3D Coordinates

Polar coordinates for the plane generalize to 3D in two different ways:

• Cylindrical coordinates
• Spherical coordinates

In this overview, we'll meet both of these coordinate systems with a focus on expressing triple integrals

over regions that are "simple" in those coordinate systems

Cylindrical coordinates

Cylindrical coordinates are like polar+$z$: specify a point in the plane via polar coordinates $(r,z)$, then its vertical displacement using $z$.

Cylindrical coords with xy

If you put $x$ and $y$ in that picture, then you can see the relationship between cylindrical and Cartesian.

Translation

Given $f(x,y,z)$, we can derive $F(r,\theta,z)$ using

That is, $F(r,\theta,z) = f(r\cos(\theta),r\sin(\theta), z)$.

Examples

If $f(x,y,z) = x^2 - \sin(yz)$, then $$F(r,\theta,z) = r^2\cos^2(\theta) - \sin(r\sin(\theta)z).$$

Any $x^2+y^2$ can be replaced with $r^2$. For example, if $f(x,y,z)=(4-(x^2+y^2))z$, then $$F(r,\theta,z) = (4-r^2)z.$$

Triple integrals in cylindrical coordinates

We can express a triple integral in cylindrical coordinates as follows:

As with polar coordinates and general change of variables, we need to translate the bounds of integration, the function, and the volume element. Note that the $r$ is part of the volume element $dV$, just as in polar coordinates.

Simple cylindrical regions

The "simplest" regions in cylindrical coordinates have the form

Such a region (where $\alpha=c=0$) is shown above.

A functional bound

Sometimes, we might want to bound the top or bottom with a function. The region above can be described by

Example

Set up $\iiint_R (x^2+y^2 + z) \, dV$ as an iterated integral in cylindrical coordinates, where $R$ is the region on the previous slide.

Solution:

Spherical coordinates

• $\rho$ represents distance to the origin,
• $\varphi$ represents angle from the positive $z$ axis,
• $\theta$ is the polar coordinate $\theta$.

Spherical to Cylindrical

To translate spherical coordinates to cylindrical, we first fix $\theta$ to determine a plane in which we can relate $\rho$ and $\varphi$ to $r$ and $z$.

Spherical to Cylindrical (cont)

Once we look at just that triangle in a plane, it's easy to read off the relationships in terms of the trig functions.

Spherical to Cartesian

We can now use cylindrical coordinates as an intermediate step to translate Cartesian to spherical.

Any $(x^2+y^2+z^2)$s that we see can be replaced by $\rho^2$.

Triple integrals in spherical coordinates

We can express a triple integral in spherical coordinates as follows:

Again, we need to translate the bounds of integration, the function, and the volume element. Now, a $\rho^2\sin(\varphi)$ appears as part of $dV$.

Simple spherical regions

The "simplest" regions in spherical coordinates have the form

Such a region is shown above.

The volume element

So where's $dV = \rho^2\sin(\varphi)\,d\rho\,d\varphi\,d\theta$ come from? Referring to the spherical coordinate transformations, we can write out the Jacobian:

Since it's super long, I'll have WolframAlpha work it out; which should return a result that looks like this.

Example

Let's compute the mass of a sphere $S$ of radius 1 with density function $f(x,y,z)=2-(x^2+y^2+z^2)$; thus, this object is most massive at its center becoming less so as we move to it's edge. The density should be: