/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 Find the rectangular equation fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 6

Find the rectangular equation for the surface by eliminating the parameters from the vector-valued function. Identify the surface and sketch its graph. $$ \mathbf{r}(u, v)=2 u \cos v \mathbf{i}+2 u \sin v \mathbf{j}+\frac{1}{2} u^{2} \mathbf{k} $$

Short Answer

Expert verified
The rectangular equation for the surface by eliminating the parameters is \( z = \frac{x^{2} + y^{2}}{8} \). The graph of this equation represents an upward-opening paraboloid.

Step by step solution

01

Isolate the parameters

To start, write the vector-valued function in terms of the Cartesian coordinates. This can be done as follows: \[x = 2u \cos(v)\] \[y = 2u \sin(v)\] \[z = \frac{1}{2} u^{2}\]
02

Eliminate the angle \(v\)

After isolating the parameters, next is to eliminate the \(v\) parameter. To do this, square both sides of the equations for x and y. We have: \[x^{2} = 4u^{2} \cos^{2}(v)\] \[y^{2} = 4u^{2} \sin^{2}(v)\] You can add these equations together and reduce. Remember the trigonometric identity that says \(sin^{2}(v) + cos^{2}(v) = 1\). Thus: \[x^{2} + y^{2} = 4u^{2}\] Now, solve for \(u^{2}\): \[u^{2} = \frac{x^{2} + y^{2}}{4}\]
03

Substitute \(u^{2}\) into z

Now, you need to substitute the above expression for \(u^{2}\) into the equation for z. This results in the equation of the surface in rectangular coordinates: \[z = \frac{1}{2}\left(\frac{x^{2} + y^{2}}{4}\right)\] This simplifies to: \[z = \frac{x^{2}+ y^{2}}{8}\] This is the final equation we are looking for.
04

Identify and sketch the surface

By observation, this equation is for a paraboloid that opens upward. To sketch the graph: Sketch a 3D parabola that is centered at the origin which opens upwards. The intersection of the paraboloid with planes parallel to the x-y plane are circles, whereas intersections with planes parallel to either the x-z or y-z planes are parabolas.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Rectangular Equation
In mathematics, converting a vector-valued function to a rectangular equation involves eliminating parameters to express the equation in terms of Cartesian coordinates. This approach simplifies the equation, removing the dependency on parameters such as angles or other variables.
  • The vector function we start with is expressed in terms of two parameters, in this case, \(u\) and \(v\).
  • We aim to express \(x\), \(y\), and \(z\) only in terms of these Cartesian coordinates, getting rid of \(u\) and \(v\).
The process involves:
  • Identifying the expressions for \(x\), \(y\), and \(z\) from the vector function \( \mathbf{r}(u, v) \).
  • Using mathematical identities to eliminate the parameters without losing the essence of the original function.
Through these steps, you can translate the complex vector expression into a more straightforward rectangular format.
Paraboloid Surface
Paraboloids are types of quadric surfaces which can be visualized as curving upwards or downwards like a parabola in 3D.A paraboloid surface can be classified further:
  • Elliptic Paraboloid: It opens upwards and has an elliptic cross-section.
  • Hyperbolic Paraboloid: Shaped like a saddle and has a hyperbolic cross-section.
The rectangular equation \[z = \frac{x^{2} + y^{2}}{8}\]and the given function suggest an elliptic paraboloid.Paraboloids have:
  • A main axis along which the parabola opens鈥攈ere, it opens along the \(z\)-axis.
  • The property that cross-sections parallel to the \(x-y\) plane create circles.
Understanding these surfaces is crucial for graphing them and analyzing their geometric properties.
Parameter Elimination
Parameter elimination is a mathematical technique to transition from parametric equations to a single equation relating the coordinates directly. The objective is to eliminate the parameters \(u\) and \(v\) without altering the surface representation.
  • Start by expressing both \(x\) and \(y\) in terms of \(u\) and \(v\).
  • Use identities like \(\sin^2(v) + \cos^2(v) = 1\) to eliminate \(v\).
  • For instance, squaring and summing\(x = 2u \cos v \) and \(y = 2u \sin v\) leads to:\[x^2 + y^2 = 4u^2\]
  • Finally, substitute back into the \(z\) equation \(z = \frac{1}{2}u^2\), which yields the simplified rectangular equation.
This practice is foundational in converting parametric surfaces into more tangible forms for analysis and visualization.
3D Graph Sketching
Sketching a 3D graph from a mathematical equation involves visualizing how the surface behaves in space.To sketch a 3D graph of a paraboloid like \(z = \frac{x^{2} + y^{2}}{8}\):
  • Recognize the vertex. Here it is at the origin (0, 0, 0).
  • Determine the symmetry, centering along the \(z\)-axis.
  • Consider intersections with planes. The intersections with \(z = \text{constant}\) planes give circles, showing the surface curves outward uniformly.
  • The sections parallel to \(x-z\) or \(y-z\) planes depict parabolic curves.
Visual aids can help grasp how these sections fit into the whole, ensuring an accurate graph sketching. Such a visualization aids in understanding the nature of enclosed and open surfaces encountered in calculus.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Evaluate \(\int_{S} \int f(x, y, z) d S .\) \(f(x, y, z)=\frac{x y}{z}\) \(S: z=x^{2}+y^{2}, \quad 4 \leq x^{2}+y^{2} \leq 16\)

Use Stokes's Theorem to evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). Use a computer algebra system to verify your results. In each case, \(C\) is oriented counterclockwise as viewed from above. \(\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+z^{2} \mathbf{j}-x y z \mathbf{k}\) \(S: z=\sqrt{4-x^{2}-y^{2}}\)

Use a computer algebra system to graph three views of the graph of the vector- valued function \(\mathbf{r}(u, v)=u \cos v \mathbf{i}+u \sin v \mathbf{j}+v \mathbf{k}, \quad 0 \leq u \leq \pi, \quad 0 \leq v \leq \pi\) from the points \((10,0,0),(0,0,10)\), and \((10,10,10)\).

Use a computer algebra system to evaluate \(\int_{S} \int x y d S\). \(S: z=\frac{1}{2} x y, \quad 0 \leq x \leq 4, \quad 0 \leq y \leq 4\)

State Stokes's Theorem.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.