   Chapter 16.3, Problem 11E

Chapter
Section
Textbook Problem

The figure shows the vector field F(x, y) = ⟨2xy, x2⟩ and three curves that start at (1, 2) and end at (3, 2).(a) Explain why ∫C F · dr has the same value for all three curves.(b) What is this common value?

(a)

To determine

To explain: The reason for CFdr has the same value of three curves.

Explanation

Given data:

Vector field is F(x,y)=2xy,x2 and limits are (1,2) and (3,2) .

Formula used:

Consider a vector field as F(x,y)=P(x,y),Q(x,y) . The condition for vector field F being a conservative field is,

Py=Qx (1)

Here,

Py is continuous first-order partial derivative of P, and

Qx is continuous first-order partial derivative of Q.

Compare the vector field F(x,y)=2xy,x2 with F(x,y)=P(x,y),Q(x,y) .

P=2xy (2)

Q=x2 (3)

Apply partial differentiation with respect to y on both sides of equation (2).

Py=y(2xy)=2xyy=2x(1) {t(t)=1}=2x

Apply partial differentiation with respect to x on both sides of equation (3)

(b)

To determine

The common value.

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