Cylindrical Coordinates

Cylindrical coordinates are a 3D coordinate system that can be thought of as the 3D extension to polar coordinates. It is especially useful when the problem has rotational symmetry around an axis.

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Definition §

The cylindrical coordinate system is notated as:

$$(\rho ,\varphi ,z)$$

• $\rho \ge 0$ : The radius, must be a non-zero number.
• $\varphi \in [0,2\pi )$: The angle starting from the $x$-axis, going anti-clockwise (towards y-axis).
• $z\in \mathbb{R}$: Can be positive, zero or negative. Same as the $z$ in Cartesian coordinates.

Conversions §

Conversion With Cartesian §

Every point in 3D Cartesian coordinates $(x,y,z)$ can be converted to cylindrical coordinates via the equation:

$$x=\rho \cos (\varphi )y=\rho \sin (\varphi )z=z$$

And vice vera:

$$\rho =\Vert (x,y)\Vert =\sqrt{x^2+y^2}\varphi =\{\begin{array}{ll}\arctan (\frac{y}{x}),& x>0,y>0(\text{Q1})\\ \pi +\arctan (\frac{y}{x}),& x<0,y>0(\text{Q2})\\ \pi -\arctan (\frac{y}{x}),& x>0,y<0(\text{Q3})\\ 2\pi -\arctan (\frac{y}{x}),& x<0,y<0(\text{Q4})\end{array}z=z$$

Key Features §

• If $\rho =c$ (radius is constant), then we have an open-capped cylinder of radius $\rho$ parallel to the $z$-axis.
• If $\varphi =c$ (angle is constant), then we have a half-plane parallel to the $XZ$ plane after a rotation $\varphi$.
• If $z=c$ (height is constant), then we have a plane parallel to the $XY$ plane, at some height $z$.

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Jacobian §

Cartesian Jacobian §

The Jacobian is given by:

$$\begin{array}{rl}\det (\mathbf{J})& =\det \left[\begin{array}{ccc}\frac{\partial x}{\partial \rho }& \frac{\partial x}{\partial \varphi }& \frac{\partial x}{\partial z}\\ \frac{\partial y}{\partial \rho }& \frac{\partial y}{\partial \varphi }& \frac{\partial y}{\partial z}\\ \frac{\partial z}{\partial \rho }& \frac{\partial z}{\partial \varphi }& \frac{\partial z}{\partial z}\end{array}\right]=\det \left[\begin{array}{ccc}\cos (\varphi )& -\rho \sin (\varphi )& 0\\ \sin (\varphi )& \rho \cos (\varphi )& 0\\ 0& 0& 1\end{array}\right]\\ & \\ & =\det \left[\begin{array}{cc}\cos (\varphi )& -\rho \sin (\varphi )\\ \sin (\varphi )& \rho \cos (\varphi )\end{array}\right]=\rho {\cos }^2(\varphi )+\rho {\sin }^2(\varphi )\\ & =\rho \end{array}$$

Note that this is the same Jacobian as that of polar coordinates!

Hence:

$${\iiint }_{D}f(x,y,z)\text{d}x\text{d}y\text{d}z={\iiint }_{D^{\ast }}f(\rho \cos (\varphi ),\rho \sin (\varphi ))\cdot \rho \text{d}\rho \text{d}\varphi \text{d}z$$