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

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

### Parametric Representations in Multivariable Calculus

In multivariable calculus, surfaces can often be represented using parametric equations. This means we describe each point on the surface using two parameters. Here are a few examples of such parametric representations:

1. **Parametric Representation 4:**
   \[
   \mathbf{r}(u, v) = \left\langle 2 \sin(v), u, 3 + 2 \cos(v) \right\rangle
   \]
   - This set of parametric equations describes a surface where:
     - The first component is \(2 \sin(v)\),
     - The second component is \(u\),
     - The third component is \(3 + 2 \cos(v)\).
   - The parameter \(u\) typically runs along one direction of the surface, and \(v\) along another.

2. **Parametric Representation 5:**
   \[
   \mathbf{r}(u, v) = \left\langle u \cos(v), u \sin(v), u \right\rangle
   \]
   - This set of parametric equations describes another surface where:
     - The first component is \(u \cos(v)\),
     - The second component is \(u \sin(v)\),
     - The third component is \(u\).
   - This representation might describe a helical or cylindrical surface, depending on the ranges of \(u\) and \(v\).

3. **Parametric Representation 6:**
   \[
   \mathbf{r}(u, v) = \left\langle u \cos(v), u \sin(v), v \right\rangle
   \]
   - This set of parametric equations describes a surface where:
     - The first component is \(u \cos(v)\),
     - The second component is \(u \sin(v)\),
     - The third component is \(v\).
   - This formulation could represent a helicoid, where \(u\) controls the radius from the axis and \(v\) the height along the axis.

Diagrams or graphs for these parametric surfaces would typically visualize how \(u\) and \(v\) map points in \(\mathbb{R}^3\). A common method is plotting these surfaces in a 3D coordinate system to understand their spatial structure.

Understanding and visualizing these parametric equations
Transcribed Image Text:### Parametric Representations in Multivariable Calculus In multivariable calculus, surfaces can often be represented using parametric equations. This means we describe each point on the surface using two parameters. Here are a few examples of such parametric representations: 1. **Parametric Representation 4:** \[ \mathbf{r}(u, v) = \left\langle 2 \sin(v), u, 3 + 2 \cos(v) \right\rangle \] - This set of parametric equations describes a surface where: - The first component is \(2 \sin(v)\), - The second component is \(u\), - The third component is \(3 + 2 \cos(v)\). - The parameter \(u\) typically runs along one direction of the surface, and \(v\) along another. 2. **Parametric Representation 5:** \[ \mathbf{r}(u, v) = \left\langle u \cos(v), u \sin(v), u \right\rangle \] - This set of parametric equations describes another surface where: - The first component is \(u \cos(v)\), - The second component is \(u \sin(v)\), - The third component is \(u\). - This representation might describe a helical or cylindrical surface, depending on the ranges of \(u\) and \(v\). 3. **Parametric Representation 6:** \[ \mathbf{r}(u, v) = \left\langle u \cos(v), u \sin(v), v \right\rangle \] - This set of parametric equations describes a surface where: - The first component is \(u \cos(v)\), - The second component is \(u \sin(v)\), - The third component is \(v\). - This formulation could represent a helicoid, where \(u\) controls the radius from the axis and \(v\) the height along the axis. Diagrams or graphs for these parametric surfaces would typically visualize how \(u\) and \(v\) map points in \(\mathbb{R}^3\). A common method is plotting these surfaces in a 3D coordinate system to understand their spatial structure. Understanding and visualizing these parametric equations
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