   Chapter 16.4, Problem 13E

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate ∫C F · dr. (Check the orientation of the curve before applying the theorem.)13. F(x, y) = ⟨y − cos y, x sin y⟩, C is the circle (x − 3)2 + (y + 4)2 = 4 oriented clockwise

To determine

To evaluate: The line integral using Green’s Theorem.

Explanation

Given data:

Vector field is F(x,y)=ycosy,xsiny and curve C is a circle (x3)2+(y+4)2=4 oriented clockwise.

Formula used:

Green’s Theorem:

Consider a positively oriented curve C which is piece-wise smooth, simple closed curve in plane with domain D. Then the line integration of vector field F(x,y)=P(x,y),Q(x,y) over curve C is,

CFdr=D(QxPy)dA (1)

Here,

Py is continuous first-order partial derivative of P,

Qx is continuous first-order partial derivative of Q, and

P and Q have continuous partial derivatives.

Write the equation of a circle.

(xa)2+(yb)2=r2

Here,

(a,b) is center, and

Compare the equations (xa)2+(yb)2=r2 and (x3)2+(y+4)2=22 .

(a,b)=(3,4)r=2

The curve C is a disk with center (3,4) and radius 2. Curve C is piecewise-smooth, and simply closed curve with domain D and curve C is in transverse clockwise direction and hence C possess the positive orientation. Therefore, the Green’s theorem is applicable.

Compare the two vector fields F(x,y)=P(x,y),Q(x,y) and F(x,y)=ycosy,xsiny .

P=ycosyQ=xsiny

Find the value of Py .

Py=y(ycosy)=y(y)y(cosy)=1(siny) {t(t)=1,t(cost)=sint}=1+siny

Find the value of Qx

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