Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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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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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