   Chapter 16.3, Problem 20E

Chapter
Section
Textbook Problem

Show that the line integral is independent of path and evaluate the integral.20. ∫C sin y dx + (x cos y − sin y) dy,C is any path from (2, 0) to (1, π)

To determine

To show: The line integral Csinydx+(xcosysiny)dy is independent of path and value of integral Csinydx+(xcosysiny)dy .

Explanation

Given data:

Line integral is Csinydx+(xcosysiny)dy .

Line integral function Csinydx+(xcosysiny)dy and curve C is path from (2,0) to (1,π) .

Formula used:

Consider a line integral as CF(x,y)=CP(x,y)dx+Q(x,y)dy . 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,

Consider vector function r(t) , atb with a smooth curve C. Consider f is a differentiable function two or three variables of gradient function f and is continuous on curve C. Then,

Cfdr=f(r(b))f(r(a)) (4)

Compare the vector field Csinydx+(xcosysiny)dy with CP(x,y)dx+Q(x,y)dy .

P=siny (2)

Q=xcosysiny (3)

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

Py=y(siny)=cosy {t(sint)=cost}=cosy

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

Qx=x(xcosysiny)=cosyx(x)sinyx(1)=cosy(1)siny(0) {t(k)=0,t(t)=1}=cosy

Substitute cosy for Py and cosy for Qx in equation (1),

cosy=cosy

Hence C2xeydx+(2yx2ey)dy is conservative vector field and hence line integral is independent of path.

Thus, the line integral Csinydx+(xcosysiny)dy is independent of path.

Consider f=fx(x,y)i+fy(x,y)j .

Compare the equations Cfdr and Csinydx+(xcosysiny)dy

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