I don't know that it is easier, but you could not that this is a conservative force field. That is, , , and . That means that the integral is independent of the path and there exists a function F(x,y,z) such that . You could find that function, F, evaluate it at (0, 2, 1) and (1, 4, 0) and subtract.Then F = (t+t+2 - (t+2)(-0.5t+1), 0.5t +t+2-0.5t(-0.5t+1), -0.5t(t+2)-3(-0.5t+1)^2)
So then F = (0.5t^2+2t, 0.25t^2+t+2, -1.25t^2+2t-3).
Therefore work = integral(0 to 2) of F dot ds
= integral(0 to 2) of (0.5t^2+2t, 0.25t^2+t+2, -1.25t^2+2t-3) dot (0.5t, t+2, -0.5t+1)
which works out to be 127/6
Is this correct? I'm not at all sure about the parametrization, is there an easier way of doing this???
Thanks in advance for any help!
so . The "constant of integration" may be a function of y and z since they are treated as constant when differentiating with respect to x.
Differentiating that with respect to y, we get so . (The terms involving x cancel out- that had to happen and is a consequence of this being a conservative force field.)
We can integrate that with respect to y go get where, now, the "constant of integration" may be a function of z.
Now we have . Differentiating that with respect to z we have
so that . From that where C really is a constant.
Evaluate that at (0, 2, 1) and (1, 4, 0) and subtract. Of course, the "C" will cancel.
By the way, this really has nothing to do with "Linear and Abstract Algebra" and should have been posted in either the "Applied Math" or "Calculus" areas.