1. Function being path independent?

I have a question about this problem. I know that I have to parametrize the circle using polar coordinates but after that I am completely lost. My book says that being path independent also means the same as being a conservative field correct? So does that mean that I have to take the partials of those terms? What am I missing here? Thanks!

I have a question about this problem. I know that I have to parametrize the circle using polar coordinates but after that I am completely lost.

My book says that being path independent also means the same as being a conservative field correct? Mr F says: Correct.

So does that mean that I have to take the partials of those terms? What am I missing here? Thanks!
If a vector field is conservative then $\displaystyle \nabla \times F = 0$. Does this happen here?

I have a question about this problem. I know that I have to parametrize the circle using polar coordinates but after that I am completely lost. My book says that being path independent also means the same as being a conservative field correct? So does that mean that I have to take the partials of those terms? What am I missing here? Thanks!

So the parametric equation for the circle is

$\displaystyle \vec{r}(t)=<4\cos(t),4\sin(t)> \implies \vec{r'}(t)=<-4\sin(t),4\cos(t)>dt$

Then the line integral is

$\displaystyle \int_{0}^{2\pi}\vec{F}(r(t))\cdot r'(t)dt=$
$\displaystyle \int_{0}^{2\pi}<3(4\cos(t))^2,(4\sin(t))>\cdot <-4\sin(t),4\cos(t)>dt=$
$\displaystyle \int_{0}^{2\pi}-192\sin(t)\cos^2(t)+16\sin(t)\cos(t)dt$

$\displaystyle =64\cos^3(t)+8\sin^2(t) \bigg|_{0}^{2\pi}=0$

You can see the vector field is conservative.

$\displaystyle \frac{\partial f}{\partial x}=3x^2 \implies f(x,y)=x^3+g(y)$

Now taking the partial with respect to y gives

$\displaystyle \frac{\partial f}{\partial y}=g'(y)=y \implies g(y)=\frac{1}{2}y^2$

So the potential function is

$\displaystyle f(x,y)=x^3-\frac{1}{2}y^2$ and notice that

$\displaystyle \nabla f =\vec{F}$

So the vector field is conservative.

4. hmmm ok then, so the question was simply asking me if it was conservative just in different wording? That's what threw me off. Thanks a ton guys