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Chapter 14: Problem 22

Find the work done by the force field \(\mathbf{F}\) in mov. ing a particle along the curve \(C\). \(\mathbf{F}(x, y, z)=(2 x-y) \mathbf{i}+2 z \mathbf{j}+(y-z) \mathbf{k} ; C\) is the line segment from (0,0,0) to (1,1,1)

Short Answer

Expert verified
The work done is \( \frac{3}{2} \).

Step by step solution

01

Parameterize the Curve

The curve \( C \) from (0,0,0) to (1,1,1) can be parameterized by the vector function \( \mathbf{r}(t) = t\mathbf{i} + t\mathbf{j} + t\mathbf{k} \), where \( 0 \leq t \leq 1 \). This effectively describes the line segment.
02

Compute the Derivative of the Parameterization

The derivative of \( \mathbf{r}(t) \) with respect to \( t \) is \( \mathbf{r}'(t) = \mathbf{i} + \mathbf{j} + \mathbf{k} \). This represents the direction vector of the path of the particle.
03

Calculate the Force Function on the Parameterized Path

Substitute \( x = t \), \( y = t \), and \( z = t \) into \( \mathbf{F}(x, y, z) \) to get \( \mathbf{F}(t, t, t) = (2t - t)\mathbf{i} + 2t\mathbf{j} + (t - t)\mathbf{k} = t\mathbf{i} + 2t\mathbf{j} \).
04

Compute the Dot Product

Find the dot product of \( \mathbf{F}(t, t, t) \) and \( \mathbf{r}'(t) \). This is \( (t\mathbf{i} + 2t\mathbf{j}) \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k}) = t + 2t = 3t \).
05

Integrate the Dot Product

Compute the work done by integrating the dot product from \( t=0 \) to \( t=1 \): \[ \int_{0}^{1} 3t \, dt \].
06

Evaluate the Integral

Evaluate the integral: \[ \int_{0}^{1} 3t \, dt = \left[ \frac{3t^2}{2} \right]_{0}^{1} \]. Substituting the limits, you get \( \frac{3 \times 1^2}{2} - \frac{3 \times 0^2}{2} = \frac{3}{2} \).

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Key Concepts

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

Parameterization of Curves
In vector calculus, parameterization of curves is a technique used to describe curves using a vector-valued function of a single parameter, usually denoted as \( t \). This approach is particularly useful in calculating line integrals and solving various problems involving curves.
  • A curve parameterized over an interval \( [a, b] \) means that as \( t \) varies from \( a \) to \( b \), the vector function traces out the entire curve.
  • Parameterization helps simplify the description of complex geometries by converting them into a set of equations that describe each dimension.
  • The curve \( C \) in our exercise is parameterized by \( \mathbf{r}(t) = t\mathbf{i} + t\mathbf{j} + t\mathbf{k} \) for \( 0 \leq t \leq 1 \). This means each point on the line segment from (0,0,0) to (1,1,1) is represented through variations of \( t \).
Parameterization is a powerful tool for transitioning from a geometric description of a curve to an algebraic one, facilitating the application of calculus techniques.
Line Integrals
Line integrals extend the concept of integrating along a path or a curve, rather than over an interval like regular integrals. In the context of vector fields, line integrals are used to calculate quantities like work done by a force field on an object moving along a curve.
  • The line integral of a vector field along a curve can be thought of as the sum of the vector field's influence along the curve path.
  • It involves the dot product of the force field \( \mathbf{F} \) and the differential path element \( d\mathbf{r} \).
  • In terms of parameterization, if \( \mathbf{r}(t) \) represents the curve from \( t=a \) to \( t=b \), the integral is typically written as \( \int_{a}^{b} \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt \).
For the given exercise, we calculated the line integral by integrating \( 3t \) from \( t=0 \) to \( t=1 \), resulting in the work done by the force field.
Dot Product
The dot product, also known as the scalar product, is an operation that takes two equal-length sequences of numbers and returns a single number. It is a fundamental concept in vector calculus and is central to the computation of line integrals.
  • The dot product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \) is calculated as \( \mathbf{A} \cdot \mathbf{B} = A_1B_1 + A_2B_2 + A_3B_3 \)
  • It provides a measure of the vectors' alignment or the extent to which two vectors point in the same direction.
  • In physical contexts, like calculating the work done by a force, the dot product helps relate force directly to movement along a path.
In our specific example, we calculated the dot product \( (t\mathbf{i} + 2t\mathbf{j}) \cdot (\mathbf{i} + \mathbf{j} + \mathbf{k}) = 3t \). This result is then used within the line integral.
Force Fields
Force fields represent distributions of forces in space and are a major application of vector calculus, especially when dealing with concepts such as line integrals and work.
  • A force field \( \mathbf{F}(x, y, z) \) assigns a vector to every point in space, indicating the force exerted at that point.
  • The key application in the context of line integrals is calculating the work done by the force field along a path or curve.
  • The configuration of the force field is defined mathematically, such as \( \mathbf{F}(x, y, z)=(2x-y) \mathbf{i}+2z \mathbf{j}+(y-z) \mathbf{k} \) in our problem, which describes a force varying with position.
Force fields are essential in understanding physical phenomena where forces are distributed in a space, and they help in quantifying influences exerted by these forces along known paths.

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Most popular questions from this chapter

Christy plans to paint both sides of a fence whose base is in the \(x y\) -plane with shape \(x=30 \cos ^{3} t, y=30 \sin ^{3} t\), \(0 \leq t \leq \pi / 2,\) and whose height at \((x, y)\) is \(1+\frac{1}{3} y,\) all measured in feet. Sketch a picture of the fence and decide how much paint she will need if a gallon covers 200 square feet.

Area of a Polygon Let \(V_{0}\left(x_{0}, y_{0}\right), V_{1}\left(x_{1}, y_{1}\right), \ldots\) \(V_{n}\left(x_{n}, y_{n}\right),\) be the vertices of a simple polygon \(P,\) labeled counterclockwise and with \(V_{0}=V_{n}\). Show each of the following: (a) \(\int_{C} x d y=\frac{1}{2}\left(x_{1}+x_{0}\right)\left(y_{1}+y_{0}\right),\) where \(C\) is the edge \(V_{0} V_{1}\) (b) Area \((P)=\sum_{i=1}^{n} \frac{x_{i}+x_{i-1}}{2}\left(y_{i}-y_{i-1}\right)\) (c) The area of a polygon with vertices having integral coordinates is always a multiple of \(\frac{1}{2}\). (d) The formula in part (b) gives the correct answer for the polygon with vertices \((2,0),(2,-2),(6,-2),(6,0),(10,4),\) and (-2,4)

In Problems \(7-12\), find \(\nabla f\). $$ f(x, y, z)=\sin (x y z) $$

In Problems \(1-6,\) sketch a sample of vectors for the given vector field F. $$ \mathbf{F}(x, y)=x \mathbf{i}-y \mathbf{j} $$

Plot the parametric surface over the indicated domain. \(\mathbf{r}(u, v)=u \mathbf{i}+3 v \mathbf{j}+\left(4-u^{2}-v^{2}\right) \mathbf{k} ; 0 \leq u \leq 2\), \(0 \leq v \leq 1\).
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