We have a vectorial field $F(x,y,z)=x i -xy j+z^2 k$. Two points: $P(1,0,0)$ and $Q(-1,0,3\pi)$. And a propeller $r(t)=\cos t i+\sin t j+t k$. I have to find the work done by F in a particle, to move that particle from P to Q across the propeller r.
I know that the formula of the work is given by an integral. But I don't know how to do it. I suppose that I would have to calculate the derivative of r and then evaluate F in r but I'm not sure. I am really confused and I need some help. Does anyone know how to do this exercise?
As you're moving along a specific path $\mathbf{r}(t)$ in space, the force on the particle is $\mathbf{F}(\mathbf{r}(t))$. The work done is then
$$W = \int_0^T \left[\mathbf{F}(\mathbf{r}(t))\cdot \mathbf{v}(t)\right]\,dt,$$
where $\mathbf{v}(t) = d\mathbf{r}/dt$. The problem here is that the final time $T$ is not given, so you must calculate it from $\mathbf{r}(T) = Q$.
Anyway, this problem makes no sense to me. There must be additional forces on the particle to make it move along $\mathbf{r}(t)$. So why calculate the work done by $\mathbf{F}$ alone? Since $\mathbf{r}(t)$ is specified, wouldn't you just calculate the total work?