Q. 13 Use聽the聽vector聽field聽F(x,聽y... [FREE SOLUTION]

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Chapter 14: Q. 13 (page 1131)

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

Short Answer

Expert verified

$The line integral as double integral is - 鈭� {鈭�}_{R} (sinx siny + x^{2} e^{y}) dA$

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)

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

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91影视

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

Short Answer

Step by step solution

Step 1. Given information is

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

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91影视

UsethevectorfieldF(x,y)=x2eyi+cos(x)sin(y)jandGreen鈥�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

Statistics

Calculus

Theoretical and Mathematical Physics

Decision Maths

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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