0.1 Cylindrical and Spherical Coordinates

Recall that in 2-space, we have different ways to represent a point. Using Cartesian coordinates, we represent a point \(P\) as \((x, y)\). In polar coordinate system, we write the point as \((r, \theta )\) where \(r\) is the radius, or distance from the origin, and \(\theta \) is the angle from the positive \(x\)-axis. In 3-space, we can represent a coordinate in a very similar way.

0.1.1 Cylindrical Coordinates

As always, let’s get started with the definition.

Just like the polar coordinates, we can see that the rectangular coordinate of \(P\) with cylindrical coordinates \((r, \theta , z)\) adhere to the following equations. \[ x = r\cos \theta , \quad y = r\sin \theta , \quad z = z \] Similarly, given point \(P\) with rectangular coordinate \((x,y,z)\), we can see that the cylindrical coordinate adhere to the following. \[ r = \sqrt {x^2 + y^2}, \quad \tan \theta = \frac {y}{x}, \quad z = z \] However, it is important to note that \(\tan \theta = \frac {y}{x}\) may not hold depending on the position of the point. It is important to see which quadrant the point lies in first before using it.

Sometimes being able to represent coordinates in cylindrical coordinates helps writing surfaces in a more simple way. Consider the surface defined by \(z = \sqrt {9x^2 + 9y^2}\). This surface is quite cumbersome to write. However using \(r = x^2 + y^2\), we can see that \(z = 3r\). Therefore the surface can be represented as \(r = \frac {z}{3}\), which in my opinion is way easier to write. Continuing, let’s discuss more on the second way of representing: spherical coordinates.

0.1.2 Spherical Coordinates

Another extension of polar coordinates for 3-space besides cylindrical coordinate is spherical coordinate. To understand the coordinates, consider the following diagram.

Using the figure above, we can define the spherical coordinate of a point \(P\).

Using this definition, let’s take a look at how the spherical coordinates can be converted to cylindrical and rectangular coordinates. From the diagram, it is evident that the following equations hold. \[ r = \rho \sin \phi , \quad x = \rho \sin \phi \cos \theta , \quad y = \rho \sin \phi \sin \theta , \quad z = \rho \cos \phi \] Similarly, the following equations holds for converting rectangular coordinates to spherical coordinates. \[ \rho = \sqrt {x^2 + y^2 + z^2}, \quad \tan \theta = \frac {y}{x}, \quad \phi = \cos ^{-1} \left ( \frac {z}{\sqrt {x^2 + y^2 + z^2}} \right ) \] Just like the cylindrical coordinates, spherical coordinates can help in representing certain surfaces. Consider the following example.

This is it for different coordinate representations of a point in 3-space! From our next notes, we will begin discussing about different functions and derivatives.