\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\units}[1]{\,\text{#1}} \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Describing Surfaces Using Parametric Equations

17Calculus

Limits

Using Limits


Derivatives

Graphing

Related Rates

Optimization

Other Applications

Integrals

Improper Integrals

Trig Integrals

Length-Area-Volume

Applications - Tools

Infinite Series

Applications

Tools

Parametrics

Conics

Polar Coordinates

Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Calculus Tools

Learning Tools

Articles

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. Here are a couple of common ways to set up the equations.

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, here are the equations.

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 these 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 [32min-48secs]

video by Dr Chris Tisdell

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]

video by Evans Lawrence

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

video by Evans Lawrence

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.

How to Ace Calculus: The Streetwise Guide

Practice

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.

Basic

\(z^2=x^2+y^2\)

Problem Statement

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

Final Answer

\( \vec{\Phi}(u,v)=\langle u\cos v, u\sin v, u \rangle \), \( u \in \mathbb{R}, ~ 0 \leq v \leq 2\pi \)

Problem Statement

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

Solution

Dr Chris Tisdell - 1877 video solution

video by Dr Chris Tisdell

Final Answer

\( \vec{\Phi}(u,v)=\langle u\cos v, u\sin v, u \rangle \), \( u \in \mathbb{R}, ~ 0 \leq v \leq 2\pi \)

Log in to rate this practice problem and to see it's current rating.

\(x^2+y^2=r^2\)

Problem Statement

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

Final Answer

\( \vec{\Phi}(u,v)=\langle r\cos u, r\sin u, v \rangle \), \( 0 \leq u \leq 2\pi, ~ v \in \mathbb{R} \)

Problem Statement

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

Solution

Dr Chris Tisdell - 1878 video solution

video by Dr Chris Tisdell

Final Answer

\( \vec{\Phi}(u,v)=\langle r\cos u, r\sin u, v \rangle \), \( 0 \leq u \leq 2\pi, ~ v \in \mathbb{R} \)

Log in to rate this practice problem and to see it's current rating.

\(x^2+y^2+z^2=r^2\)

Problem Statement

Parameterize the sphere of fixed radius r, \(x^2+y^2+z^2=r^2\). Remember that there are multiple ways to parameterize a surface, so your answer may not be the same as given in the solution.

Final Answer

\( \vec{\Phi}(u,v)=\langle r \sin(u)\cos(v), r\sin(u)\sin(v), r\cos(u) \rangle \), \( 0 \leq u \leq \pi, 0 \leq v \leq 2\pi \)

Problem Statement

Parameterize the sphere of fixed radius r, \(x^2+y^2+z^2=r^2\). Remember that there are multiple ways to parameterize a surface, so your answer may not be the same as given in the solution.

Solution

Dr Chris Tisdell - 1879 video solution

video by Dr Chris Tisdell

Final Answer

\( \vec{\Phi}(u,v)=\langle r \sin(u)\cos(v), r\sin(u)\sin(v), r\cos(u) \rangle \), \( 0 \leq u \leq \pi, 0 \leq v \leq 2\pi \)

Log in to rate this practice problem and to see it's current rating.

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

Problem Statement

Parameterize the plane \( 2x-4y+3z=16 \). Remember that there are multiple ways to parameterize a surface, so your answer may not be the same as given in the solution.

Hint

To get the same answer as we have, let \(u=x\) and \(v=y\).

Problem Statement

Parameterize the plane \( 2x-4y+3z=16 \). Remember that there are multiple ways to parameterize a surface, so your answer may not be the same as given in the solution.

Final Answer

\( \vec{r}(u,v)=\langle u,v, (16-2u+4v)/3 \rangle \), \(u \in \mathbb{R}\), \(v \in \mathbb{R}\)

Problem Statement

Parameterize the plane \( 2x-4y+3z=16 \). Remember that there are multiple ways to parameterize a surface, so your answer may not be the same as given in the solution.

Hint

To get the same answer as we have, let \(u=x\) and \(v=y\).

Solution

For this one, it is easiest to let \(u=x\) and \(v=y\). Then we can substitute into the equation of the plane and solve for z to get the last component. Also, since there is no restriction on the plane. Therefore, x and y can take on all values of the real numbers. We can either specify \(x \in \mathbb{R}\) and \(y \in \mathbb{R}\) or just not say anything about the range of values. In most cases, it is probably better to specify the values so that we communicate that we didn't just forget about the ranges. However, check with your instructor to see what they require.

Final Answer

\( \vec{r}(u,v)=\langle u,v, (16-2u+4v)/3 \rangle \), \(u \in \mathbb{R}\), \(v \in \mathbb{R}\)

Log in to rate this practice problem and to see it's current rating.

Intermediate

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

Problem Statement

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

Final Answer

\(x=2\sin(u)\cos(v), y=2\sin(u)\sin(v), z=2\cos(u) \)
\( 0 \leq u \leq \pi/4, 0 \leq v \leq 2\pi\)
in vector form: \(\vec{r}(u,v)=2\sin(u)\cos(v)\hat{i}+2\sin(u)\sin(v)\hat{j}+2\cos(u)\hat{k}\)

Problem Statement

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

Solution

Although her final answer is correct in this video, it would be better to use the variables u and v instead of \(\phi\) and \(\theta\) in the final form of the parameterized surface, especially if you are going to be doing a surface integral using this parametric surface. So setting \(u=\phi\) and \(v=\theta\) we get the final answer.

Krista King Math - 1876 video solution

video by Krista King Math

Final Answer

\(x=2\sin(u)\cos(v), y=2\sin(u)\sin(v), z=2\cos(u) \)
\( 0 \leq u \leq \pi/4, 0 \leq v \leq 2\pi\)
in vector form: \(\vec{r}(u,v)=2\sin(u)\cos(v)\hat{i}+2\sin(u)\sin(v)\hat{j}+2\cos(u)\hat{k}\)

Log in to rate this practice problem and to see it's current rating.

parametric surfaces 17calculus youtube playlist

Here is a playlist of the videos on this page.

Really UNDERSTAND Calculus

Topics You Need To Understand For This Page

Related Topics and Links

To bookmark this page and practice problems, log in to your account or set up a free account.

Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

memorize to learn

Join Amazon Prime - Watch Thousands of Movies & TV Shows Anytime - Start Free Trial Now

As an Amazon Associate I earn from qualifying purchases.

Practice

next: surface integrals →

Page Sections

next: surface integrals →

Practice 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.

Do NOT follow this link or you will be banned from the site!

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.

Links and banners on this page are affiliate links. We carefully choose only the affiliates that we think will help you learn. Clicking on them and making purchases help you support 17Calculus at no extra charge to you. However, only you can decide what will actually help you learn. So think carefully about what you need and purchase only what you think will help you.

We use cookies on this site to enhance your learning experience.

17calculus

Copyright © 2010-2022 17Calculus, All Rights Reserved     [Privacy Policy]     [Support]     [About]

mathjax.org
Real Time Web Analytics