# Lecture 012

\begin{cases} r^2 = x^2 + y^2\\ \tan \theta = \frac{y}{x}\\ \theta = \arctan(\frac{y}{x}) + \pi \text{ if (x,y) in quadrant 2 or 3}\\ \theta = \arctan(\frac{y}{x}) \text{ if (x,y) in quadrant 1 or 4, output negative theta on quadrant 4}\\ x = r\cos \theta\\ y = r\sin \theta\\ \rho = \sqrt{x^2 + y^2 + z^2}\\ \cos \varphi = \frac{z}{\sqrt{x^2 + y^2 + z^2}}\\ x = r \cos \theta = \rho \sin \varphi \cos \theta\\ y = \rho \sin \varphi \sin \theta\\ z = \rho \cos \varphi\\ r = \rho \sin \varphi\\ \end{cases}

Remember:

• $x = r \cos x$ because initial angle $\theta$ is at $x$ direction

• $r = \rho \sin \varphi$ because initial angle $\varphi$ is NOT at $r$ direction

Cylinder Coordinates: $(r, \theta, z)$

• $z = k$: a plane

• $r = k \iff x^2 + y^2 = k^2$: a cylinder

• $\theta = k$: half plane

• $z = r^2 \cos^2 \theta \iff z = x^2$: cylinder over a parabola

• $r = 3 \sin \theta$: circular cylinder over circle of radius $\frac{3}{2}$ centered at $(0, \frac{3}{2})$

• $r = z \iff x^2 + y^2 = z^2 (z \geq 0)$: half-cone

• $r^2 + z^2 = 9$: sphere

Spherical Coordinates: $(\rho, \theta, \varphi) | \rho > 0 \land 0 \leq \varphi \leq \pi$

• $p = k$: sphere at radius $\sqrt{k}$ centered at $(0, 0, 0)$

• $\theta = k$: half plane

• $\varphi = k$: half cone

• $\rho = 2 \cos \varphi \implies x^2 + y^2 + z^2 = 2z$: sphere centered at $(0, 0, 1)$

• $\rho \sin \varphi = 4 \implies r = 4$: cylinder

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