Prove that integral curvilinear:
$\displaystyle \int_C yzdx + xzdy + yx^2dz$
is dependent on the path
The line integral for conservative vector fields are independent of path. Note that we can describe the vector field here as $\displaystyle \bold{F} = \left< yz, xz, yx^2 \right>$. show that this vector field is conservative, that is, find a function $\displaystyle f(x,y,z)$ such that $\displaystyle \nabla f = \bold{F}$. that will show independents of path.
it is not enough to pick two random paths and show the line integral is the same.
i suppose it is the same for you. curl is defined as follows:
let $\displaystyle \bold{F} = P \bold{i} + Q \bold{j} + R \bold{k}$
then $\displaystyle \text{curl}\bold{F} = \nabla \times \bold{F} = \left(\frac {\partial R}{\partial y} - \frac {\partial Q}{\partial z} \right) \bold{i} + \left(\frac {\partial P}{\partial z} - \frac {\partial R}{\partial x} \right) \bold{j} + \left(\frac {\partial Q}{\partial x} - \frac {\partial P}{\partial y} \right) \bold{k}$