   Chapter 16.3, Problem 19E

Chapter
Section
Textbook Problem

Show that the line integral is independent of path and evaluate the integral.19. ∫C 2xe−y dx + (2y – x2e−y) dy, C is any path from (1, 0) to (2, 1)

To determine

To show: The line integral C2xeydx+(2yx2ey)dy is independent of path and value of integral C2xeydx+(2yx2ey)dy .

Explanation

Given data:

Line integral is C2xeydx+(2yx2ey)dy .

Line integral function C2xeydx+(2yx2ey)dy and curve C is path from (1,0) to (2,1) .

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 C2xeydx+(2yx2ey)dy with CP(x,y)dx+Q(x,y)dy .

P=2xey (2)

Q=2yx2ey (3)

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

Py=y(2xey)=2xy(ey)=2x(ey) {t(et)=et}=2xey

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

Qx=x(2yx2ey)=2yx(1)eyx(x2)=2y(0)ey(2x) {t(k)=0,t(t)=1}=2xey

Substitute 2xey for Py and 2xey for Qx in equation (1),

2xey=2xey

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

Thus, the line integral C2xeydx+(2yx2ey)dy is independent of path.

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

Compare the equations Cfdr and C2xeydx+(2yx2ey)dy .

f=2xeyi+(2yx2ey)j

Compare the equation f=fx(x,y)i+fy(x,y)j with f=2xeyi+2yx2eyj

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