   Chapter 16.4, Problem 11E

Chapter
Section
Textbook Problem

Use Green’s Theorem to evaluate ∫C F · dr. (Check the orientation of the curve before applying the theorem.)11. F(x, y) = ⟨y cos x − xy sin x, xy + x cos x⟩, C is the triangle from (0, 0) to (0, 4) to (2, 0) to (0, 0)

To determine

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

Explanation

Given data:

Vector field is F(x,y)=ycosxxysinx,xy+xcosx and curve C is a triangle from (0,0) to (0,4) to (2,0) to (0,0) .

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.

The curve C is piecewise-smooth, and simply closed curve with domain D={0x20y42x 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)=ycosxxysinx,xy+xcosx .

P=ycosxxysinxQ=xy+xcosx

Find the value of Py .

Py=y(ycosxxysinx)=y(ycosx)y(xysinx)=cosx(1)xsinx(1) {t(t)=1}=cosxxsinx

Find the value of Qx .

Qx=x(xy+xcosx)=x(xy)+x(xcosx)=y(1)+(1)cosx+x(sinx) {t(t)=1,t(cost)=sint,t(uv)=uv+uv}=y+cosxxsinx

Re-modify the equation (1)

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