Vector Calculus

Cylindrical and Spherical Coordinates

Cylindrical

Useful with shapes that contain axial symmetry e.g. cylinders or cones. Represented as $(r,\theta ,z)$.

• $x=rcos\theta$
• $y=rsin\theta$
• $z=z$
• $r^2=x^2+y^2$
• $\theta =ta{n}^{-}1\left(\frac{y}{x}\right)$

Spherical

Useful with shapes that contain spherical symmetry i.e. symmetric everywhere about a point. Represented as $(\rho ,\theta ,\varphi )$.

• $z=\rho cos\varphi$
• $r=\rho sin\varphi$
• ${\rho }^2=x^2+y^2+z^2$

Note: Recall from above that $x=rcos\theta$ and $y=rsin\theta$.

• $x=\rho sin\varphi cos\theta$
• $y=\rho sin\varphi sin\theta$