/*! 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 1 Explain the meaning of the integ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 1

Explain the meaning of the integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) in Stokes' Theorem.

Short Answer

Expert verified
Answer: In the context of Stokes' Theorem, the integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) represents the circulation of the vector field \(\mathbf{F}\) around a closed curve \(C\), quantifying the twisting or rotational aspects of the vector field. According to Stokes' Theorem, this circulation is equal to the surface integral of the curl of the vector field over the surface bounded by the curve.

Step by step solution

01

Define the line integral

The line integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) is the integral of the dot product of a vector field \(\mathbf{F}\) and a differential displacement vector \(d \mathbf{r}\) along a closed curve \(C\). It represents the work done by the vector field \(\mathbf{F}\) along the curve \(C\).
02

Introduce Stokes' Theorem

Stokes' Theorem relates the line integral of a vector field \(\mathbf{F}\) around a closed curve \(C\) to the surface integral of the curl of the vector field over the smooth surface \(S\) bounded by the curve \(C\). Mathematically, the theorem is expressed as: $$\oint_{C} \mathbf{F} \cdot d \mathbf{r} = \iint_{S} \nabla \times \mathbf{F} \cdot d \mathbf{S}$$
03

Explain the meaning of the line integral in Stokes' Theorem

In the context of Stokes' Theorem, the line integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) represents the circulation of the vector field \(\mathbf{F}\) around the closed curve \(C\). Circulation can be understood as a measure of how much the vector field twists around the curve. If the circulation is zero, it means that the vector field is conservative (i.e., it has no rotational components) and the work done by the vector field around the curve is zero.
04

Relate the line integral to the surface integral

Stokes' Theorem states that the circulation of a vector field \(\mathbf{F}\) around a closed curve \(C\) is equal to the surface integral of the curl of the vector field over the smooth surface \(S\) bounded by the curve \(C\). This relation implies that, in regions where the curl of the vector field is significant, there is more likely to be a significant circulation around the curve \(C\). In other words, the line integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) quantifies the swirling or rotational properties of the vector field around the curve. In summary, the integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) in Stokes' Theorem represents the circulation of the vector field \(\mathbf{F}\) around a closed curve \(C\), quantifying the twisting or rotational aspects of the vector field. According to Stokes' Theorem, this circulation is equal to the surface integral of the curl of the vector field over the surface bounded by the curve.

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!

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

Find a vector field \(\mathbf{F}\) with the given curl. In each case, is the vector field you found unique? $$\operatorname{curl} \mathbf{F}=\langle 0,1,0\rangle$$

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

The potential function for the force field due to a charge \(q\) at the origin is \(\varphi=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{|\mathbf{r}|},\) where \(\mathbf{r}=\langle x, y, z\rangle\) is the position vector of a point in the field and \(\varepsilon_{0}\) is the permittivity of free space. a. Compute the force field \(\mathbf{F}=-\nabla \varphi\). b. Show that the field is irrotational; that is \(\nabla \times \mathbf{F}=\mathbf{0}\).

Prove the following properties of the divergence and curl. Assume \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields and \(c\) is a real number. a. \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\) b. \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\) c. \(\nabla \cdot(c \mathbf{F})=c(\nabla \cdot \mathbf{F})\) d. \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\)

The gravitational force between two point masses \(M\) and \(m\) is $$ \mathbf{F}=G M m \frac{\mathbf{r}}{|\mathbf{r}|^{3}}=G M m \frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}} $$ where \(G\) is the gravitational constant. a. Verify that this force field is conservative on any region excluding the origin. b. Find a potential function \(\varphi\) for this force field such that \(\mathbf{F}=-\nabla \varphi\) c. Suppose the object with mass \(m\) is moved from a point \(A\) to a point \(B,\) where \(A\) is a distance \(r_{1}\) from \(M\) and \(B\) is a distance \(r_{2}\) from \(M .\) Show that the work done in moving the object is $$ G M m\left(\frac{1}{r_{2}}-\frac{1}{r_{1}}\right) $$ d. Does the work depend on the path between \(A\) and \(B\) ? Explain.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.