By changing our co-ordinate system using parametric surfaces we can represent other types of shapes more easily. Recall that if we want to represent a circle in cartesian co-ordinates we would have to use an implicit function, such as:
Such functions are very inconvenient to work with, because if we were to represent the function in terms of or it would have square roots in it, which can have either a positive or negative value.
Instead, it is much easier to represent circular geometry using a polar co-ordinates in terms of (the radius) and (the angle between 1 and ). Then we can can use a the following change of co-ordinates to express our circle:
Or the other way around:
So, converting from cartesian co-ordinates to polar co-ordinates, our circle is:
Representing our co-ordinate system in this way has important advantages. Since there are no square roots, it is a lot easier to take the integral of a circle and find its area:
This should look familiar!
We can now extend this to three dimensions to define a cylinder using cylindrical co-ordinates. Cylindrical co-ordinates are effective a cartesian extension of polar co-ordinates, so there is a radius, and a co-ordinate.
The translation from cylindrical to cartesian co-ordinates is as follows:
And from cartesian to cylindrical we have:
For instance, a cylinder with height 1, and radius 2 is defined as:
If we want to find the volume of a cylinder, we must use a triple integral as the input space has three parameters.
Note also that we need to apply a density correction which is the determinant of the Jacobian matrix for each component. In the case of a cylinder, that is , as we have:
Finding the surface area of a cylinder requires us to find a parameterization in terms of two parameters. Because this cylinder has a constant radius, we can use a parameterization that keeps constant.
From there we can use the same approach to find the length of the normal vector at every point.
So from those points, we can find the surface area.