Chapter 14: Q. 13 (page 1131)

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

Short Answer

Expert verified

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

Step by step solution

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

Short Answer

Step by step solution

Step 1. Given information is

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

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UsethevectorfieldF(x,y)=x2eyi+cos(x)sin(y)jandGreen’sTheoremtowritethelineintegralofF(x,y)abouttheunitcircle,traversedcounterclockwise,asadoubleintegral.Donotevaluatetheintegral.

Short Answer

Step by step solution

Step 1. Given information is

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

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