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Find the work done by the force field f(x, y, z) = y + z, x + z, x + y on a particle that moves along the line segment from (1, 0, 0) to (5, 3, 2).

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5 votes
Note that the vector field is irrotational, since


\\abla*\mathbf f(x,y,z)=\mathbf 0

which means
\mathbf f is conservative. This means there is some scalar potential function
f(x,y,z) such that
\\abla f(x,y,z)=\mathbf f(x,y,z). If we can find such a function, then we only need to evaluate the difference of
f(5,3,2) and
f(1,0,0) (because the gradient theorem holds in this case).

We're looking for a function
f(x,y,z) that satisfies


(\partial f)/(\partial x)=y+z

(\partial f)/(\partial y)=x+z

(\partial f)/(\partial z)=x+y

Integrate the first equation with respect to
x to get


f(x,y,z)=xy+xz+g(y,z)

Differentiate with respect to
y, then we must have


(\partial f)/(\partial y)=x+z=x+(\partial g)/(\partial y)

\implies(\partial g)/(\partial y)=z

\implies g(y,z)=yz+h(z)

\implies f(x,y,z)=xy+xz+yz+h(z)

Differentiate with respect to
z, and we have to have


(\partial f)/(\partial z)=x+y=x+y+(\mathrm dh)/(\mathrm dz)

\implies(\mathrm dh)/(\mathrm dz)=0

\implies h(z)=C

\implies f(x,y,z)=xy+xz+yz+C

Now, by the gradient theorem, we have


\text{work}=\displaystyle\int_(\mathcal C)\mathbf f\cdot\mathrm d\mathbf r=f(5,3,2)-f(1,0,0)=31

where
\mathcal C is the line segment.
User Prashant Borde
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