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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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F191441f0-66db-457d-b504-dd20fe1485fa%2F1c09cc0d-98e1-476d-96e9-d9d725730bb6%2Fz4tuxkc_processed.png&w=3840&q=75)
Transcribed Image Text:### 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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