/*! 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 7 In Problems 6 through 10, use St... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 15: Problem 7

In Problems 6 through 10, use Stokes' theorem to evaluate $$\oint_{C} \mathbf{F} \cdot \mathbf{T} d s.$$ \(\mathbf{F}=2 z \mathbf{i}+x \mathbf{j}+3 y \mathbf{k} ; \quad C\) is the ellipse in which the plane \(z=x\) meets the cylinder \(x^{2}+y^{2}=4\), oriented counterclockwise as viewed from above.

Short Answer

Expert verified
The value is \(8\pi\).

Step by step solution

01

Understand Stokes' Theorem

Stokes' Theorem relates a surface integral over a surface \(S\) to a line integral around the boundary \(C\) of \(S\). It is given by \( \oint_{C} \mathbf{F} \cdot \mathbf{T} \, ds = \iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \), where \( abla \times \mathbf{F} \) is the curl of \( \mathbf{F} \) and \( \mathbf{n} \) is the unit normal to \(S\).
02

Compute the Curl of F

First, find the curl of \(\mathbf{F}\). Given \( \mathbf{F} = 2z\mathbf{i} + x \mathbf{j} + 3y \mathbf{k} \), the curl \( abla \times \mathbf{F} \) is computed using the determinant:\[abla \times \mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \2z & x & 3y\end{vmatrix}.\]After expansion, \( abla \times \mathbf{F} = (3 - 0)\mathbf{i} - (0 - 2)\mathbf{j} + (1 - 0)\mathbf{k} = 3\mathbf{i} + 2\mathbf{j} + \mathbf{k}.\)
03

Describe the Surface S

The surface \( S \) is part of the plane \( z = x \) within the cylinder \( x^2 + y^2 = 4 \). Thus \( z \) can be expressed as \( z = x \) for the points on the surface. The boundary of this surface is the curve \( C \), which is an ellipse.
04

Determine the Normal Vector for the Surface

The plane \( z = x \) has a normal vector \( \mathbf{n} = \langle 1, 0, -1 \rangle \) because it is oriented with respect to \( z - x = 0 \). However, since our surface is part of this plane within a cylinder, we ensure it's outward oriented with \( \mathbf{n} = \langle 1, 0, -1 \rangle \).
05

Evaluate the Surface Integral

The integral becomes \( \iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \). Substitute \( abla \times \mathbf{F} = \langle 3, 2, 1 \rangle \) and \( \mathbf{n} = \langle 1, 0, -1 \rangle \), so the dot product is:\[\langle 3, 2, 1 \rangle \cdot \langle 1, 0, -1 \rangle = 3\cdot1 + 2\cdot0 + 1\cdot(-1) = 2.\]Now evaluate \( \iint_{S} 2 \, dS \), where the area of \( S \) is the area of the elliptical region \( x^2 + y^2 \leq 4 \). The area of this circle is \( \pi \times 4 = 4\pi \). Hence the integral is \( 2 \times 4\pi = 8\pi \).
06

Final Result

By Stokes鈥� theorem, \( \oint_{C} \mathbf{F} \cdot \mathbf{T} \, ds = 8\pi \). Therefore, the value of the line integral around curve \( C \) is \( 8\pi \).

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.

Line Integral
A line integral helps determine the work done by a force field along a curve. The concept involves tracing out a path and summing contributions from every tiny segment of this path. In mathematical terms, the line integral of a vector field \( \mathbf{F} \) along a curve \( C \) is denoted as \( \oint_{C} \mathbf{F} \cdot \mathbf{T} \, ds \). This requires:
  • Understanding \( \mathbf{F} \), the vector field.
  • \( \mathbf{T} \), the unit tangent vector at each point on the curve.
  • \( ds \), the differential element of the curve, representing a miniature piece of the path.
The line integral sums up how much the vector field is aligned with the curve \( C \) as you traverse it. In the context of Stokes' Theorem, this can be related back to a surface integral, often making calculation more straightforward in complex scenarios.
Curl of a Vector Field
The curl of a vector field is a measure of the field's rotation or swirling behavior at a given point. It can be thought of as how much the field spins around the point. For a vector field \( \mathbf{F} = 2z\mathbf{i} + x \mathbf{j} + 3y \mathbf{k} \), the curl is calculated using a cross product of the del operator \( abla \) and the vector field \( \mathbf{F} \):
  • Visualize \( abla \times \mathbf{F} \) as the 3D rotation of the field.
  • The resulting vector, \( \mathbf{curl} \), represents the axis of rotation and magnitude.
  • The specific steps involve setting up and evaluating a determinant which captures partial derivatives with respect to each coordinate direction.
In our exercise, the curl came out to be \( 3\mathbf{i} + 2\mathbf{j} + \mathbf{k} \), depicting the rotation of field \( \mathbf{F} \) throughout the space of interest.
Surface Integral
A surface integral extends line integrals to two dimensions, integrating over a surface instead of a curve. For vector fields, the surface integral is used to measure the flux - or total flow - across the surface. The expression \( \iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS \) tells you:
  • The curl of the field \( abla \times \mathbf{F} \), an indicator of how the field rotates around the surface.
  • \( \mathbf{n} \), the normal vector, which provides the maximum point of surface for considering the flow direction.
  • \( dS \), an area element of the surface \( S \), comparable to \( ds \) in line integrals.
In the given exercise, after substituting the curl and the normal vector, the surface integral simplifies to \( 8\pi \). It represents the field's cumulative rotational effect over the elliptical surface.
Normal Vector
The normal vector is essential for understanding the orientation of a surface in space. It is a vector perpendicular to the surface at any given point, and it plays a crucial role in surface integrals. Here鈥檚 what to remember about normal vectors:
  • The normal vector \( \mathbf{n} \) determines the direction "outwards" from a surface.
  • It interacts with other vectors through dot products to convey the extent to which fields pierce the surface.
  • In our problem, since the plane is defined by \( z = x \), the normal vector is \( \langle 1, 0, -1 \rangle \), ensuring we regarded the orientation correctly for Stokes' Theorem.
  • The chosen normal vector ensures that our calculations account for the surface's correct side which affects how we analyze the surface integral.
Understanding normal vectors is key, especially when verifying orientation consistency for applications like Stokes' Theorem.

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 the line integral $$\int_{C} \mathbf{F} \cdot d \mathbf{r}=\int_{C} \mathbf{F} \cdot \mathbf{T} d s$$ along the indicated path \(C\). \(\mathbf{F}(x, y, z)=(2 x+3 y) \mathbf{i}+(3 x+2 y) \mathbf{j}+3 z^{2} \mathbf{k}: \quad C\) is the path from \((0.0,0)\) to \((4,2,3)\) that consists of three line segments parallel to the \(x\) -axis, the \(y\) -axis, and the \(z\) -axis, in that order.

Determine whether the vector fields are conservative. Find potential functions for those that are conservative (either by inspection or by using the method of Example 4 ). \(\mathbf{F}(x, y)=\left(\frac{2 x}{y}-\frac{3 y^{2}}{x^{4}}\right) \mathbf{i}+\left(\frac{2 y}{x^{3}}-\frac{x^{2}}{y^{2}}+\frac{1}{\sqrt{y}}\right) \mathbf{j}\)

Evaluate the surface integral \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d S\), where \(\mathbf{n}\) is the upward-pointing unit normal vector to the given surface \(S\). \(\mathbf{F}=2 y \mathbf{i}+3 z \mathbf{k}: \quad S\) is the part of the plane \(z=3 x+2\) that lies within the cylinder \(x^{2}+y^{2}=4\).

Consider an infinitely long vertical straight wire with a uniform positive charge of \(q\) coulombs per meter. Assume by symmetry that the electric field vector \(\mathbf{E}\) is at each point a horizontal radial vector directed away from the wire. Apply Gauss's law in ( 14 ). with \(S\) being a cylinder with the wire as its axis, to show that \(|\mathbf{E}|=q /\left(2 \pi \epsilon_{i} r\right)\). Thus the electric field intensity is inversely proportional to distance \(r\) from the wire.

Let \(f\) and \(g\) be functions with continuous second-order partial derivatives in the region \(R\) bounded by the piecewise smooth simple closed curve \(C\). Apply Green's theorem in vector form to show that $$\oint_{C} f \nabla g \cdot \mathbf{n} d s=\iint_{R}[f \nabla \cdot \nabla g+\nabla f \cdot \nabla g] d A$$ It was this formula rather than Green's theorem itself that appeared in Green's book of 1828 .
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied 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.