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Chapter 16: Problem 12

Use a computer to graph the parametric surface. Get a printout and indicate on it which grid curves have \(u\) constant and which have \(v\) constant. $$ \begin{array}{l}{x=\cos u, \quad y=\sin u \sin v, \quad z=\cos v} \\ {0 \leqslant u \leqslant 2 \pi, \quad 0 \leqslant v \leqslant 2 \pi}\end{array} $$

Short Answer

Expert verified
Use software to plot the surface and label grid curves where \(u\) and \(v\) are constant.

Step by step solution

01

Define the Parametric Equations

The surface is defined by the parametric equations: \[ x = \cos u, \quad y = \sin u \sin v, \quad z = \cos v \]where \( 0 \leq u \leq 2\pi \) and \( 0 \leq v \leq 2\pi \). These equations describe the coordinates of points on the surface in 3D space.
02

Understand Grid Curves

Grid curves on a parametric surface are lines where one of the parameters \(u\) or \(v\) is held constant while the other varies. Curves with \(u\) constant vary \(v\), and curves with \(v\) constant vary \(u\).
03

Visualize the Surface

Use a computer software like Mathematica, Wolfram Alpha, or Python with Matplotlib to plot the parametric surface using the given equations. Ensure that the software can interpret the parametric equations and boundary conditions.
04

Plotting the Surface

Input the parametric equations into the software. For example, using Python with Matplotlib: ```python import numpy as np import matplotlib.pyplot as plt from mpl_toolkits.mplot3d import Axes3D u = np.linspace(0, 2 * np.pi, 50) v = np.linspace(0, 2 * np.pi, 50) u, v = np.meshgrid(u, v) x = np.cos(u) y = np.sin(u) * np.sin(v) z = np.cos(v) fig = plt.figure() ax = fig.add_subplot(111, projection='3d') ax.plot_surface(x, y, z, rstride=1, cstride=1, color='c', alpha=0.3, edgecolor='none') plt.show() ```
05

Identify Grid Curves on the Printout

Once the surface is plotted, print the graph. Identify which grid curves correspond to \(u\) being constant and which correspond to \(v\) being constant. Label these curves directly on the printout; typically, \(u\) constant curves will show the motion in the \(v\) direction and vice versa.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametric Equations
Parametric equations are a powerful tool in mathematics. They allow us to describe complex surfaces and curves in forms that are easy to manipulate and visualize. For a parametric surface, like the one in the exercise, each point on the surface is described by a combination of equations. These equations produce a map from two parameters, commonly denoted as \( u \) and \( v \), to a point in 3D space \(( x(u, v), y(u, v), z(u, v) )\). Here, the specific equations are \( x = \cos u \), \( y = \sin u \sin v \), and \( z = \cos v \).
By varying \( u \) and \( v \) within their respective ranges, you can sweep out a parametric surface in the 3D space.
  • \( u \) varies from \( 0 \) to \( 2\pi \).
  • \( v \) also varies from \( 0 \) to \( 2\pi \).
These boundaries ensure the surface forms a complete shape, typically a recognizable geometric figure like a torus or a sphere. Understanding these kinds of equations empowers you to design and analyze complex surfaces beyond basic Cartesian-coordinate systems.
Grid Curves
Grid curves are essential features of parametric surfaces that help visualize and understand the surface's structure. On a parametric surface, a grid curve is formed when one of the parameters is kept constant while the other varies. This results in a series of curves that look like a grid laid across the surface.
For the exercise, think of holding \( u \) constant; this transforms the equations into lines that only depend on \( v \), specifically \( y = \sin u \sin v \) and \( z = \cos v \). These curves run along one direction of the surface. Similarly, holding \( v \) constant gives lines depending solely on \( u \), with equations \( x = \cos u \) and \( y = \sin u \sin v' \).
  • Curves with constant \( u \) vary in direction of \( v \).
  • Curves with constant \( v \) vary in direction of \( u \).
Recognizing which is which can help you interpret how the surface behaves or how it might be manipulated in modeling.
3D Plotting
Visualizing parametric surfaces through 3D plotting is crucial to understanding their geometry and behavior. This involves using graphing software or tools like Matplotlib or Wolfram Alpha. These tools render the equations into a 3D space where you can inspect the forms directly.
Using software like Python with Matplotlib, you translate mathematical equations into visual plots. You might input code that defines \( u \) and \( v \) as arrays and computes corresponding \( x \), \( y \), and \( z \) values from the parametric equations.
Once this is done, the software constructs a visual image of the surface, allowing one to see how variables relate visually.
  • Plotting helps verify theoretical deductions about the surface.
  • It also allows identification of key features such as grid curves easily.
This visualization is crucial in applications like computer graphics, CAD modeling, and even in scientific simulations, where precise models of complex surfaces are necessary.

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Most popular questions from this chapter

Show that any vector field of the form $$ \mathbf{F}(x, y, z)=f(y, z) \mathbf{i}+g(x, z) \mathbf{j}+h(x, y) \mathbf{k} $$ is incompressible.

Find a parametric representation for the surface. The part of the sphere \(x^{2}+y^{2}+z^{2}=4\) that lies above the cone \(z=\sqrt{x^{2}+y^{2}}\)

Find the exact value of \(\int_{C} x^{3} y^{2} z d s,\) where \(C\) is the curve with parametric equations \(x=e^{-t} \cos 4 t, y=e^{-t} \sin 4 t,\) \(z=e^{-t}, 0 \leqslant t \leqslant 2 \pi\)

(a) Show that the parametric equations \(x=a \cosh u \cos v\) \(y=b \cosh u \sin v, z=c \sinh u,\) represent a hyperboloid of one sheet. (b) Use the parametric equations in part (a) to graph the hyperboloid for the case \(a=1, b=2, c=3\). (c) Set up, but do not evaluate, a double integral for the surface area of the part of the hyperboloid in part (b) that lies between the planes \(z=-3\) and \(z=3 .\)

Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane. $$ \mathbf{r}(u, v)=\left(1-u^{2}-v^{2}\right) \mathbf{i}-v \mathbf{j}-u \mathbf{k} ; \quad(-1,-1,-1) $$
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