?In Exercises 1-6, match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]

In Exercises 1-6, match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] \(\mathbf{r}(u, v)=u \mathbf{i}+v \mathbf{j}+u v \mathbf{k}\) Text Transcription: r(u, v) = ui + vj + uvk

In Exercises 1-6, match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] \(\mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+u \mathbf{k}\) Text Transcription: r(u, v) = u cos vi + u sin vj + uk

In Exercises 1-6, match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] \(\mathbf{r}(u, v)=u \mathbf{i}+\frac{1}{2}(u+v) \mathbf{j}+v \mathbf{k}\) Text Transcription: r(u, v) = ui + 1 / 2 (u + v)j + vk

In Exercises 1-6, match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] \(\mathbf{r}(u, v)=u \mathbf{i}+\frac{1}{4} v^{3} \mathbf{j}+v \mathbf{k}\) Text Transcription: r(u, v) = ui + 1 / 4 v^{3}j + vk

In Exercises 1-6, match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] \mathbf{r}(u, v)=2 \cos v \cos u \mathbf{i}+2 \cos v \sin u \mathbf{j}+2 \sin v \mathbf{k}$ Text Transcription: r(u, v) = 2 cos v cos ui + 2 cos v sin uj + 2 sin vk

In Exercises 1-6, match the vector-valued function with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] \(\mathbf{r}(u, v)=4 \cos u \mathbf{i}+4 \sin u \mathbf{j}+v \mathbf{k}\) Text Transcription: r(u, v) = 4 cos ui + 4 sin uj + vk

In Exercises 7 - 10, find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. Identify the surface and sketch its graph. \(\mathbf{r}(u, v)=u \mathbf{i}+v \mathbf{j}+\frac{v}{2} \mathbf{k}\) Text Transcription: r(u, v) = ui + vj + v / 2 k

In Exercises 7 - 10, find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. Identify the surface and sketch its graph. \(\mathbf{r}(u, v)=2 u \cos v \mathbf{i}+2 u \sin v \mathbf{j}+\frac{1}{2} u^{2} \mathbf{k}\) Text Transcription: r(u, v) = 2u cos vi + 2u sin vj + 1 / 2 u^{2}k

In Exercises 7 - 10, find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. Identify the surface and sketch its graph. \(\mathbf{r}(u, v)=2 \cos u \mathbf{i}+v \mathbf{j}+2 \sin u \mathbf{k}\) Text Transcription: r(u, v) = 2 cos ui + vj + 2 sin uk

In Exercises 7 - 10, find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. Identify the surface and sketch its graph. \(\mathbf{r}(u, v)=3 \cos v \cos u \mathbf{i}+3 \cos v \sin u \mathbf{j}+5 \sin v \mathbf{k}\) Text Transcription: r(u, v) = 3 cos v cos ui + 3 cos v sin uj + 5 sin vk

In Exercises 11 - 16, use a computer algebra system to graph the surface represented by the vector-valued function. \(\mathbf{r}(u, v)=2 u \cos v \mathbf{i}+2 u \sin v \mathbf{j}+u^{4} \mathbf{k}\) \(0 \leq u \leq 1, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = 2u cos vi + 2u sin vj + u^{4}k 0 leq u leq 1, 0 leq v leq 2 pi

In Exercises 11 - 16, use a computer algebra system to graph the surface represented by the vector-valued function. \(\mathbf{r}(u, v)=2 \cos v \cos u \mathbf{i}+4 \cos v \sin u \mathbf{j}+\sin v \mathbf{k}\) \(0 \leq u \leq 2 \pi, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = 2 cos v cos ui + 4 cos v sin uj + sin vk 0 leq u leq 2 pi, 0 leq v leq 2 pi

In Exercises 11 - 16, use a computer algebra system to graph the surface represented by the vector-valued function. \(\mathbf{r}(u, v)=2 \sinh u \cos v \mathbf{i}+\sinh u \sin v \mathbf{j}+\cosh u \mathbf{k}\) \(0 \leq u \leq 2, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = 2 sinh u cos vi + sinh u sin vj + cosh uk 0 leq u leq 2, 0 leq v leq 2 pi

In Exercises 11 - 16, use a computer algebra system to graph the surface represented by the vector-valued function. \(\mathbf{r}(u, v)=2 u \cos v \mathbf{i}+2 u \sin v \mathbf{j}+v \mathbf{k}\) \(0 \leq u \leq 1, \quad 0 \leq v \leq 3 \pi\) Text Transcription: r(u, v) = 2u cos vi + 2u sin vj + vk 0 leq u leq 1, 0 leq v leq 3 pi

In Exercises 11 - 16, use a computer algebra system to graph the surface represented by the vector-valued function. \(\mathbf{r}(u, v)=(u-\sin u) \cos v \mathbf{i}+(1-\cos u) \sin v \mathbf{j}+u \mathbf{k}\) \(0 \leq u \leq \pi, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = (u - sin u) cos vi + (1 - cos u) sin vj + uk 0 leq u leq pi, 0 leq v leq 2 pi

In Exercises 11 - 16, use a computer algebra system to graph the surface represented by the vector-valued function. \(\mathbf{r}(u, v)=\cos ^{3} u \cos v \mathbf{i}+\sin ^{3} u \sin v \mathbf{j}+u \mathbf{k}\) \(0 \leq u \leq \frac{\pi}{2}, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = cos^{3} u cos vi + sin^{3} u sin vj + uk 0 leq u leq pi / 2, 0 leq v leq 2 pi

In Exercises 17 - 20, determine how the graph of the surface s(u, v) differs from the graph of \(r(u, v)= u \cos v \mathrm{i}+u \sin v \mathrm{j}+u^{2} \mathrm{k}\) (see figure), where \(0 \leq u \leq 2\) and \(0 \leq v \leq 2 \pi\). (It is not necessary to graphs.) \(\mathbf{s}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}-u^{2} \mathbf{k}\) \(0 \leq u \leq 2, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = u cos vi + u sin vj + u^{2}k 0 leq u leq 2 0 leq v leq 2 pi s(u, v) = u cos vi + u sin vj - u^{2}k 0 leq u leq 2, 0 leq v leq 2 pi

In Exercises 17 - 20, determine how the graph of the surface s(u, v) differs from the graph of \(r(u, v)= u \cos v \mathrm{i}+u \sin v \mathrm{j}+u^{2} \mathrm{k}\) (see figure), where \(0 \leq u \leq 2\) and \(0 \leq v \leq 2 \pi\). (It is not necessary to graphs.) \(\mathbf{s}(u, v)=u \cos v \mathbf{i}+u^{2} \mathbf{j}+u \sin v \mathbf{k}\) \(0 \leq u \leq 2, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = u cos vi + u sin vj + u^{2}k 0 leq u leq 2 0 leq v leq 2 pi s(u, v) = u cos vi + u^{2}j + u sin vk 0 leq u leq 2, 0 leq v leq 2 pi

In Exercises 17 - 20, determine how the graph of the surface s(u, v) differs from the graph of \(r(u, v)= u \cos v \mathrm{i}+u \sin v \mathrm{j}+u^{2} \mathrm{k}\) (see figure), where \(0 \leq u \leq 2\) and \(0 \leq v \leq 2 \pi\). (It is not necessary to graphs.) \(\mathbf{s}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+u^{2} \mathbf{k}\) \(0 \leq u \leq 3, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = u cos vi + u sin vj + u^{2}k 0 leq u leq 2 0 leq v leq 2 pi s(u, v) = u cos vi + u sin vj + u^{2}k 0 leq u leq 3, 0 leq v leq 2 pi

In Exercises 17 - 20, determine how the graph of the surface s(u, v) differs from the graph of \(r(u, v)= u \cos v \mathrm{i}+u \sin v \mathrm{j}+u^{2} \mathrm{k}\) (see figure), where \(0 \leq u \leq 2\) and \(0 \leq v \leq 2 \pi\). (It is not necessary to graphs.) \(\mathbf{s}(u, v)=4 u \cos v \mathbf{i}+4 u \sin v \mathbf{j}+u^{2} \mathbf{k}\) \(0 \leq u \leq 2, \quad 0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = u cos vi + u sin vj + u^{2}k 0 leq u leq 2 0 leq v leq 2 pi s(u, v) = 4u cos vi + 4u sin vj + u^{2}k 0 leq u leq 2, 0 leq v leq 2 pi

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The plane z = y

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The plane x + y + z = 6

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The cone \(y=\sqrt{4 x^{2}+9 z^{2}}\) Text Transcription: y = sqrt{4x^2 + 9z^2}

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The cone \(x=\sqrt{16 y^{2}+z^{2}}\) Text Transcription: x = sqrt{16y^2 + z^2}

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The cylinder \(x^{2}+y^{2}=25\) Text Transcription: x^2 + y^2 = 25

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The cylinder \(4 x^{2}+y^{2}=16\) Text Transcription: 4x^2 + y^2 = 16

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The cylinder \(z=x^{2}\) Text Transcription: z = x^2

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The ellipsoid \(\frac{x^{2}}{9}+\frac{y^{2}}{4}+\frac{z^{2}}{1}=1\) Text Transcription: x^2 / 9 + y^2 / 4 + z^2 / 1 = 1

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The part of the plane z = 4 that lies inside the cylinder \(x^{2}+y^{2}=9\) Text Transcription: x^2 + y^2 = 9

In Exercises 21 - 30, find a vector-valued function whose graph is the indicated surface. The part of the paraboloid \(z=x^{2}+y^{2}\) that lies inside the cylinder \(x^{2}+y^{2}=9\) Text Transcription: z = x^2 + y^2 x^2 + y^2 = 9

In Exercises 31 - 34, write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. Function Axis of Revolution \(y=\frac{x}{2}, \quad 0 \leq x \leq 6\) x-axis Text Transcription: y = x / 2, 0 leq x leq 6

In Exercises 31 - 34, write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. Function Axis of Revolution \(y=\sqrt{x}, \quad 0 \leq x \leq 4\) x-axis Text Transcription: y = sqrt{x}, 0 leq x leq 4

In Exercises 31 - 34, write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. Function Axis of Revolution \(x=\sin z, \quad 0 \leq z \leq \pi\) z-axis Text Transcription: x = sin z, 0 leq z leq pi

In Exercises 31 - 34, write a set of parametric equations for the surface of revolution obtained by revolving the graph of the function about the given axis. Function Axis of Revolution \(z=y^{2}+1, \quad 0 \leq y \leq 2\) y-axis Text Transcription: z = y^2 + 1, 0 leq y leq 2

In Exercises 35 - 38, find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. \(\mathbf{r}(u, v)=(u+v) \mathbf{i}+(u-v) \mathbf{j}+v \mathbf{k}, \quad(1,-1,1)\) Text Transcription: r(u, v) = (u + v)i + (u - v)j + vk, (1, -1, 1)

In Exercises 35 - 38, find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. \(\mathbf{r}(u, v)=u \mathbf{i}+v \mathbf{j}+\sqrt{u v} \mathbf{k}, \quad(1,1,1)\) Text Transcription: r(u, v) = ui + vj + sqrt{uv}k, (1, 1, 1)

In Exercises 35 - 38, find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. \(\mathbf{r}(u, v)=2 u \cos v \mathbf{i}+3 u \sin v \mathbf{j}+u^{2} \mathbf{k}, \quad(0,6,4)\) Text Transcription: r(u, v) = 2u cos vi + 3u sin vj +u^{2}k, (0, 6, 4)

In Exercises 35 - 38, find an equation of the tangent plane to the surface represented by the vector-valued function at the given point. \(\mathbf{r}(u, v)=2 u \cosh v \mathbf{i}+2 u \sinh v \mathbf{j}+\frac{1}{2} u^{2} \mathbf{k}, \quad(-4,0,2)\) Text Transcription: r(u, v) = 2u cosh vi + 2 u sinh vj + 1 / 2 u^{2}k, (-4, 0, 2)

In Exercises 39 - 46, find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the plane \(\mathbf{r}(u, v)=4 u \mathbf{i}-v \mathbf{j}+v \mathbf{k}\), where \(0 \leq u \leq 2\) and \(0 \leq v \leq 1\) Text Transcription: r(u, v) = 4ui - vj + vk 0 leq u leq 2 0 leq v leq 1

In Exercises 39 - 46, find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the paraboloid \(\mathbf{r}(u, v)=2 u \cos v \mathbf{i}+2 u \sin v \mathbf{j}+ u^{2} \mathbf{k}\), where \(0 \leq u \leq 2\) and \(0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = 2u cos vi + 2u sin vj + u^{2}k 0 leq u leq 2 0 leq v leq 2 pi

In Exercises 39 - 46, find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the cylinder \(\mathbf{r}(u, v)=a \cos u \mathbf{i}+a \sin u \mathbf{j}+v \mathbf{k}\), where \(0 \leq u \leq 2 \pi\) and \(0 \leq v \leq b\) Text Transcription: r(u, v) = a cos ui + a sin uj + vk 0 leq u leq 2 pi 0 leq v leq b

In Exercises 39 - 46, find the area of the surface over the given region. Use a computer algebra system to verify your results. The sphere \(\mathbf{r}(u, v)=a \sin u \cos v \mathbf{i}+a \sin u \sin v \mathbf{j}+a \cos u \mathbf{k}\), where \(0 \leq u \leq \pi\) and \(0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = a sin u cos vi + a sin u sin vj + a cos uk 0 leq u leq pi 0 leq v leq 2 pi

In Exercises 39 - 46, find the area of the surface over the given region. Use a computer algebra system to verify your results. The part of the cone \(\mathbf{r}(u, v)=a u \cos v \mathbf{i}+a u \sin v \mathbf{j}+u \mathbf{k}\), where \(0 \leq u \leq b\) and \(0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = au cos vi + au sin vj + uk 0 leq u leq b 0 leq v leq 2 pi

In Exercises 39 - 46, find the area of the surface over the given region. Use a computer algebra system to verify your results. The torus \(\mathbf{r}(u, v)=(a+b \cos v) \cos u \mathbf{i}+(a+b \cos v) \sin u \mathbf{j}+ b \sin v \mathbf{k}\), where \(a>b, 0 \leq u \leq 2 \pi\), and \(0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = (a + b cos v) cos ui + (a + b cos v) sin uj + b sin vk a > b, 0 leq u leq 2 pi 0 leq v leq 2 pi

In Exercises 39 - 46, find the area of the surface over the given region. Use a computer algebra system to verify your results. The surface of revolution \(\mathbf{r}(u, v)=\sqrt{u} \cos v \mathbf{i}+\sqrt{u} \sin v \mathbf{j}+ u \mathbf{k}\), where \(0 \leq u \leq 4\) and \(0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = sqrt{u} cos vi + sqrt{u} sin vj + uk 0 leq u leq 4 0 leq v leq 2 pi

In Exercises 39 - 46, find the area of the surface over the given region. Use a computer algebra system to verify your results. The surface of revolution \(\mathbf{r}(u, v)=\sin u \cos v \mathbf{i}+u \mathbf{j}+ \sin u \sin v \mathbf{k}\), where \(0 \leq u \leq \pi\) and \(0 \leq v \leq 2 \pi\) Text Transcription: r(u, v) = sin u cos vi + uj + sin u sin vk 0 leq u leq pi 0 leq v leq 2 pi

Parametric Surface Define a parametric surface.

Give the double integral that yields the surface area of a parametric surface over an open region D.

Show that the cone in Example 3 can be represented parametrically by \(\mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+u \mathbf{k}\), where \(0 \leq u\) and \(0 \leq v \leq 2 \pi\). Text Transcription: r(u, v) = u cos vi+u sin vj + uk 0 leq u 0 leq v leq 2 pi

The figures below are graphs of \(\mathbf{r}(u, v)=u \mathbf{i}+\sin u \cos v \mathbf{j}+\sin u \sin v \mathbf{k}\), where \(0 \leq u \leq \pi / 2\) and \(0 \leq v \leq 2 \pi\). Match each of the graphs with the point in space from which the surface is viewed. The points are (10, 0, 0), (-10, 10, 0), (0, 10, 0), and (10, 10, 10). Text Transcription: r(u, v) = ui + sin u cos vj + sin u sin vk 0 leq u leq pi / 2 0 leq v leq 2 pi

An equation of an astroidal sphere in x, y, and z is \(x^{2 / 3}+y^{2 / 3}+z^{2 / 3}=a^{2 / 3}\) . A graph of an astroidal sphere is shown below. Show that this surface can be represented parametrically by \(\mathbf{r}(u, v)=a \sin ^{3} u \cos ^{3} v \mathbf{i}+a \sin ^{3} u \sin ^{3} v \mathbf{j}+a \cos ^{3} u \mathbf{k}\) where \(0 \leq u \leq \pi\) and \(0 \leq v \leq 2 \pi\). Text Transcription: x^{2/3} + y^{2/3} + z^{2/3} = a^{2/3} r(u, v) = a sin^3 u cos^3 vi + a sin^3 u sin^3 vj + a cos^3 uk 0 leq u leq pi 0 leq v leq 2 pi

Use a computer algebra system to graph three views of the graph of the vector-valued function \(\mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+v \mathbf{k}, 0 \leq u \leq \pi, 0 \leq v \leq \pi\) from the points (10, 0, 0), (0, 0, 10), and (10, 10, 10). Text Transcription: r(u, v) = u cos vi + u sin vj + vk, 0 leq u leq pi, 0 leq v leq pi

Use a computer algebra system to graph the torus \(\mathbf{r}(u, v)=(a+b \cos v) \cos u \mathbf{i}+\) \((a+b \cos v) \sin u \mathbf{j}+b \sin v \mathbf{k}\) for each set of values of a and b, where \(0 \leq u \leq 2 \pi\) and \(0 \leq v \leq 2 \pi\). Use the results to describe the effects of a and b on the shape of the torus. (a) a = 4, b = 1 (b) a = 4, b = 2 (c) a = 8, b = 1 (d) a = 8, b = 3 Text Transcription: r(u, v) = (a + b cos v) cos ui + (a + b cos v) sin uj + b sin vk 0 leq u leq 2 pi 0 leq v leq 2 pi

Consider the function in Exercise 14. (a) Sketch a graph of the function where u is held constant at u = 1. Identify the graph. (b) Sketch a graph of the function where v is held constant at \(v=2 \pi / 3\). Identify the graph. (c) Assume that a surface is represented by the vector-valued function \(\mathbf{r}=\mathbf{r}(u, v)\). What generalization can you make about the graph of the function when one of the parameters is held constant? Text Transcription: v = 2 pi/3 r = r(u, v)

The surface of the dome on a new museum is given by \(\mathbf{r}(u, v)=20 \sin u \cos v \mathbf{i}+20 \sin u \sin v \mathbf{j}+20 \cos u \mathbf{k}\) where \(0 \leq u \leq \pi / 3,0 \leq v \leq 2 \pi\), and r is in meters. Find the surface area of the dome. Text Transcription: r(u, v) = 20 sin u cos vi + 20 sin u sin vj + 20 cos uk 0 leq u leq pi/3, 0 leq v leq 2 pi

Find a vector-valued function for the hyperboloid \(x^{2}+y^{2}-z^{2}=1\) and determine the tangent plane at (1, 0, 0). Text Transcription: x^2 + y^2 - z^2 = 1

Graph and find the area of one turn of the spiral ramp \(\mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+2 v \mathbf{k}\) where \(0 \leq u \leq 3\) and \(0 \leq v \leq 2 \pi\). Text Transcription: r(u, v) = u cos vi + u sin vj + 2vk 0 leq u leq 3 0 leq v leq 2 pi

Area Let f be a nonnegative function such that f ‘ is continuous over the interval [a, b]. Let S be the surface of revolution formed by revolving the graph of f, where \(a \leq x \leq b\), about the x-axis. Let x = u, y = f(u) cos v, and z = f(u) sin v, where \(a \leq u \leq b\) and \(0 \leq v \leq 2 \pi\). Then, S is represented parametrically by \(\mathbf{r}(u, v)=u \mathbf{i}+f(u) \cos v \mathbf{j} +f(u) \sin v \mathbf{k}\). Show that the following formulas are equivalent. Surface area \(=2 \pi \int_{a}^{b} f(x) \sqrt{1+\left[f^{\prime}(x)\right]^{2}} d x\) Surface area \(=\int_{D} \int\left\|\mathbf{r}_{u} \times \mathbf{r}_{v}\right\| d A\) Text Transcription: a leq x leq b a leq u leq b 0 leq v leq 2 r(u, v) = ui + f(u) cos vj + f(u) sin vk = 2 pi int_{a}^{b} f(x) sqrt{1 + [f ‘(x)]^2} dx = int_D int || r_u X r_v || dA

The parametric equations x = 3 + sin u[7 - cos (3u - 2v) -2 cos (3u + v)] y = 3 + cos u[7 - cos (3u - 2v) - 2 cos (3u + v)] z = sin (3u - 2v) + 2 sin (3u + v) where \(-\pi \leq u \leq \pi\) and \(-\pi \leq v \leq \pi\), represent the surface shown below. Try to create your own parametric surface using a computer algebra system. Text Transcription: -pi leq u leq pi -pi leq v leq pi

The surface shown in the figure is called a Möbius strip and can be represented by the parametric equations \(x=\left(a+u \cos \frac{v}{2}\right) \cos v, y=\left(a+u \cos \frac{v}{2}\right) \sin v, z=u \sin \frac{v}{2}\) where \(-1 \leq u \leq 1,0 \leq v \leq 2 \pi\), and a = 3. Try to graph other Möbius strips for different values of a using a computer algebra system. Text Transcription: x = (a + u cos v / 2) cos v, y = (a + u cos v / 2) sin v, z = u sin v / 2 -1 leq u leq 1, 0 leq v leq 2 pi