Draw the parameterized surface. (Decide on a reasonable domain for u and v.) ### Parametric Equations for SurfacesIn the study of vector calculus and surface parametrization, we often use parametric equations to describe surfaces in three-dimensional space. Here are three examples of such parametric equations:1. **First Parametric Equation:**\[ \mathbf{r}(u, v) = \langle u, v, u^2 + v^2 \rangle \] This equation defines a surface where the coordinates $(u, v)$ map to a three-dimensional point. The $z$-coordinate is given by the sum of the squares of $u$ and $v$.2. **Second Parametric Equation:**\[ \mathbf{r}(u, v) = \langle u^2 + v^2, u, v \rangle \] In this case, the surface is described such that the $x$-coordinate is the sum of the squares of $u$ and $v$, while the $u$ and $v$ coordinates directly map to the $y$ and $z$-coordinates, respectively.3. **Third Parametric Equation:**\[ \mathbf{r}(u, v) = \langle \cos(u), \sin(u), v \rangle \] This surface involves trigonometric functions, where the $x$- and $y$-coordinates are defined by cosine and sine of $u$, respectively, creating a circular or helical pattern in the $xy$-plane, depending on the variation of $v$-coordinate.These parametrizations give insight into how different types of surfaces can be described mathematically in a three-dimensional space, each providing unique geometric properties and visual representations.

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Answered: Draw the parameterized surface

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Calculus

Draw the parameterized surface

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Draw the parameterized surface. (Decide on a reasonable domain for u and v.)

### Parametric Equations for Surfaces

In the study of vector calculus and surface parametrization, we often use parametric equations to describe surfaces in three-dimensional space. Here are three examples of such parametric equations:

1. **First Parametric Equation:**
\[ \mathbf{r}(u, v) = \langle u, v, u^2 + v^2 \rangle \]

This equation defines a surface where the coordinates $(u, v)$ map to a three-dimensional point. The $z$-coordinate is given by the sum of the squares of $u$ and $v$.

2. **Second Parametric Equation:**
\[ \mathbf{r}(u, v) = \langle u^2 + v^2, u, v \rangle \]

In this case, the surface is described such that the $x$-coordinate is the sum of the squares of $u$ and $v$, while the $u$ and $v$ coordinates directly map to the $y$ and $z$-coordinates, respectively.

3. **Third Parametric Equation:**
\[ \mathbf{r}(u, v) = \langle \cos(u), \sin(u), v \rangle \]

This surface involves trigonometric functions, where the $x$- and $y$-coordinates are defined by cosine and sine of $u$, respectively, creating a circular or helical pattern in the $xy$-plane, depending on the variation of $v$-coordinate.

These parametrizations give insight into how different types of surfaces can be described mathematically in a three-dimensional space, each providing unique geometric properties and visual representations.

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