You CAN Ace Calculus

### Topics You Need To Understand For This Page

 precalculus parametrics

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### 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.

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 upt 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, 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 [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.

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

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.

Basic Problems

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

Problem Statement

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

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

Problem Statement

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

Solution

### 1877 solution video

video by Dr Chris Tisdell

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

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

Problem Statement

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

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

Problem Statement

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

Solution

### 1878 solution video

video by Dr Chris Tisdell

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

Sphere of fixed radius r, $$x^2+y^2+z^2=r^2$$.

Problem Statement

Sphere of fixed radius r, $$x^2+y^2+z^2=r^2$$.

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

Problem Statement

Sphere of fixed radius r, $$x^2+y^2+z^2=r^2$$.

Solution

### 1879 solution video

video by Dr Chris Tisdell

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

Plane $$2x-4y+3z=16$$.

Problem Statement

Plane $$2x-4y+3z=16$$.

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

Problem Statement

Plane $$2x-4y+3z=16$$.

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.

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

Intermediate Problems

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 $$z=\sqrt{x^2+y^2}$$.

$$x=2\sin(u)\cos(v), y=2\sin(u)\sin(v), z=2\cos(u)$$
$$0 \leq u \leq \pi/4, 0 \leq v \leq 2\pi$$
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 below.

### 1876 solution video

video by Krista King Math

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