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Chapter 16: Problem 10

Use Stokes' Theorem to evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r} .\) In each case \(C\) is oriented counterclockwise as viewed from above. $$\begin{array}{l}{\mathbf{F}(x, y, z)=x y \mathbf{i}+2 z \mathbf{j}+3 y \mathbf{k}, \quad C \text { is the curve of intersection }} \\ {\text { of the plane } x+z=5 \text { and the cylinder } x^{2}+y^{2}=9}\end{array}$$

Short Answer

Expert verified
The integral evaluates to \( 18\pi/\sqrt{2} \).

Step by step solution

01

Identify the Vector Field and Surface

Given the vector field \( \mathbf{F}(x, y, z) = x y \mathbf{i} + 2z \mathbf{j} + 3y \mathbf{k} \), we need to identify a surface \( S \) whose boundary is the curve \( C \). The curve \( C \) is formed by the intersection of the plane \( x+z=5 \) and the cylinder \( x^2 + y^2 = 9 \). The surface \( S \) can be taken as the part of the plane bounded by the cylinder.
02

Apply Stokes' Theorem

Stokes' Theorem states that \( \int_{C} \mathbf{F} \cdot d \mathbf{r} = \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} \), where \( abla \times \mathbf{F} \) is the curl of \( \mathbf{F} \) and \( d\mathbf{S} \) is the vector area element on \( S \).
03

Calculate the Curl \( \nabla \times \mathbf{F} \)

The curl \( abla \times \mathbf{F} \) for the vector field \( \mathbf{F}(x, y, z) = x y \mathbf{i} + 2z \mathbf{j} + 3y \mathbf{k} \) is calculated as: \[abla \times \mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \xy & 2z & 3y\end{vmatrix} = \left(-3\right)\mathbf{i} - (0)\mathbf{j} + (-x)\mathbf{k} = -3\mathbf{i} - x\mathbf{k}.\]
04

Parametrize the Surface \( S \)

We use the plane equation \( x+z=5 \) and the cylinder's equation \( x^2 + y^2 = 9 \) to generate the surface. Parametrize \( S \) over \( x \) and \( y \): \( z = 5 - x \) and \( x = 3\cos(\theta), y = 3\sin(\theta) \) where \( \theta \) goes from \( 0 \) to \( 2\pi \).
05

Calculate \( d\mathbf{S} \) and the Surface Integral

From the parametrization, the area vector element \( d\mathbf{S} \) is \( (\mathbf{n}) dS \). For the plane \( x+z=5 \), the unit normal is \( \mathbf{n} = (1, 0, -1)/\sqrt{2} \). Then, \[ \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} = \int_{S} (-3, 0, -x) \cdot \left( (1, 0, -1) \frac{1}{\sqrt{2}} \right) \ dS. \] Calculating the dot product, we get each contribution and evaluate the integral over the bounded surface.
06

Evaluate the Surface Integral

The integral simplifies to \[ \int_{S} \left( -3 \cdot \frac{1}{\sqrt{2}} + x \cdot \frac{1}{\sqrt{2}}\right) dS = \frac{1}{\sqrt{2}} \int_{S} (-3 + 3\cos(\theta)) dS. \] Substituting for \( dS = dxdy \) over the disk, we finally compute the integral \[ = \frac{1}{\sqrt{2}} \left( \int_{r=0}^{3}\int_{\theta=0}^{2\pi} 3(1 - \cos(\theta))\, r dr d\theta \right). \]

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Key Concepts

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

Vector Calculus
Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \( \mathbb{R}^3 \). It involves understanding how vectors change in space, which is crucial in physics and engineering. In this context, vectors can have both magnitude and direction.

The primary operations in vector calculus include:
  • Gradient: Measures how a scalar field changes at every point.
  • Divergence: Indicates how much a vector field spreads out from a point.
  • Curl: Tells us how a vector field rotates around a point.
  • Line Integrals: Add up vector fields along a curve.
  • Surface Integrals: Extend the concept of line integrals over a surface.
In our problem, we use line integrals and the curl operation, which are part of these vector calculus concepts.
Parametrization
Parametrization is a technique that represents a curve or a surface using one or more parameters. This method simplifies complex surfaces or curves into simpler, understandable forms. It is essential in the process of computing integrals on those surfaces or curves.

In our exercise, the surface \( S \) where Stokes' Theorem is applied needs to be parametrized. This involves representing the cylinder and plane intersection as a parameterized surface. For instance:
  • Use the equation of the plane \( x+z=5 \).
  • Employ polar coordinates to describe the cylinder's circular intersection, such as \( x = 3\cos(\theta), \, y = 3\sin(\theta) \).
With these steps, the parameterization fully captures the region of interest, which is crucial for performing integrations on this surface.
Surface Integrals
Surface integrals extend the concept of line integrals to two dimensions, allowing you to integrate over surfaces rather than curves. When performing a surface integral, the aim is to sum a quantity over a 2D surface in space. This is often represented as \( \int_{S} \mathbf{F} \cdot d\mathbf{S} \).

Here's how it ties to our problem:
  • The surface integral involves integrating the curl of the vector field \( abla \times \mathbf{F} \) over a surface \( S \), rather than the curve \( C \).
  • We compute \( d\mathbf{S} \), which includes a normal vector to the surface and its area element.
  • It involves contributions from the field across each segment of the surface.
By calculating these integrals, you incorporate the contributions of the vector field across the defined parameterized surface.
Curl of a Vector Field
The curl of a vector field is a measure of its rotation around a point. It is one of the fundamental concepts in vector calculus represented by \( abla \times \mathbf{F} \). Curl indicates how much and in which direction the field tends to rotate.

To calculate the curl:
  • Use the determinant of a 3 脳 3 matrix involving unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), partial derivative operators \( \partial/\partial x, \partial/\partial y, \partial/\partial z \), and the components of the vector field \( \mathbf{F} \).
  • Plug the specific components of \( \mathbf{F}(x, y, z) = xy\mathbf{i} + 2z\mathbf{j} + 3y\mathbf{k} \) into this matrix to compute each part of the curl.
In our case, the result \( abla \times \mathbf{F} = -3\mathbf{i} - x\mathbf{k} \) helps to set up the surface integral portion of Stokes' Theorem, crucial for solving the problem by calculating the field's effect over a surface.

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Most popular questions from this chapter

(a) Set up, but do not evaluate, a double integral for the area of the surface with parametric equations \(x=a u \cos v\) , \(y=b u \sin v, z=u^{2}, 0 \leqslant u \leqslant 2,0 \leqslant v \leqslant 2 \pi\) (b) Eliminate the parameters to show that the surface is an elliptic paraboloid and set up another double integral for the surface area. (c) Use the parametric equations in part (a) with \(a=2\) and \(b=3\) to graph the surface. (d) For the case \(a=2, b=3,\) use a computer algebra system to find the surface area correct to four decimal places.

A 160 -lb man carries a \(25-1 \mathrm{b}\) can of paint up a helical staircase that encircles a silo with a radius of 20 \(\mathrm{ft}\) . If the silo is 90 \(\mathrm{ft}\) high and the man makes exactly three complete revolutions, how much work is done by the man against gravity in climbing to the top?

Let $$\mathbf{F}(x, y)=\frac{-y \mathbf{i}+x \mathbf{j}}{x^{2}+y^{2}}$$ (a) Show that \(\partial P / \partial y=\partial Q / \partial x\) (b) Show that \(\int_{c} \mathbf{F} \cdot d \mathbf{r}\) is not independent of path.[Hint: Compute \(\int_{c} \mathbf{F} \cdot d \mathbf{r}\) and \(\int_{c} \mathbf{F} \cdot d \mathbf{r},\) where \(C_{1}\) and \(C_{2}\) are the upper and lower halves of the circle \(x^{2}+y^{2}=1\) from \((1,0)\) to \((-1,0) . ]\) Does this contradict Theorem 6\(?\)

Use the Divergence Theorem to evaluate \(\iint_{S} \mathbf{F} \cdot d \mathbf{S},\) where \(\mathbf{F}(x, y, z)=z^{2} x \mathbf{i}+\left(\frac{1}{3} y^{3}+\tan z\right) \mathbf{j}+\left(x^{2}+y^{2}\right) \mathbf{k}\) and \(S\) is the top half of the sphere \(x^{2}+y^{2}+z^{2}=1\) [Hint: Note that \(S\) is not a closed surface. First compute integrals over \(S_{1}\) and \(S_{2},\) where \(S_{1}\) is the disk \(x^{2}+y^{2} \leqslant 1\) oriented downward, and \(S_{2}=S \cup S_{1 .}\)]

The temperature at the point \((x, y, z)\) in a substance with conductivity \(K=6.5\) is \(u(x, y, z)=2 y^{2}+2 z^{2} .\) Find the rate of heat flow inward across the cylindrical surface \(y^{2}+z^{2}=6\) 0\(\leqslant x \leqslant 4 .\)
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