   Chapter 16.4, Problem 28E

Chapter
Section
Textbook Problem

Calculate ∫C F · dr, where F(x, y) = ⟨x2 + y, 3x − y2⟩ and C is the positively oriented boundary curve of a region D that has area 6.

To determine

To calculate: The value of CFdr for F(x,y)=x2+y,3xy2 .

Explanation

Given data:

Field vector is F(x,y)=x2+y,3xy2 and area is 6.

Formula used:

Green’s Theorem:

Consider a positively oriented curve C which is piece-wise smooth, simple closed curve in plane with region 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,

A is area,

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.

Compare the two vector fields F(x,y)=P(x,y),Q(x,y) and F(x,y)=x2+y,3xy2 .

P=x2+y (2)

Q=3xy2 (3)

Find the value of Py .

Take partial differentiation for equation (2) with respect to y.

Py=y(x2+y)=0+1 {t(k)=0,t(t)=1}=1

Find the value of Qx

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