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1. For the vector field

F = yz ˆx + zx ˆy + xy ˆz

(^x means the unit vector of x)

find the integral of F • dl from (0, 0, 0) to (1, 2, 3) in Cartesian coordinates

in each of the following ways:

(a) along a straight line path from (0, 0, 0,) to (1, 2, 3)

(b) along straight line paths from (0, 0, 0,) to (1, 0, 0), then from (1, 0, 0)

to (1, 2, 0), then from (1, 2, 0) to (1, 2, 3)

(c) without choosing a particular path

for part a), i used the dl as (^x + 2 ^y + 3 ^z)dx since vector r= ^x + 2 ^y + 3 ^z

thus, integral of F dot dl = yzx + z(x^2) + 3/2 (x^2)y evaluated from x=0 to x=1, and get yz+z+3/2y

for part b), i added the three different solutions from path 1, path 2, and path 3.

path1: using dl1=dx ^x, integral of path 1 of F*dl1 = yz

path2: using dl2=dy ^y, integral of path 2 of F*dl2 = 2zx

path3: using dl3=dz ^z, integral of path 3 of F*dl3 = 3xy

so my solution to part b) is yz+2zx+3xy

i was wondering, are my solutions to part a) and part b) correct? If not, please guide me through on how to get the solutions to part a,b, and c. thanks alot!

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# Homework Help: Finding integral of dot product of F and dl

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