   Chapter 16.3, Problem 18E

Chapter
Section
Textbook Problem

(a) Find a function f such that F = ∇ f and (b) use part (a) to evaluate ∫C F · dr along the given curve C.18. F(x, y, z) = sin y i + (x cos y + cos z) j − y sin z k,C: r(t) = sin t i + t j + 2t k, 0 ⩽ t ⩽ π/2

(a)

To determine

To find: The potential function f such that F=f .

Explanation

Given data:

Vector field is F(x,y,z)=sinyi+(xcosy+cosz)jysinzk .

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

Write the relation between the potential function f and vector field F .

f=F

Substitute fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k for f ,

F=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k

Compare the equation F=fx(x,y,z)i+fy(x,y,z)j+fz(x,y,z)k with F(x,y,z)=sinyi+(xcosy+cosz)jysinzk .

fx(x,y,z)=siny (1)

fy(x,y,z)=xcosy+cosz (2)

fz(x,y,z)=ysinz (3)

Integrate equation (1) with respect to x.

f(x,y,z)=(siny)dx=sinydx=siny(x)+g(y,z) {dt=t}

f(x,y,z)=xsiny+g(y,z) (4)

Apply partial differentiation with respect to y on both sides of equation (4).

fy(x,y,z)=y(xsiny+g(y,z))=y(xsiny)+y(g(y,z))=xy(siny)+gy(y,z)=x(cosy)+gy(y,z) {t(sint)=1}

fy(x,y,z)=xcosy+gy(y,z) (5)

Compare the equations (2) and (5)

(b)

To determine

The value of Cfdr along the curve C.

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