/*! 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 28 A vector force field \(\mathbf{F... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 28

A vector force field \(\mathbf{F}\) is defined in Cartesian coordinates by $$ \mathbf{F}=F_{0}\left[\left(\frac{y^{3}}{3 a^{3}}+\frac{y}{a} e^{x y / a^{2}}+1\right) \mathbf{i}+\left(\frac{x y^{2}}{a^{3}}+\frac{x+y}{a} e^{x y / a^{2}}\right) \mathbf{j}+\frac{z}{a} e^{x y / a^{2}} \mathbf{k}\right] $$ Use Stokes' theorem to calculate $$ \oint_{L} \mathbf{F} \cdot d \mathbf{r} $$ where \(L\) is the perimeter of the rectangle \(A B C D\) given by \(A=(0,1,0), B=(1,1,0)\), \(C=(1,3,0)\) and \(D=(0,3,0)\)

Short Answer

Expert verified
By Stokes' theorem, the line integral \( \oint_{L} \mathbf{F} \cdot d \mathbf{r} = -2 F_{0} \).

Step by step solution

01

Understand the Problem

The goal is to apply Stokes' theorem to calculate the line integral \( \oint_{L} \mathbf{F} \cdot d \mathbf{r} \) around the given rectangle in the plane \( z=0 \).
02

State Stokes' Theorem

Stokes' theorem for a vector field \( \mathbf{F} \) and surface \( S \) with boundary \( L \) is \( \oint_{L} \mathbf{F} \cdot d \mathbf{r} = \iint_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \). Here, \( L \) is the boundary of the rectangle \( A, B, C, D \), and \( S \) is the surface in the \( z = 0 \) plane.
03

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

The curl of \( \mathbf{F} \) is \( abla \times \mathbf{F} = \left( \frac{\partial F_{k}}{\partial y} - \frac{\partial F_{j}}{\partial z}\right) \mathbf{i} + \left( \frac{\partial F_{i}}{\partial z} - \frac{\partial F_{k}}{\partial x} \right) \mathbf{j} + \left( \frac{\partial F_{j}}{\partial x} - \frac{\partial F_{i}}{\partial y} \right) \mathbf{k} \), where \( \mathbf{F} = F_{i} \mathbf{i} + F_{j} \mathbf{j} + F_{k} \mathbf{k} \). Plugging in the components of \( \mathbf{F} \), we get: \( F_{i} = F_{0} \left( \frac{y^{3}}{3 a^{3}} + \frac{y}{a} e^{xy / a^2} + 1 \right) \), \( F_{j} = F_{0} \left( \frac{x y^{2}}{a^{3}} + \frac{x + y}{a} e^{xy / a^{2}} \right) \), \( F_{k} = F_{0} \left( \frac{z}{a} e^{xy / a^{2}} \right) \). Since \( F_{i} \) and \( F_{j} \) do not depend on \( z \), \( \partial F_{i} / \partial z = \partial F_{j} / \partial z = 0 \). Additionally, \( F_{k} = 0 \) when \( z = 0 \), so no contributions from its derivatives. Thus, the curl simplifies to \( abla \times \mathbf{F} = \mathbf{k} \left( \frac{\partial F_{j}}{\partial x} - \frac{\partial F_{i}}{\partial y} \right) \). Calculating partial derivatives: \( \frac{\partial F_{j}}{\partial x} = F_{0} \left( \frac{y^{2}}{a^{3}} + \frac{(y + x) y}{a^3} e^{xy / a^{2}} \right) \), \( \frac{\partial F_{i}}{\partial y} = F_{0} \left( \frac{y^{2}}{a^{2}} + (x y / a^{3}) e^{xy / a^{2}} + \frac{1}{a} e^{xy / a^{2}} \right) \).
04

Simplify the Curl

Simplify the expression: \( \partial F_{j}/\partial x - \partial F_{i}/\partial y \). Notice many terms cancel out: \( F_{0} (\frac{y^{2}}{a^{2}} + \frac{yx}{a^{3}} e^{xy / a^{2}}) - F_{0} (\frac{y^{2}}{a^{3}} + \frac{y^{2}}{a^{3}} e^{xy / a^{2}} + \frac{1}{a} e^{xy / a^{2}}) = - F_{0} \).
05

Integral Over the Surface

On the surface \( S \), where \( z = 0 \), the area element \( d\mathbf{S} = dx dy \mathbf{k} \). Thus, \( \iint_{S} abla \times \mathbf{F} \cdot d\mathbf{S} = -F_{0} \iint_{S} dx dy = -F_{0} \times (1 \times 2) = -2 F_{0} \).
06

Apply Stokes' Theorem

Using Stokes' theorem, \( \oint_{L} \mathbf{F} \cdot d \mathbf{r} = \iint_{S} (abla \times \mathbf{F}) \cdot d \mathbf{S} = -2 F_{0} \).

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Key Concepts

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

Vector Force Field
A vector force field is a function that assigns a vector to every point in a space. In the context of this problem, the force field \(\textbf{F}\) is given by the expression: \[ \mathbf{F}=F_{0}\left[\left(\frac{y^{3}}{3 a^{3}}+\frac{y}{a} e^{x y / a^{2}}+1\right) \mathbf{i}+\left(\frac{x y^{2}}{a^{3}}+\frac{x+y}{a} e^{x y / a^{2}}\right) \mathbf{j}+\frac{z}{a} e^{x y / a^{2}} \mathbf{k}\right] \].
The components of this vector field are:
  • \(F_i = F_{0} \left( \frac{y^{3}}{3 a^{3}}+\frac{y}{a} e^{x y / a^{2}} + 1\right)\)
  • \(F_j = F_{0} \left( \frac{x y^{2}}{a^{3}}+\frac{x+y}{a} e^{x y / a^{2}}\right)\)
  • \(F_k = F_{0} \left( \frac{z}{a} e^{x y / a^{2}}\right) \).
Understanding the force field's structure helps in solving the problem using Stokes' theorem. Notice how the components involve different combinations of the variables x, y, and z. Here, \(\textbf{i}\), \(\textbf{j}\), and \(\textbf{k}\) are the unit vectors along the x, y, and z axes, respectively.
Curl of a Vector Field
The curl of a vector field represents the rotational or swirling behavior of the field at a given point. For a vector field \(\textbf{F} = F_i \textbf{i} + F_j \textbf{j} + F_k \textbf{k}\), the curl is mathematically defined as:
\[ \abla \times \mathbf{F} = \left( \frac{ \partial F_k }{ \partial y} - \frac{ \partial F_j }{ \partial z}\right) \mathbf{i} + \left( \frac{ \partial F_i }{ \partial z} - \frac{ \partial F_k }{ \partial x} \right) \mathbf{j} + \left( \frac{ \partial F_j }{ \partial x} - \frac{ \partial F_i }{ \partial y} \right) \mathbf{k} \].
In this problem, we are specifically interested in the curl of \(\textbf{F}\) in the plane \(z = 0 \) (which means \(F_k = 0\)). This simplifies our calculation, and we only need to find the expression:
\[abla \times \mathbf{F} = \mathbf{k}\left( \frac{ \partial F_j }{ \partial x} - \frac{ \partial F_i }{ \partial y} \right)\].
When you compute partial derivatives and simplify, many terms cancel out, notably simplifying the expression to:
  • \(abla \times \mathbf{F})_k = -F_{0}\).
This simplification helps in the application of Stokes' theorem.
Line Integral
A line integral, particularly in the context of vector fields, measures the field's component along a specified path. In mathematical terms, it is denoted by: \[ \oint_{L} \mathbf{F} \cdot d\mathbf{r} \], where
  • \( \mathbf{F} \) is the vector field
  • \( L \) is the path or closed curve
  • \( d\mathbf{r} \) is a vector representing an infinitesimal line segment of the path.
Stokes' theorem relates the line integral around a closed curve to a surface integral over the surface bounded by the curve. Mathematically, Stokes' theorem states:
\[ \oint_{L} \mathbf{F}\cdot d\mathbf{r} = \iint_{S} (abla \times \mathbf{F})\cdot d\mathbf{S} \], where:
  • The left-hand side represents the line integral.
  • The right-hand side represents the surface integral of the curl of \( \mathbf{F}\times d\textbf{S}\) over the surface \( S \).
For our problem, the surface \( S \) is the rectangle \( ABCD\) in the \( z=0 \) plane with area element \( d\mathbf{S} = dx \ dy \mathbf{k}. \) Calculating this gives:
\[ \iint_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} = -F_{0}\int dx\ dy = -2 F_{0} \] Using Stokes' theorem, \[ \oint_{L} \mathbf{F} \cdot d\mathbf{r} = -2 F_{0}. \] This clearly relates the concept of line integrals to surface integrals and the curl of a vector field.

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Most popular questions from this chapter

A vector field \(\mathbf{F}\) is defined in cylindrical polar coordinates \(\rho, \theta, z\) by $$ \mathbf{F}=F_{0}\left(\frac{x \cos \lambda z}{a} \mathbf{i}+\frac{y \cos \lambda z}{a} \mathbf{j}+(\sin \lambda z) \mathbf{k}\right) \equiv \frac{\rho}{a}(\cos \lambda z) \mathbf{e}_{\rho}+(\sin \lambda z) \mathbf{k} $$ where \(\mathbf{i}, \mathbf{j}\) and \(\mathbf{k}\) are the unit vectors along the Cartesian axes and \(\mathrm{e}_{\rho}\) is the unit vector \((x / \rho) \mathbf{i}+(y / \rho) \mathbf{j}\) (a) Calculate, as a surface integral, the flux of \(\mathbf{F}\) through the closed surface bounded by the cylinders \(\rho=a\) and \(\rho=2 a\) and the planes \(z=\pm a \pi / 2\) (b) Evaluate the same integral using the divergence theorem.

\(\mathbf{F}\) is a vector field \(x y^{2} \mathbf{i}+2 \mathbf{j}+x \mathbf{k}\), and \(L\) is a path parameterised by \(x=c t, y=c / t\), \(z=d\) for the range \(1 \leq t \leq 2\). Evaluate (a) \(\int_{L} \mathbf{F} d t\), (b) \(\int_{L} \mathbf{F} d y\) and (c) \(\int_{L} \mathbf{F} \cdot d \mathbf{r}\).

A vector field \(\mathbf{a}\) is given by \(-z x r^{-3} \mathbf{i}-z y r^{-3} \mathbf{j}+\left(x^{2}+y^{2}\right) r^{-3} \mathbf{k}\), where \(r^{2}=x^{2}+y^{2}+z^{2}\). Establish that the field is conservative (a) by showing \(\nabla \times a=0\) and (b) by constructing its potential function \(\phi .\)

Obtain an expression for the value \(\phi_{P}\) at a point \(P\) of a scalar function \(\phi\) that satisfies \(\nabla^{2} \phi=0\) in terms of its value and normal derivative on a surface \(S\) that encloses it, by proceeding as follows. (a) In Green's second theorem take \(\psi\) at any particular point \(Q\) as \(1 / r\), where \(r\) is the distance of \(Q\) from \(P\). Show that \(\nabla^{2} \varphi=0\) except at \(r=0\) (b) Apply the result to the doubly connected region bounded by \(S\) and a small sphere \(\Sigma\) of radius \(\delta\) centred on \(\mathrm{P}\). (c) Apply the divergence theorem to show that the surface integral over \(\Sigma\) involving \(1 / \delta\) vanishes, and prove that the term involving \(1 / \delta^{2}\) has the value \(4 \pi \phi_{P}\) (d) Conclude that $$ \phi p=-\frac{1}{4 \pi} \int_{S} \phi \frac{\partial}{\partial n}\left(\frac{1}{r}\right) d S+\frac{1}{4 \pi} \int_{S} \frac{1}{r} \frac{\partial \phi}{\partial n} d S $$ This important result shows that the value at a point \(P\) of a function \(\phi\) that satisfies \(\nabla^{2} \phi=0\) everywhere within a closed surface \(S\) that encloses \(P\) may be expressed entirely in terms of its value and normal derivative on \(S .\) This matter is taken up more generally in connection with Green's functions in chapter 19 and in connection with functions of a complex variable in section \(20.12\)

Evaluate the surface integral \(\int \mathbf{r} \cdot d \mathbf{S}\), where \(\mathbf{r}\) is the position vector, over that part of the surface \(z=a^{2}-x^{2}-y^{2}\) for which \(z \geq 0\), by each of the following methods: (a) parameterize the surface as \(x=a \sin \theta \cos \phi, y=a \sin \theta \sin \phi, z=a^{2} \cos ^{2} \theta\), and show that $$ \mathbf{r} \cdot d \mathbf{S}=a^{4}\left(2 \sin ^{3} \theta \cos \theta+\cos ^{3} \theta \sin \theta\right) d \theta d \phi $$ (b) apply the divergence theorem to the volume bounded by the surface and the plane \(z=0\)
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