   Chapter 16.4, Problem 17E

Chapter
Section
Textbook Problem

Use Green’s Theorem to find the work done by the force F(x, y) = x(x + y) i + xy2 j in moving a particle from the origin along the x-axis to (1, 0), then along the line segment to (0, 1), and then back to the origin along the y-axis.

To determine

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

Explanation

Given data:

Force is F(x,y)=x(x+y)i+xy2j and particle moving from origin along x-axis to (1,0) , then line segment to (0,1) and finally return to origin along the y-axis.

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)i+Q(x,y)j 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 triangle, which is piecewise-smooth, and simply closed curve with domain D={0x10y1x and curve C is positively oriented. Therefore, the Green’s theorem is applicable.

Compare the two vector fields F(x,y)=P(x,y)i+Q(x,y)j and F(x,y)=x(x+y)i+xy2j .

P=x(x+y)Q=xy2

Find the value of Py .

Py=y(x(x+y))=xy(x+y)=x(0+1) {t(k)=0,t(t)=1}=x

Find the value of Qx .

Qx=x(xy2)=y2x(x)=y2(1) {t(t)=1}=y2

Re-modify the equation (1)

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