Question

Find the work done by the force field f on a particle moving along the given path. f(x, y, z) = yzi + xzj + xyk

c.line from (0, 0, 0) to (9, 7, 4)

Answer 1

Computing the line integral directly would be easy enough, but in general it's worth checking to see if the vector field is conservative; that is, whether there exists a scalar function
`f` whose gradient corresponds exactly to the given vector field
`\mathbf f`, or
`\\abla f=\mathbf f`. Equivalently, we're looking for
`f` such that


`(\partial f)/(\partial x)=yz`

`(\partial f)/(\partial y)=xz`

`(\partial f)/(\partial z)=xy`

We find that



`f_x=yz\implies f(x,y,z)=xyz+g(y,z)`

`f_y=xz+g_y=xz\implies g_y=0\implies g(y,z)=h(z)`

`f_z=xy+h_z=xy\implies h_z=0\implies h(z)=C`

and so there is indeed such a function
`f`, with
`f(x,y,z)=xyz+C`.
Thus by the fundamental theorem of calculus, the line integral is path-independent, and


`\displaystyle\int_(\mathcal C)\mathbf f\cdot\mathrm d\mathbf r=\int_(\mathcal C)\\abla f\cdot\mathrm d\mathbf r=f(\mathbf b)-f(\mathbf a)`

where
`\mathcal C` is any path starting at
`\mathbf a` and ending at
`\mathbf b`. Here,


`\displaystyle\int_(\mathcal C)\mathbf f\cdot\mathrm d\mathbf r=f(9,7,4)-f(0,0,0)=252`