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Chapter 14: Q 56. (page 1108)

Find the work done by the given vector field $F$ along specified curve $C$ .
$F (x, y) = x e^{y} i + 2 y j$ where $C$ is work $y = \ln x$ from $(e, 1) to (e^{2}, 2)$

Short Answer

Expert verified

Required work done is ${鈭�}_{C} F (x, y) 路 d r = \frac{1}{3} (e^{6} - e^{3}) + 3 .$

Step by step solution

01

Given Information

It is given that $F (x, y) = x e^{y} i + 2 y j$

We need to find work done along curve $C$ $y = \ln x from (e, 1) to (e^{2}, 2)$

We need to evaluate ${鈭�}_{C} F (x, y) 路 d r$

02

Parametrization of Curve

The given curve is parametrized as

$r (t) = <e^{t}, t> for 1 鈮� t 鈮� 2$

The derivative is $r^{'} (t) = \frac{d}{d t} r (t)$

$= <\frac{d}{d t} e^{t}, \frac{d}{d t} t>$

$= <e^{t}, 1> .$

03

Rewriting vector field

Vector field becomes

$F (x, y) = y i - x j with r (t) = <e^{t}, t>$

$F (x (t), y (t)) = e^{t} 路 e^{t} i + 2 t j$

$= e^{2 t} i + 2 t j$

$= <e^{2 t}, 2 t>$

04

Simplification

Line integral will be:

${鈭�}_{C} F (x, y) 路 d r$

$= {鈭�}_{C} F (x (t), y (t)) 路 r^{'} (t) d t$

Putting values

$= {鈭�}_{1}^{2} <e^{2 t}, 2 t> 路 <e^{t}, 1> d t$

Apply dot product

$= {鈭�}_{1}^{2} [e^{2 t} 路 e^{t} + 2 t 路 1] d t$

$= {鈭�}_{1}^{2} [e^{3 t} + 2 t] d t$

$= {[\frac{e^{3 t}}{3} + t^{2}]}_{1}^{2}$

$= (\frac{e^{3 (2)}}{3} + 2^{2}) - (\frac{e^{3 (1)}}{3} + 1^{2})$

$= (\frac{e^{6}}{3} + 4) - (\frac{e^{3}}{3} + 1)$

$= \frac{e^{6}}{3} + 4 - \frac{e^{3}}{3} - 1$

$= \frac{1}{3} (e^{6} - e^{3}) + 3$

Hence, required work done is

${鈭�}_{C} F (x, y) 路 d r = \frac{1}{3} (e^{6} - e^{3}) + 3$

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Most popular questions from this chapter

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Two different surfaces with the same area. (Try to make these very different, not just shifted copies of each other.)

(b) Let S be the surface parametrized by $r (u, z) = x (u, z) i + y (u, v) j + z (u, v) k .$

Give two different unit normal vectors to S at the point $r (u_{0}, v_{0}) .$

(c) A smooth surface that can be smoothly parametrized as $r (x, z) = <x, f (x), z> .$

Find

$\begin{aligned} {鈭�}_{S} 鈥� F (x, y, z) 鈰� n d S if \\ F (x, y, z) = 2 x z i + 2 y z j 鈭� 18 k \end{aligned}$

and S is the portion of the hyperboloid $x^{2} + y^{2} - 9 = z^{2}$ that lies between the planes

z = 鈭�4 and z = 0, with n pointing outwards.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The result of integrating a vector field over a surface is a vector.

(b) True or False: The result of integrating a function over a surface is a scalar.

(c) True or False: For a region R in the $x y - p l a n e, d S = d A .$

(d) True or False: In computing ${鈭�}_{S} f (x, y, z) d S$ , the direction of the normal vector is irrelevant.

(e) True or False: If f (x, y, z) is defined on an open region containing a smooth surface S, then ${鈭�}_{S} f (x, y, z) d S$ measures the flow through S in the positive z direction determined by f (x, y, z).

(f) True or False: If F(x, y, z) is defined on an open region containing a smooth surface S , then ${鈭�}_{S} F (x, y, z) . n d S$ measures the flow through S in the direction of n determined by the field F(x, y, z).

(g) True or False: In computing ${鈭�}_{S} F (x, y, z) . n d S$ ,the direction of the normal vector is irrelevant.

(h) True or False: In computing ${鈭�}_{S} F (x, y, z) . n d S$ ,with n pointing in the correct direction, we could use a scalar multiple of n, since the length will cancel in the $d S$ term.

$C o m p u t e t h e c u r l o f t h e v e c t o r f i e l d s i n E x e r c i s e s 23 鈥� 28 . G (x, y, z) = (x^{2} + y 鈭� z) i 鈭� 2 y \cos zj + e^{x^{2} + y^{2} + z^{2}} k$

Give a formula for a normal vector to the surface S determined by y = g(x,z), where g(x,z) is a function with continuous partial derivatives.

See all solutions

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