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17calculus > parametrics > parametric surfaces

Parametric Surfaces

Building parametric equations of surfaces can appear to be confusing. There is an art to it but the basic techniques to get started are fairly straightforward. When you first learned parametrics, you probably used t as your parametric variable. When describing surfaces with parametric equations, we need to use two variables. We will use u and v, which is common in vector calculus.

Techniques

Something to keep in mind when building parametric representations is that there are infinite ways to describe any surface.

1. If the description of the surface is in the form \(z=g(x,y)\), the easiest parametric equations are
\(u=x\), \(v=y\) and \(z=g(u,v)\). In vector form, \(\vec{r}(u,v)=u\hat{i}+v\hat{j}+g(u,v)\hat{k}\).

2. A second basic technique is to describe your surface in cylindrical or spherical coordinates and then set u and v to the appropriate variable. For example, let's say we have a spherical shell of radius 3 that we want to describe in parametric form.
In general, conversion from rectangular to spherical coordinates, the equations are

rectangular → spherical coordinates

\(x=\rho \sin \phi \cos \theta\)

\(y=\rho \sin \phi \sin \theta\)

\(z=\rho \cos \phi\)

\(0 \leq \phi \leq \pi\)       \(0 \leq \theta \leq 2\pi\)

Since our example has a radius of 3, \(\rho = 3\) in the above equations. We can set \(u=\phi\) and \(v=\theta\) giving us the parametric equations

a parametric description of a spherical shell of radius 3

\(x=3 \sin(u) \cos(v)\)

\(y=3 \sin(u) \sin(v)\)

\(z=3 \cos(u)\)

\(0 \leq u \leq \pi\)       \(0 \leq v \leq 2\pi\)

\(\vec{r}(u,v)=3\sin(u)\cos(v)\hat{i} + 3\sin(u)\sin(v)\hat{j} + 3\cos(u)\hat{k}\)

It is important to use the variables u and v for the parametric equations, not because u and v have any significance themselves but because they need to be completely different than the spherical coordinates variables. When you work with surface integrals, you will see why.

Okay, time for a video. Here is a gentle introduction to the idea of parameterizing curves in space.

Dr Chris Tisdell - Parametrised surfaces

Here are a couple of video clips discussing how to set up parametric surfaces. He shows some nice special cases which give you more of a feel for how to set them up.

Evans Lawrence - Lecture 31 - Parametric Surfaces, Surface Integrals [22min-10secs]

Evans Lawrence - Lecture 32 - More on Parametric Surfaces, Surface Integrals [8min-36secs]


Now you are ready to learn how to set up and evaluate surface integrals. However, try your hand at some practice problems first, to make sure you really understand and are able to set up various types of parameterized surfaces.

surface integrals →

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Practice Problems

Instructions - - Unless otherwise instructed, parameterize these surfaces. Remember that there are multiple ways to parameterize a surface, so your answer may not be the same as given in the solution.

Level A - Basic

Practice A01

Find a set of parametric equations for the surface of the cone \(z^2=x^2+y^2\).

answer

solution

Practice A02

Find a set of parametric equations for the surface of the cylinder \(x^2+y^2=r^2\).

answer

solution

Practice A03

Sphere of fixed radius r, \(x^2+y^2+z^2=r^2\).

answer

solution

Practice A04

Plane \( 2x-4y+3z=16 \).

answer

solution


Level B - Intermediate

Practice B01

Find a parametric representation of the part of the sphere \(x^2+y^2+z^2=4\) that lies above the cone \(z=\sqrt{x^2+y^2}\).

answer

solution

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