   Chapter 16, Problem 14RE

Chapter
Section
Textbook Problem

Show that F is a conservative and use this fact to evaluate ∫C F · dr along the given curve.14. F(x, y, z) = ey i + (xey + ez) j + yez k, C is the line segment from (0, 2, 0) to (4, 0, 3)

To determine

To show : F is a conservative vector field and The value of CFdr , where C is the line segment from (0,2,0) and terminal point (4,0,3) .

Explanation

Given data:

F(x,y,z)=eyi+(xey+ez)j+yezk (1)

F(x,y,z)=eyi+(xey+ez)j+yezk with C is the line segment from (0,2,0) to (4,0,3) .

Formula used:

Consider the standard equation of an curl F for F=Pi+Qj+Rk

curlF=|ijkxyzPQR| (2)

Find the value of curlF .

Substitute ey for P , (xey+ez) for Q and yez for R in equation (2),

curlF=|ijkxyz(ey)(xey+ez)yez|=[y(yez)z(xey+ez)]i[x(yez)z(ey)]j+[x(xey+ez)y(ey)]k=(ezez)i(00)j+(eyey)k=(0)i(0)j+(0)k

Simplify expression as follows.

curlF=0

Since curlF=0 , the F is a conservative vector and the domain of F is 3

Thus, F is a conservative vector field is shown.

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 (3)

Compare the equation (3) and equation (1).

fx(x,y,z)=ey (4)

fy(x,y,z)=xey+ez (5)

fz(x,y,z)=yez (6)

Integrate equation (5) with respect to y

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