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Chapter 9: Problem 29

Use Stokes' Theorem to evaluate the integral $$ \int_{C}-y^{3} d x+x^{3} d y-z^{3} d z $$ for \(C\) the (positively oriented) curve of intersection between the cylinder \(x^{2}+y^{2}=1\) and the plane \(x+y+z=1\)

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Step by step solution

01

Find the Vector Field

The vector field \(\mathbf{F}\) can be written as: \[\mathbf{F} = -y^{3} \mathbf{i} + x^{3} \mathbf{j} - z^{3} \mathbf{k}\] By weighting the differential elements \(dx\), \(dy\), and \(dz\) by the physical quantities \(-y^3\), \(x^3\), and \(-z^3\) respectively.
02

Curl of the Vector Field

The curl of the field \(\mathbf{F}\) is calculated using \(\nabla \times \mathbf{F}\). The curl results to: \[\nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ -y^{3} & x^{3} & -z^{3} \end{vmatrix} = (3z^2\mathbf{i} - 0\mathbf{j} - 3x^{2}\mathbf{k}) = 3z^{2}\mathbf{i} - 3x^{2}\mathbf{k}\]
03

Application of Stokes' Theorem

Integrate the flux of the curl over a disk \(D\) that lies in the plane \(z = 1 - x - y\), and is enclosed by the given curve \(C\): \[\int_{C} \mathbf{F} \cdot d\mathbf{r} = \iint_D (\nabla \times \mathbf{F}) \cdot d\mathbf{S}\] where the surface \(S\) is positively oriented towards the upward direction \(n = \mathbf{k}\), the disk \(D\) has polar coordinates \(x = r\cos(\theta)\), \(y = r\sin(\theta)\), and \(z = 1 - r\). So our double integral simplifies into: \[\iint_D (\nabla \times \mathbf{F}) \cdot d\mathbf{S} =\iint_D [3r^2(1-r)\mathbf{i} - 3r^2\cos^2(\theta)\mathbf{k} ] \cdot \mathbf{k} \, rdrd\theta = -3 \int^{2\pi}_{0} \int^1_0 r^4 \cos^2(\theta) \, dr d\theta\]
04

Solve the Integral

Since the integrand does not depend on \(r\), we get \[-3 \int^{2\pi}_{0} \cos^2(\theta) d\theta \int^1_0 r^4 \, dr = -3 \cdot \frac{1}{2} \cdot \frac{1}{5} = -3/10 \] Note here that the integral of \(r^4\) dr from 0 to 1 is \(1/5\), and the integral of \(\cos^2(\theta)\) d\theta from 0 to \(2\pi\) is \(\pi\), hence you get the factor of \(\pi/2\).

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Most popular questions from this chapter

For the given vector field, find the divergence and curl of the field. a. \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}\) b. \(\mathbf{F}=\frac{y}{r} \mathbf{i}-\frac{x}{r} \mathbf{j}\), for \(r=\sqrt{x^{2}+y^{2}}\). c. \(\mathbf{F}=x^{2} y \mathbf{i}+z \mathbf{j}+x y z \mathbf{k} .\)

Find the lengths of the following curves: a. \(y(x)=x\) for \(x \in[0,2]\). b. \((x, y, z)=(t, \ln t, 2 \sqrt{2} t)\) for \(1 \leq t \leq 2\). c. \(y(x)=\cosh x, x \in[-2,2]\). (Recall the hanging chain example from classical dynamics.)

Let \(S\) be a closed surface and \(V\) the enclosed volume. Prove Green's first and second identities, respectively. a. \(\int_{S} \phi \nabla \psi \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d V\). b. \(\int_{S}[\phi \nabla \psi-\psi \nabla \phi] \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V\).

The moments of inertia for a system of point masses are given by sums instead of integrals. For example, \(I_{x x}=\sum_{i} m_{i}\left(y_{i}^{2}+z_{i}^{2}\right)\) and \(I_{x y}=\) \(-\sum_{i} m_{i} x_{i} y_{i}\). Find the inertia tensor about the origin for \(m_{1}=2.0 \mathrm{~kg}\) at \((1.0,0,1.0), m_{2}=5.0 \mathrm{~kg}\) at \((1.0,-1.0,0)\), and \(m_{3}=1.0 \mathrm{~kg}\) at \((1.0,1.0,1.0)\) where the coordinate units are in meters.

A particle moves under the force field \(\mathbf{F}=-\nabla V\), where the potential function is given by \(V(x, y, z)=x^{3}+y^{3}-3 x y+5\). Find the equilibrium points of \(\mathbf{F}\) and determine if the equilibria are stable or unstable.
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