Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Find the work done by the vector field
$F (x, y) = (\cos x - 3 y e^{x}) i + \sin x \sin y j$
in moving an object around the periphery of the rectangle with vertices $(0, 0), (2, 0), (2, π)$ , and $(0, π)$ , starting and ending at $(2, π)$ .

Ans: The required work done is $2 \sin 2 + 3 π (e^{2} - 1)$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information:

$F (x, y) = (\cos x - 3 y e^{x}) i + \sin x \sin y j$
Moving an object around the periphery of the rectangle with vertices $(0, 0), (2, 0), (2, π)$ and $(0, π)$
starting and ending at $(2, π)$

Step 2. Finding the work done by the vector filed:

The work done by a vector field F moving a particle along the curve C is computed by the following line integral:

$\int_{C} F \cdot d r .$

Hence, evaluate the line integral $\int_{C} F \cdot d r$ to evaluate the required work done.

Step 3. Using Green's Theorem to evaluate the required the line integral:

Green's Theorem states that,

"Let R be a region in the plane with smooth boundary curve C oriented counterclockwise by $r (t) = ⟨ (x (t), y (t)) ⟩$ for $a \leq t \leq b$ .

If a vector field $F (x, y) = <F_{1} (x, y), F_{2} (x, y)>$ is defined on R, then,

\int_{C} F \cdot d r = \iint_{R} (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) d A . " \dots \dots (1)

Step 4. Finding the vector:

For the vector field $F (x, y) = (\cos x - 3 y e^{x}) i + \sin x \sin y j$ ,

$F_{1} (x, y) = \cos x - 3 y e^{x}$ and $F_{2} (x, y) = \sin x \sin y$ .

Now, first find $\frac{\partial F_{2}}{\partial x}$ and $\frac{\partial F_{1}}{\partial y}$ .

Then,

$\frac{\partial F_{2}}{\partial x} = \frac{\partial}{\partial x} (\sin x \sin y) = \cos x \sin y$

and,

$\frac{\partial F_{1}}{\partial y} = \frac{\partial}{\partial y} (\cos x - 3 y e^{x}) = - 3 e^{x}$

Step 5. Using Green's Theorem (1) to evaluate the integral:

Now, use Green's Theorem (1) to evaluate the integral $\int_{C} F \cdot d r$ as follows:

$\int_{C} F \cdot d r = \iint_{R} (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) d A = \iint_{R} (\cos x \sin y - (- 3 e^{x})) d A = \iint_{R} (\cos x \sin y + 3 e^{x}) d A \dots \dots . (2)$

Step 6. Finding the region of integration:

Here, the boundary curve C is a rectangle with vertices $(0, 0), (2, 0), (2, π), (0, π)$ ,

starting and ending at $(2, π)$ .

So the region R bounded by this rectangle is shown in the following figure.

Viewed it as an x - or y-simple region, then the region of integration will be,

$R = {(x, y) ∣ 0 \leq x \leq 2, 0 \leq y \leq π}$ .

Step 7. Evaluating the integral (2):

Then, evaluate the integral (2) as follows:

$\int_{C} F \cdot d r = \int_{R} (\cos x \sin y + 3 e^{x}) d A = \int_{0}^{π} \int_{0}^{2} (\cos x \sin y + 3 e^{x}) d x d y = \int_{0}^{π} [\int_{0}^{2} (\cos x \sin y + 3 e^{x}) d x] d y = \int_{0}^{π} {[\sin x \sin y + 3 e^{x}]}_{0}^{2} d y = \int_{0}^{π} [(\sin 2 \sin y + 3 e^{2}) - (\sin 0 \sin y + 3 e^{0})] d y = \int_{0}^{π} [(\sin 2 \sin y + 3 e^{2}) - (0 \cdot \sin y + 3 \cdot 1)] d y = \int_{0}^{π} (\sin 2 \sin y + 3 e^{2} - 3) d y = {[- \sin 2 \cos y + 3 e^{2} y - 3 y]}_{0}^{π} = (- \sin 2 \cdot (- 1) + 3 e^{2} π - 3 π) - (- \sin 2 \cos 0 + 3 e^{2} \cdot 0 - 3 \cdot 0) = (\sin 2 + 3 π (e^{2} - 1)) - (- \sin 2 \cdot 1 + 0) = \sin 2 + 3 π (e^{2} - 1) + \sin 2 = 2 \sin 2 + 3 π (e^{2} - 1)$

Therefore, the required work done is $2 \sin 2 + 3 π (e^{2} - 1)$ .

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Calculus

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Short Answer

Find the work done by the vector field
$F (x, y) = (\cos x - 3 y e^{x}) i + \sin x \sin y j$
in moving an object around the periphery of the rectangle with vertices $(0, 0), (2, 0), (2, π)$ , and $(0, π)$ , starting and ending at $(2, π)$ .

Step by Step Solution

Step 1. Given information:

Step 2. Finding the work done by the vector filed:

Step 3. Using Green's Theorem to evaluate the required the line integral:

Step 4. Finding the vector:

Step 5. Using Green's Theorem (1) to evaluate the integral:

Step 6. Finding the region of integration:

Step 7. Evaluating the integral (2):

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Find the work done by the vector fieldF(x,y)=cosx-3yexi+sinxsinyjin moving an object around the periphery of the rectangle with vertices (0,0),(2,0),(2,π), and (0,π), starting and ending at (2,π).

Step by Step Solution

Step 1. Given information:

Step 2. Finding the work done by the vector filed:

Step 3. Using Green's Theorem to evaluate the required the line integral:

Step 4. Finding the vector:

Step 5. Using Green's Theorem (1) to evaluate the integral:

Step 6. Finding the region of integration:

Step 7. Evaluating the integral (2):

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Find the work done by the vector field
$F (x, y) = (\cos x - 3 y e^{x}) i + \sin x \sin y j$
in moving an object around the periphery of the rectangle with vertices $(0, 0), (2, 0), (2, π)$ , and $(0, π)$ , starting and ending at $(2, π)$ .