \end{array}\right|$

The Jacobian determinant is used extensively during integral transformation as it is an important component of the Gauss’s theorem that we discussed in section 7.2.1. Returning to the Jacobian found for the transformation of Cartesian to cylindrical coordinates (see Eq. 7.70), the Jacobian determinant can be calculated to be

$detJ=\left|\begin{array}{lll}cos\phi \hfill & -rsin\phi \hfill & 0\hfill \\ sin\phi \hfill & rcos\phi \hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right|=r{cos}^2\phi +r{sin}^2\phi =r$

So in fact the Jacobian matrix of Eq. 7.70 has a nonzero determinant and thus the inverse can be constructed. It is given by Eq. 7.72.

7.5.3 Cartesian/Polar Coordinates Conversion

Again, we can simplify the derivation of the Jacobian and the inverse Jacobian for the conversion of Cartesian to polar coordinates by noting that polar coordinates are merely a special case of cylindrical coordinates. Both matrices are 2 × 2 matrices given by

$\begin{array}{ll}J\hfill & =\left(\begin{array}{cc}\hfill \frac{\partial x}{\partial r}\hfill & \hfill \frac{\partial x}{\partial \phi }\hfill \\ \hfill \frac{\partial y}{\partial r}\hfill & \hfill \frac{\partial y}{\partial \phi }\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}\hfill \frac{\partial rcos\phi }{\partial r}\hfill & \hfill \frac{\partial rcos\phi }{\partial \phi }\hfill \\ \hfill \frac{\partial rsin\phi }{\partial r}\hfill & \hfill \frac{\partial rsin\phi }{\partial \phi }\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{cc}\hfill cos\phi \hfill & \hfill -rsin\phi \hfill \\ \hfill sin\phi \hfill & \hfill rcos\phi \hfill \end{array}\right)\hfill \end{array}$

and

$\begin{array}{lll}{J}^{-1}& =& \left(\begin{array}{cc}\frac{\partial r}{\partial x}& \frac{\partial r}{\partial y}\\ \frac{\partial \phi }{\partial x}& \frac{\partial \phi }{\partial y}\end{array}\right)\\ & =& \left(\begin{array}{cc}cos\phi & sin\phi \\ -\frac{sin\phi }{r}& \frac{cos\phi }{r}\end{array}\right)\end{array}$

Jacobian Determinant. The Jacobian determinant is given by

$\begin{array}{ll}det\mathbf{J}\hfill & =\left|\begin{array}{ll}cos\phi \hfill & -rsin\phi \hfill \\ sin\phi \hfill & rcos\phi \hfill \end{array}\right|=r{cos}^2\phi +rsi{n}^2\phi =r\hfill \end{array}$

7.5.4 Cartesian/Spherical Coordinates Conversion

We will now derive the Jacobian matrix for the conversion of Cartesian to spherical coordinates using the position vector (see Eq. 7.78) in Cartesian coordinates given by

$\overrightarrow{F}\left(\rho ,\theta ,\phi \right)=\rho sin\theta cos\phi {\overrightarrow{e}}_x+\rho sin\theta sin\phi {\overrightarrow{e}}_y+\rho cos\theta {\overrightarrow{e}}_z$

Now we built the Jacobian for this vector given according to Eq. 7.69 given by

$\begin{array}{ll}J\hfill & =\left(\begin{array}{lll}\frac{\partial x}{\partial \rho }\hfill & \frac{\partial x}{\partial \theta }\hfill & \frac{\partial x}{\partial \phi }\hfill \\ \frac{\partial y}{\partial \rho }\hfill & \frac{\partial y}{\partial \theta }\hfill & \frac{\partial y}{\partial \phi }\hfill \\ \frac{\partial z}{\partial \rho }\hfill & \frac{\partial z}{\partial \theta }\hfill & \frac{\partial z}{\partial \phi }\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{ccc}\hfill \frac{\partial \left(\rho sin\theta cos\phi \right)}{\partial \rho }\hfill & \hfill \frac{\partial \left(\rho sin\theta cos\phi \right)}{\partial \theta }\hfill & \hfill \frac{\partial \left(\rho sin\theta cos\phi \right)}{\partial \phi }\hfill \\ \hfill \frac{\partial \left(\rho sin\theta sin\phi \right)}{\partial \rho }\hfill & \hfill \frac{\partial \left(\rho sin\theta sin\phi \right)}{\partial \theta }\hfill & \hfill \frac{\partial \left(\rho sin\theta sin\phi \right)}{\partial \phi }\hfill \\ \hfill \frac{\partial \left(\rho cos\theta \right)}{\partial \rho }\hfill & \hfill \frac{\partial \left(\rho cos\theta \right)}{\partial \theta }\hfill & \hfill \frac{\partial \left(\rho cos\theta \right)}{\partial \phi }\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{ccc}\hfill sin\theta cos\phi \hfill & \hfill \rho cos\theta cos\phi \hfill & \hfill -\rho sin\theta sin\phi \hfill \\ \hfill sin\theta sin\phi \hfill & \hfill \rho cos\theta sin\phi \hfill & \hfill \rho sin\theta cos\phi \hfill \\ \hfill cos\theta \hfill & \hfill -\rho sin\theta \hfill & \hfill 0\hfill \end{array}\right)\hfill \end{array}$

The inverse matrix is given by

$\begin{array}{ll}{\mathrm{J}}^{-1}\hfill & =\left(\begin{array}{lll}\frac{\partial \rho }{\partial x}\hfill & \frac{\partial \rho }{\partial y}\hfill & \frac{\partial \rho }{\partial z}\hfill \\ \frac{\partial \theta }{\partial x}\hfill & \frac{\partial \theta }{\partial y}\hfill & \frac{\partial \theta }{\partial z}\hfill \\ \frac{\partial \phi }{\partial x}\hfill & \frac{\partial \phi }{\partial y}\hfill & \frac{\partial \phi }{\partial z}\hfill \end{array}\right)\hfill \\ \hfill & =\left(\begin{array}{ccc}\hfill sin\theta cos\phi \hfill & \hfill sin\theta sin\phi \hfill & \hfill cos\theta \hfill \\ \hfill \frac{1}{\rho }cos\theta cos\phi \hfill & \hfill \frac{1}{\rho }cos\theta sin\phi \hfill & \hfill -\frac{1}{\rho }\mathit{sin\theta }\hfill \\ \hfill -\frac{1}{\rho }\frac{sin\phi }{sin\theta }\hfill & \hfill \frac{1}{\rho }\frac{\mathit{cos\phi }}{sin\theta }\hfill & \hfill 0\hfill \end{array}\right)\hfill \end{array}$

Jacobian Determinant. The determinant of Eq. 7.79 is given by

$\begin{array}{ll}detJ\hfill & =\left|\begin{array}{ccc}\hfill sin\theta cos\phi \hfill & \hfill \rho cos\theta cos\phi \hfill & \hfill -\rho sin\theta sin\phi \hfill \\ \hfill sin\theta sin\phi \hfill & \hfill \rho cos\theta sin\phi \hfill & \hfill \rho sin\theta cos\phi \hfill \\ \hfill cos\theta \hfill & \hfill -\rho sin\theta \hfill & \hfill 0\hfill \end{array}\right|\hfill \\ \hfill & ={\rho }^2\left({cos}^2\theta {cos}^2\phi sin\theta +{sin}^2\theta {sin}^2\phi sin\theta -\left(-sin\theta {sin}^2\phi {cos}^2\theta -{sin}^2\theta co{s}^2\mathit{\phi sin\theta }\right)\right)\hfill \\ \hfill & ={\rho }^2\left(\left({cos}^2\theta +{sin}^2\theta \right)co{s}^2\phi sin\theta +\left({sin}^2\theta +co{s}^2\theta \right){sin}^2\phi \text{sin}\theta \right)\hfill \\ \hfill & ={\rho }^{2}sin\theta \left({cos}^2\phi +{sin}^2\phi \right)\hfill \\ \hfill & ={\rho }^{2}sin\theta \hfill \end{array}$