vector field question Consider the vector field $$F(x,y,z)=(zy+\sin x, zx-2y, yx-z)$$ (a) Is there a scalar field $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ whose gradient is $F$?
(b) Compute $\int _C F\cdot dr \neq 0$ where the curve $C$ is given by $x=y=z^2$ between $(0,0,0)$ and $(0,0,1)$.
I have no idea how to do the first one and for the second one, is there any typo on the curve equation because I have no idea how to parameterize it...

The way to approach this is to consider $F(x,y,z) = F_x \hat{x} + F_y \hat{y} + F_z \hat{z}$, and compute the following integrals:
$$
\int F_x(x,y,z)dx \hspace{3pc} \int F_y(x,y,z) dy \hspace{3pc} \int F_z(x,y,z)dz
$$
I'll compute the first one for you: $\int (yz + \sin x)dx = xyz - \cos x + c(y,z)$ where we observe that the "constant" term is actually a function of $y$ and $z$ since if we take the partial derivative of this function with respect to $x$ the $c$ term will drop out.  You need to compute the corresponding integrals for $F_y$ and $F_z$ and ask yourself can the constant terms I get from integrating each equation fit into the other equations?  For instance, integrating the second equation will yield $xyz - y^2 + d(x,z)$.  We therefore see that the first and second integrals have $xyz$ in common, and it's completely reasonable that $-y^2$ might be a part of $c(x,y)$.  Additionally, it's also quite possible that $-\cos x$ is a part of $d(x,z)$.  
Once you compute the scalar field, the second part can be done by the fundamental theorem of line integrals: $\int F\cdot \textbf{dr} = f(x,y,z) |_{\textbf{r}_0}^{\textbf{r}_1}$.
