/*! 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 20 Sketch the following vector fiel... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 17: Problem 20

Sketch the following vector fields. $$\mathbf{F}=\langle 0,0,1\rangle$$

Short Answer

Expert verified
The sketch of the vector field \(\mathbf{F}=\langle 0,0,1\rangle\) consists of constant vectors pointing strictly in the positive z-direction, all parallel to each other, and independent of the points in the \((x, y, z)\) space.

Step by step solution

01

Understanding vector fields

A vector field is a function that assigns a vector to each point in the domain (usually \(\mathbb{R}^n\)). In this exercise, the vector field is given by \(\mathbf{F}(x, y, z) = \langle 0, 0, 1 \rangle\). This means that at any point \((x, y, z)\), the value of the vector field is \(\langle 0, 0, 1 \rangle\). In other words, the field is constant with respect to \(x\), \(y\), and \(z\).
02

Sketch the vector field

To sketch the vector field \(\mathbf{F}=\langle 0, 0, 1 \rangle\), we must plot the vectors at different points in the three-dimensional space, keeping in mind that the vector will always be constant and pointing strictly in the positive z-direction. 1. Draw the \(x\), \(y\), and \(z\) axes in a three-dimensional coordinate system. 2. At several points in the \((x, y, z)\) space, draw vectors with their tails at the selected points and directed along the positive z-axis. The magnitude of all the vectors should be kept the same since the vector field is constant. 3. Observe that all vectors are parallel to each other and point in the positive z-direction. Verify that the vector field \(\mathbf{F}\) appears like a constant field, independent of the points in space. Now we have successfully sketched the vector field \(\mathbf{F}=\langle 0,0,1\rangle\).

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.

Sketching Vector Fields
Vector fields are graphical representations of a function that assigns a vector to every point in a space. Think of it like a weather map showing wind speed and direction at various points. When sketching vector fields, it's essential to keep in mind that vectors are displayed with both direction and magnitude (length).

In order to sketch a vector field efficiently, follow these steps:
  • Choose a representative set of points in the domain, such as the three-dimensional coordinate system.
  • At each point, draw a vector with the appropriate magnitude and direction as determined by the vector function.
  • Ensure consistency in spacing and scale to accurately depict how the vectors differ across the space.
Sketching helps visualize how a vector field behaves, whether it shows rotational movement, flow in one direction, or chaotic behavior. It is an invaluable tool for understanding complex dynamical systems in fields such as physics and engineering.
Constant Vector Fields
A constant vector field is one where the vectors remain the same throughout the entire domain. This is a simpler form of a vector field where, regardless of the position in space, the magnitude and direction of the vector are constant.

Here's an analogy: Imagine a field with a steady wind blowing from the north at the same speed everywhere. No matter where you stand in this field, the wind's effect on you would be identical. In mathematical terms, if you were mapping this constant vector field, every vector arrow would look the same and point in the same direction.

A constant vector field is visually uniform and can be represented by drawing evenly spaced, parallel vectors with the same length. In physics, such fields are sometimes idealizations used in theoretical models, as they can simplify complex calculations and provide a clear basic understanding of a force field's influence on its surroundings.
Three-dimensional Coordinate System
A three-dimensional coordinate system helps us describe the position of points in space and is essential for representing vector fields in three dimensions. It's comprised of three mutually perpendicular axes, commonly labeled as the x-axis, y-axis, and z-axis.

When plotting a vector field in this system, each vector's tail starts at a point defined by an (x, y, z) coordinate. Meanwhile, the vector's length and direction extend from the point according to the field's function. By using a three-dimensional coordinate system, it's possible to visualize vector fields in the context of physical space, aiding in the understanding of how forces like gravity or electromagnetic fields interact with objects within a given volume.

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

Prove the following identities. Assume \(\varphi\) is a differentiable scalar- valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G})$$

Stokes' Theorem on closed surfaces Prove that if \(\mathbf{F}\) satisfies the conditions of Stokes' Theorem, then \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S=0\) where \(S\) is a smooth surface that encloses a region.

Find the exact points on the circle \(x^{2}+y^{2}=2\) at which the field \(\mathbf{F}=\langle f, g\rangle=\left\langle x^{2}, y\right\rangle\) switches from pointing inward to pointing outward on the circle, or vice versa.

Fourier's Law of heat transfer (or heat conduction ) states that the heat flow vector \(\mathbf{F}\) at a point is proportional to the negative gradient of the temperature; that is, \(\mathbf{F}=-k \nabla T,\) which means that heat energy flows from hot regions to cold regions. The constant \(k>0\) is called the conductivity, which has metric units of \(J /(m-s-K)\) A temperature function for a region \(D\) is given. Find the net outward heat flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=-k \iint_{S} \nabla T \cdot \mathbf{n} d S\) across the boundary S of \(D\) In some cases, it may be easier to use the Divergence Theorem and evaluate a triple integral. Assume \(k=1 .\) $$\begin{aligned} &T(x, y, z)=100+x+2 y+z\\\ &D=\\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\\} \end{aligned}$$

Miscellaneous integral identities Prove the following identities. a. \(\iiint_{D} \nabla \times \mathbf{F} d V=\iint_{S}(\mathbf{n} \times \mathbf{F}) d S\) (Hint: Apply the Divergence Theorem to each component of the identity.) b. \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\) (Hint: Apply Stokes 'Theorem to each component of the identity.)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Applied Mathematics

Read Explanation

Logic and Functions

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.