   Chapter 16.4, Problem 18E

Chapter
Section
Textbook Problem

A particle starts at the origin, moves along the x-axis to (5, 0), then along the quarter-circle x2 + y2 = 25, x ⩾ 0, y ⩾ 0 to the point (0, 5), and then down the y-axis back to the origin. Use Green’s Theorem to find the work done on this particle by the force field F(x, y) = (sin x, sin y + xy2 + 1 3 x3⟩.

To determine

To find: The work done by force field using Green’s Theorem.

Explanation

Given data:

Force is F(x,y)=sinx,siny+xy2+13x3 and particle moving from origin along x-axis to (5,0) , then along the quarter circle of x2+y2=25 to point (0,5) and finally down the y-axis to origin.

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 work done by force F(x,y)=P(x,y),Q(x,y) moving over curve C is,

W=CFdr

W=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 typically a quarter-disk, which is piecewise-smooth, and simply closed curve with domain D={0r50θπ2 in polar coordinate system and curve C is positively oriented. Therefore, the Green’s theorem is applicable.

Consider curve parametric equations as x=r , y=0 , and dA=rdrdθ .

Compare the two vector fields F(x,y)=P(x,y),Q(x,y) and F(x,y)=sinx,siny+xy2+13x3 .

P=sinxQ=siny+xy2+13x3

Find the value of Py .

Py=y(sinx)=0 {t(k)=0}

Find the value of Qx

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