Short Answer

$Use the vector field F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j and Green ’ s Theorem to write the line integral of F (x, y) about the unit circle, traversed counterclockwise, as a double integral . Do not evaluate the integral .$

$The line integral as double integral is - \int \int_{R} (sinx siny + x^{2} e^{y}) dA$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information is

$F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j$

Step 2. Line integral of F(x, y)

$For the vector field, F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j F_{1} (x, y) = x^{2} e^{y} F_{2} (x, y) = \cos (x) \sin (y) Now, first find \frac{\partial F_{2}}{\partial x} and \frac{\partial F_{1}}{\partial y} Then, \frac{\partial F_{2}}{\partial x} = \frac{\partial (\cos x \sin y)}{\partial x} = - sinx siny and, \frac{\partial F_{1}}{\partial y} = \frac{\partial (x^{2} e^{y})}{\partial y} = x^{2} e^{y} Now, using Green' s theorem to evaluate the integral, \int_{c}^{} F . dx = \int \int_{R} (\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}) dA = \int \int_{R} (- sinx siny - x^{2} e^{y}) dA = - \int \int_{R} (sinx siny + x^{2} e^{y}) dA Therefore, the line integral as double integral is - \int \int_{R} (sinx siny + x^{2} e^{y}) dA$

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Calculus

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Short Answer

$Use the vector field F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j and Green ’ s Theorem to write the line integral of F (x, y) about the unit circle, traversed counterclockwise, as a double integral . Do not evaluate the integral .$

Step by Step Solution

Step 1. Given information is

Step 2. Line integral of F(x, y)

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Use the vector field F(x, y) = x2eyi +cos(x)sin(y)j and Green’s Theorem to writethe line integral of F(x, y) about the unit circle, traversed counterclockwise, as adouble integral. Do not evaluate the integral.

Step by Step Solution

Step 1. Given information is

Step 2. Line integral of F(x, y)

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$Use the vector field F (x, y) = x^{2} e^{y} i + \cos (x) \sin (y) j and Green ’ s Theorem to write the line integral of F (x, y) about the unit circle, traversed counterclockwise, as a double integral . Do not evaluate the integral .$