Conservative Vector Fields

krtica

If C is the curve given by r(t)=<1+3sin(t), 1+5sin^2(t), 1+5sin^3(t)>, 0≤t≤π/2 and F is the radial vector field F(x, y, z)=<x, y, z>, compute the work done by F on a particle moving along C.

Work= int (F dot dr)

If F is the potential function(?), do I integrate F with respect to each variable, then substitute the values of x, y, and z in r(t)? Would this then just be dotted into 1 since d/dt sin(t) is cos(t), which is 0 at π/2? Would my answer be something like (4^2/2)+(6^2/2)+(6^2/2)?

gabbagabbahey

Hint: [itex]d\textbf{r}=\textbf{r}'(t)dt[/itex]

krtica

So the integral is gross?

gabbagabbahey

No, the point of my hint is that you should differentiate your position function with respect to [itex]t[/itex], and then take the dot product with the position function and finally integrate the result. You'll have something like 6 terms to integrate, but they should all be straightforward.