   Chapter 16.3, Problem 15E

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.15. F(x, y, z) = yz i + xz j + (xy + 2z) k,C is the line segment from (1, 0, −2) to (4, 6, 3)

(a)

To determine

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

Explanation

Given data:

Vector field is F(x,y,z)=yzi+xzj+(xy+2z)k .

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)=yzi+xzj+(xy+2z)k .

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

fy(x,y,z)=xz (2)

fz(x,y,z)=xy+2z (3)

Integrate equation (1) with respect to x.

f(x,y,z)=yzdx=yzdx=yz(x)+g(y,z) {dt=t}

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

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

fy(x,y,z)=y(xyz+g(y,z))=y(xyz)+y(g(y,z))=xzy(y)+g(y,z)=xz(1)+g(y,z) {t(t)=1}

fy(x,y,z)=xz+g(y,z) (5)

Compare the equations (2) and (5)

(b)

To determine

The value of Cfdr along the curve C.

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