wondering if anyone can help me with the following:
"Prove that the force $\mathbf{F} = 2xe^{x^{2}}\sin y \ \mathbf{i} + e^{x^{2}}\cos y \ \mathbf{j}$ is conservative, find the potentail and use it to calculate the work by the force along the path from point $(0,0)$ to $(1,{\pi/2})$."
A quick calculation shows $\text{curl} \ \mathbf{F} = 0$, so the force is conservative. Then $\nabla \Phi=\mathbf{F}$, therefore the potential is given by $\Phi=e^{{x}^{2}}\sin y +c$.
The equation of the line joining both points is $\mathbf{r} = t \ \mathbf{i} + \dfrac{\pi}{2}t \ \mathbf{j}$.
So far so good, but for the last question, the usual way to find the work is to calculate:
$\displaystyle \int \mathbf{F} d\mathbf{r}$
What I don't understand is how to find the work using the potential given that it is a scalar function and not a vector one.
Since $\Phi$ is a potential of $F$, the work is given by
$$ \Phi(1, \frac{\pi}{2})- \Phi(0,0).$$