/*! 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 Illustrate situations where, by ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 6

Illustrate situations where, by bringing in a surface integral, Stokes's theorem can be used to obtain information about line integrals that would be hard to find directly. Let \(C_{1}\) and \(C_{2}\) be piecewise smooth simple closed curves contained in the cylinder \(x^{2}+y^{2}=1\) that do not intersect one another, both oriented counterclockwise when viewed from high above the \(x y\) -plane looking down, as illustrated in Figure \(10.28 .\) If \(\mathbf{F}\) is the vector field \(\mathbf{F}(x, y, z)=\left(z-2 y^{3}, 2 y z+x^{5}, y^{2}+x\right),\) show that \(\int_{C_{1}} \mathbf{F} \cdot d \mathbf{s}=\int_{C_{2}} \mathbf{F} \cdot d \mathbf{s}\)

Short Answer

Expert verified
By Stokes's theorem, \(\oint_{C_1} \mathbf{F} \cdot d\mathbf{s} = \oint_{C_2} \mathbf{F} \cdot d\mathbf{s}\) as the curl vanishes over the closed surfaces created by joining \(C_1\) and \(C_2\).

Step by step solution

01

Understand Stokes's Theorem

Stokes's Theorem relates a surface integral over a surface \(S\) to a line integral over the boundary curve \(C\) of that surface. The theorem is given by the formula: \[ \oint_{C} \mathbf{F} \cdot d\mathbf{s} = \iint_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \] where \(\mathbf{F}\) is a vector field and \(abla \times \mathbf{F}\) is the curl of \(\mathbf{F}\).
02

Calculate the Curl of \(\mathbf{F}\)

Compute \(abla \times \mathbf{F}\) by taking the curl of the vector field \(\mathbf{F}(x, y, z) = (z-2y^3, 2yz+x^5, y^2+x)\): \[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ z-2y^3 & 2yz+x^5 & y^2+x \end{vmatrix} \]By evaluating the determinant, we find: \[ abla \times \mathbf{F} = (1, 0, -5x^4 - 2) \]
03

Analyze the Symmetry of \(C_1\) and \(C_2\)

Since \(C_1\) and \(C_2\) are both simple closed curves contained within the cylinder \(x^2+y^2=1\), they both lie on the same cylindrical surface which bridges these curves. They have no intersections with each other.
04

Apply Stokes's Theorem to Each Curve

By Stokes's Theorem, for each curve \(C_1\) and \(C_2\): \[ \oint_{C_1} \mathbf{F} \cdot d\mathbf{s} = \iint_{S_1} (abla \times \mathbf{F}) \cdot d\mathbf{S} \] \[ \oint_{C_2} \mathbf{F} \cdot d\mathbf{s} = \iint_{S_2} (abla \times \mathbf{F}) \cdot d\mathbf{S} \] Since \(S_1\) and \(S_2\) are surfaces connected by the same vector field and curl, they result in equivalent integrals. Both surfaces together form a closed boundary over which the curl integrates to zero since the divergence of curl is a fundamental identity in vectors (curl over a closed surface is zero).
05

Conclude Equality of Line Integrals

Since the calculation between the two surfaces \(S_1\) and \(S_2\) results in zero change in the curl over the entire closed surface between \(C_1\) and \(C_2\), it follows that:\[ \int_{C_1} \mathbf{F} \cdot d\mathbf{s} = \int_{C_2} \mathbf{F} \cdot d\mathbf{s} \] Thus, by Stokes's theorem, the line integrals are equal.

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.

Vector Field
A vector field is a function that assigns a vector to every point in space. In three dimensions, this involves a three-component vector ### Examples:
  • For every point \( (x, y, z) \), you might have a vector \( \mathbf{F}(x, y, z) = (F_1(x, y, z), F_2(x, y, z), F_3(x, y, z)) \).
  • The vector field given in our problem is \( \mathbf{F}(x, y, z) = (z-2y^3, 2yz+x^5, y^2+x) \).
Vector fields are used in various applications to represent, for example, gravitational, electric, or magnetic fields. They describe how a certain quantity changes simultaneously in space at different points. This provides us with a tool to model real-world phenomena. In the context of the problem, the vector field \( \mathbf{F} \) describes forces or directions acting in a 3D space 鈥� which Stokes鈥檚 Theorem helps to analyze.
Surface Integral
Surface integrals extend the concept of line integrals to higher dimensions. When working with surfaces rather than curves, surface integrals calculate the flow of a vector field across a surface. ### What is a Surface?
  • A surface \( S \) might be curved or flat, enclosed by boundary curves such as \( C_1 \) and \( C_2 \) in our exercise.
  • In this context, both curves are on a cylindrical surface described by \( x^2 + y^2 = 1 \).
### How does it Work?
  • The surface integral involves computing the "net flow" of a vector field through \( S \), detailed by the formula \( \iint_{S} \mathbf{F} \cdot d\mathbf{S} \), indicating how much of the field penetrates the surface.
  • In Stokes鈥檚 Theorem, this is equated to a line integral around the boundary of the surface.
Surface integrals provide profound insights into how vector fields behave over different spatial regions, emphasizing areas where the field has greater or lesser intensity.
Line Integral
A line integral is a type of integral where a function is evaluated along a curve. This concept is employed when calculating quantities such as work done by a force field along a path. ### How it Works:
  • A line integral computes the accumulation of a vector field \( \mathbf{F} \) as you "move along" a curve \( C \).
  • Mathematically, this is expressed as \( \oint_{C} \mathbf{F} \cdot d\mathbf{s} \), meaning you integrate the dot product of \( \mathbf{F} \) and the differential path direction \( d\mathbf{s} \).
In our problem, the task is to evaluate the line integrals \( \int_{C_1} \mathbf{F} \cdot d\mathbf{s} \) and \( \int_{C_2} \mathbf{F} \cdot d\mathbf{s} \). Using Stokes's Theorem, these line integrals can be translated into surface integrals, making complex calculations more manageable.Thus, line integrals help us understand the effect of a vector field along a specific path, critical for applications like determining work or circulation in vector calculus.
Curl of a Vector Field
The curl of a vector field is a measure of how much the field tends to rotate around a point. It is an essential concept in vector calculus that provides insight into the twisting and rotational behavior of a field. ### Definition:
  • The curl is computed using the formula \( abla \times \mathbf{F} \), where \( \mathbf{F} \) is a vector field.
  • For a given field \( \mathbf{F}(x, y, z) = (z-2y^3, 2yz+x^5, y^2+x) \), it results in \( abla \times \mathbf{F} = (1, 0, -5x^4 - 2) \).
### Application:
  • Curl is used in Stokes's Theorem to relate surface integrals and line integrals.
  • When a vector field \( \mathbf{F} \) has a non-zero curl, it suggests vortex-like behavior or circulation.
Understanding the curl is crucial for applying Stokes鈥檚 Theorem, as it determines how the vector field behaves around the boundary of a surface.

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

Let \(\mathbf{F}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) be a vector field such that \(\mathbf{F}(-\mathbf{x})=-\mathbf{F}(\mathbf{x})\) for all \(\mathbf{x}\) in \(\mathbb{R}^{3},\) and let \(S\) be the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\), oriented by the outward normal. Is it necessarily true that \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}=0 ?\) Explain.

Let \(\mathbf{F}(x, y, z)=\left(x z, y z, z^{2}\right)\) (a) Let \(S\) be the cylinder \(x^{2}+y^{2}=4,1 \leq z \leq 3\), oriented by the outward normal. Find \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) (b) Suppose that you "close up" the cylinder \(S\) in part (a) by adding the disks \(x^{2}+y^{2} \leq 4\), \(z=3,\) and \(x^{2}+y^{2} \leq 4, z=1,\) at the top and bottom, respectively, where the disk at the top is oriented by the upward normal and the one at the bottom is oriented by the downward normal. Let \(S_{1}\) be the resulting surface: cylinder, top, and bottom. Find \(\iint_{S_{1}} \mathbf{F} \cdot d \mathbf{S}\)

Determine whether the given vector field \(\mathbf{F}\) is a curl field. If it is, find a vector field \(\mathbf{G}\) whose curl is \(\mathbf{F}\). If not, explain why not. $$ \mathbf{F}(x, y, z)=(y, z, x) $$

Suppose that a surface \(S\) in \(\mathbb{R}^{3}\) is the graph \(z=f(x, y)\) of a smooth real-valued function \(f: D \rightarrow \mathbb{R}\) whose domain \(D\) is a bounded subset of \(\mathbb{R}^{2}\). Let \(S\) be oriented by the upward normal. Show that: $$ \iint_{S} 1 d x \wedge d y=\operatorname{Area}(D) $$ In other words, the integral is the area of the projection of \(S\) on the \(x y\) -plane. Similar interpretations hold for \(\iint_{S} 1 d y \wedge d z\) and \(\iint_{S} 1 d z \wedge d x\) under analogous hypotheses.

One way to produce a torus is to take the circle \(C\) in the \(x z\) -plane of radius \(b\) and center \((a, 0,0)\) and rotate it about the \(z\) -axis. We assume that \(a>b\). The surface that is swept out is a torus. The circle \(C\) can be parametrized by \(\alpha(\psi)=(a+b \cos \psi, 0, b \sin \psi), 0 \leq \psi \leq 2 \pi\). The distance of \(\alpha(\psi)\) to the \(z\) -axis is \(a+b \cos \psi\), so rotating \(\alpha(\psi)\) counterclockwise by \(\theta\) about the \(z\) -axis brings it to the point \(((a+b \cos \psi) \cos \theta,(a+b \cos \psi) \sin \theta, b \sin \psi)\). This gives the following parametrization of the torus with \(\psi\) and \(\theta\) as parameters, illustrated in Figure 10.29: $$ \sigma(\psi, \theta)=((a+b \cos \psi) \cos \theta,(a+b \cos \psi) \sin \theta, b \sin \psi), \quad 0 \leq \psi \leq 2 \pi, \quad 0 \leq \theta \leq 2 \pi $$ Let \(S\) denote the resulting torus, and assume that it is oriented by the unit normal vector that points towards the exterior of the torus. (a) Show that: $$ \frac{\partial \sigma}{\partial \psi} \times \frac{\partial \sigma}{\partial \theta}=-b(a+b \cos \psi)(\cos \psi \cos \theta, \cos \psi \sin \theta, \sin \psi) $$ (b) Is \(\sigma\) orientation-preserving or orientation-reversing? (c) Find the surface area of the torus. (d) Let \(\mathbf{F}(x, y, z)=(x, y, z)\). Use the parametrization of \(S\) and the definition of the surface integral to find \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\). Then, use your answer and the result of Example 10.16 to find the volume of the solid torus, that is, the three-dimensional region enclosed by the torus. (e) Find the integral of the vector field \(\mathbf{F}(x, y, z)=(0,0,1)\) over \(S\). Use whatever method seems best. What does your answer say about the flow of \(\mathbf{F}\) through \(S ?\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Decision Maths

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.