You CAN Ace Calculus
external links you may find helpful |
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Single Variable Calculus |
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Multi-Variable Calculus |
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Acceleration Vector |
Arc Length (Vector Functions) |
Arc Length Function |
Arc Length Parameter |
Conservative Vector Fields |
Cross Product |
Curl |
Curvature |
Cylindrical Coordinates |
Lagrange Multipliers |
Line Integrals |
Partial Derivatives |
Partial Integrals |
Path Integrals |
Potential Functions |
Principal Unit Normal Vector |
Differential Equations |
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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.
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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 |
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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 |
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\(x=\rho \sin \phi \cos \theta\) |
\(y=\rho \sin \phi \sin \theta\) |
\(z=\rho \cos \phi\) |
\(0 \leq \phi \leq \pi\) \(0 \leq \theta \leq 2\pi\) |
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 |
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\(x=3 \sin(u) \cos(v)\) |
\(y=3 \sin(u) \sin(v)\) |
\(z=3 \cos(u)\) |
\(0 \leq u \leq \pi\) \(0 \leq v \leq 2\pi\) |
\(\vec{r}(u,v)=3\sin(u)\cos(v)\hat{i} + 3\sin(u)\sin(v)\hat{j} + 3\cos(u)\hat{k}\) |
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.
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.
video by Evans Lawrence
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.
surface integrals → |
Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems |
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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. |
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 |
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Find a set of parametric equations for the surface of the cone \(z^2=x^2+y^2\).
Problem Statement |
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Find a set of parametric equations for the surface of the cone \(z^2=x^2+y^2\).
Final Answer |
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\( \vec{\Phi}(u,v)=\langle u\cos v, u\sin v, u \rangle \), \( u \in \mathbb{R}, ~ 0 \leq v \leq 2\pi \) |
Problem Statement |
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Find a set of parametric equations for the surface of the cone \(z^2=x^2+y^2\).
Solution |
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video by Dr Chris Tisdell
Final Answer |
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\( \vec{\Phi}(u,v)=\langle u\cos v, u\sin v, u \rangle \), \( u \in \mathbb{R}, ~ 0 \leq v \leq 2\pi \) |
close solution |
Find a set of parametric equations for the surface of the cylinder \(x^2+y^2=r^2\).
Problem Statement |
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Find a set of parametric equations for the surface of the cylinder \(x^2+y^2=r^2\).
Final Answer |
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\( \vec{\Phi}(u,v)=\langle r\cos u, r\sin u, v \rangle \), \( 0 \leq u \leq 2\pi, ~ v \in \mathbb{R} \) |
Problem Statement |
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Find a set of parametric equations for the surface of the cylinder \(x^2+y^2=r^2\).
Solution |
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video by Dr Chris Tisdell
Final Answer |
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\( \vec{\Phi}(u,v)=\langle r\cos u, r\sin u, v \rangle \), \( 0 \leq u \leq 2\pi, ~ v \in \mathbb{R} \) |
close solution |
Sphere of fixed radius r, \(x^2+y^2+z^2=r^2\).
Problem Statement |
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Sphere of fixed radius r, \(x^2+y^2+z^2=r^2\).
Final Answer |
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\( \vec{\Phi}(u,v)=\langle r \sin(u)\cos(v), r\sin(u)\sin(v), r\cos(u) \rangle \), \( 0 \leq u \leq \pi, 0 \leq v \leq 2\pi \) |
Problem Statement |
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Sphere of fixed radius r, \(x^2+y^2+z^2=r^2\).
Solution |
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video by Dr Chris Tisdell
Final Answer |
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\( \vec{\Phi}(u,v)=\langle r \sin(u)\cos(v), r\sin(u)\sin(v), r\cos(u) \rangle \), \( 0 \leq u \leq \pi, 0 \leq v \leq 2\pi \) |
close solution |
Plane \( 2x-4y+3z=16 \).
Problem Statement |
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Plane \( 2x-4y+3z=16 \).
Final Answer |
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\( \vec{r}(u,v)=\langle u,v, (16-2u+4v)/3 \rangle \), \(u \in \mathbb{R}\), \(v \in \mathbb{R}\) |
Problem Statement |
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Plane \( 2x-4y+3z=16 \).
Solution |
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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 |
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\( \vec{r}(u,v)=\langle u,v, (16-2u+4v)/3 \rangle \), \(u \in \mathbb{R}\), \(v \in \mathbb{R}\) |
close solution |
Intermediate Problems |
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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 |
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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 |
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\(x=2\sin(u)\cos(v), y=2\sin(u)\sin(v), z=2\cos(u) \) |
Problem Statement |
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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 |
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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.
video by Krista King Math
Final Answer |
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\(x=2\sin(u)\cos(v), y=2\sin(u)\sin(v), z=2\cos(u) \) |
close solution |