Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the parametric surface over the indicated domain. \(\mathbf{r}(u, v)=u \mathbf{i}+3 v \mathbf{j}+\left(4-u^{2}-v^{2}\right) \mathbf{k} ; 0 \leq u \leq 2\), \(0 \leq v \leq 1\)

Step-by-Step Solution

Answer

Plot the parametric surface using the equations \( x = u \), \( y = 3v \), \( z = 4 - u^2 - v^2 \) over \( 0 \leq u \leq 2 \) and \( 0 \leq v \leq 1 \).

Step 1: Understand the Parametric Surface Equation

The given parametric surface is defined by the vector function \( \mathbf{r}(u, v) = u \mathbf{i} + 3v \mathbf{j} + \left(4 - u^2 - v^2\right) \mathbf{k} \). This represents a surface in three-dimensional space, where the parameters \( u \) and \( v \) determine the position on the surface. The domain for \( u \) is from 0 to 2 and for \( v \) is from 0 to 1.

Step 2: Parametrize the Surface

The surface is described in terms of the parameters \( u \) and \( v \). For this surface, the coordinates in the x-direction, y-direction, and z-direction are given by \( x = u \), \( y = 3v \), and \( z = 4 - u^2 - v^2 \) respectively. This means for any point on the surface, its position in space is determined by these equations.

Step 3: Identify the Domain of the Parameters

Check the bounds for \( u \) and \( v \). Here, \( u \) ranges from 0 to 2 and \( v \) ranges from 0 to 1. These limits will define the extent of the surface in the parametric plot.

Step 4: Plot the Surface

Use a plotting software like Python with libraries such as Matplotlib or a graphing calculator that supports 3D plotting. Plot the surface by sweeping \( u \, \) from 0 to 2 and \( v \, \) from 0 to 1. Substituting into the parametric equations, the plot will utilize the equations \( x = u \), \( y = 3v \), and \( z = 4 - u^2 - v^2 \) for the corresponding ranges of \( u \) and \( v \).

Key Concepts

3D plotting

Creating visualizations of mathematical concepts in three-dimensional space can be both fascinating and challenging. 3D plotting involves representing data or equations, like parametric surfaces, in a way that reveals their spatial structure and relationships. Imagine translating the mathematical equation of a parametric surface into an interactive graph you can manipulate and explore. The process takes the abstract and makes it tangible. When you're plotting a surface, you're essentially mapping out each point defined by its coordinates in space, resulting from the vector function.

• With the vector function, define the surface using parameters like variables or constants.
• Tools like Python's Matplotlib or online graphing calculators enable the visualization of these surfaces, offering functions to view the surface from different angles, zoom in, or even rotate it.

Starting with a systematic plan, like identifying parameter domains and using software for rendering, ensures clarity and comprehension in interpreting 3D plots.

vector functions

Vector functions are vital in understanding and representing various mathematically defined surfaces in three dimensions. A vector function in this context assigns a three-dimensional vector to two parameters, commonly denoted as \(u\) and \(v\). The vector function \(\mathbf{r}(u, v) = u \mathbf{i} + 3v \mathbf{j} + (4 - u^2 - v^2) \mathbf{k}\) reveals how each point on a surface corresponds to a specific combination of parameter values. Such a function does a splendid job of encoding complex geometric surfaces in a compact form.

• Each unit vector (\(\mathbf{i}, \mathbf{j}, \mathbf{k}\)) represents a spatial direction: x, y, and z, respectively.
• By altering the parameters \(u\) and \(v\), you determine the exact position of any point on the surface, providing control over the visualization's resolution and detail.

Understanding these vector functions is crucial for mathematicians and engineers who need to model and analyze real-world patterns and phenomena.

domain of parameters

The domain of parameters refers to the allowed set of values that the parameters can take for defining the surface within the vector function. Grasping this concept is key to understanding how far and wide the surface stretches across space. In our given exercise, the domain for \(u\) is from 0 to 2, and \(v\) is from 0 to 1. These boundaries dictate the region over which the surface is plotted.

• The domain defines the stretch of the surface, limiting the plotting area and thereby controlling the practical plot's size.
• Thoughtful determination of parameter constraints allows for efficient computation and visualization, avoiding unnecessary calculations outside the domain.

In practice, analyzing and setting these domains help visualize parametric surfaces accurately without running into computational limits or errors.