How to use the normal form of the Green's Theorem?

[randomcat]

Homework Statement

Suppose that F = ∇f for some scalar potential function f(x, y) = 1/2(x² + y²)
Let C denote the positively oriented unit circle, parametrized by r(t) = (cos t, sin t), 0 ≤ t ≤ 2∏. Compute the flux integral of $\oint$F$\bullet$N ds, where N is the outward unit normal to C.

Homework Equations

I know that the theorem is basically $\oint$F$\bullet$N ds = double integral of ($\frac{\partial M}{\partial x}$ + $\frac{\partial N}{\partial y}$) dx dy

My question is, how do you find F(x, y)? I'm given f(x, y) in the problem, but do I need to use it to solve for the answer?

The Attempt at a Solution

What I did (which was wrong) was that I set M = 1/2x² and N = 1/2y² and took the partial derivatives of those. That gives me (x, y). With the parametrization, I get (cos t, sin t). Then I integrate them over the unit circle and got 0, which was wrong. The correct answer is 2pi.

My professor wrote that F = ∇f = F(x, y) in the solutions, but where did F(x, y) come from?

 

[LCKurtz]

Homework Statement

Suppose that F = ∇f for some scalar potential function f(x, y) = 1/2(x² + y²)
Let C denote the positively oriented unit circle, parametrized by r(t) = (cos t, sin t), 0 ≤ t ≤ 2∏. Compute the flux integral of $\oint$F$\bullet$N ds, where N is the outward unit normal to C.

Homework Equations

I know that the theorem is basically $\oint$F$\bullet$N ds = double integral of ($\frac{\partial M}{\partial x}$ + $\frac{\partial N}{\partial y}$) dx dy

My question is, how do you find F(x, y)? I'm given f(x, y) in the problem, but do I need to use it to solve for the answer?

The Attempt at a Solution

What I did (which was wrong) was that I set M = 1/2x² and N = 1/2y² and took the partial derivatives of those. That gives me (x, y). With the parametrization, I get (cos t, sin t). Then I integrate them over the unit circle and got 0, which was wrong. The correct answer is 2pi.

My professor wrote that F = ∇f = F(x, y) in the solutions, but where did F(x, y) come from?

You are given $f=(x^2+y^2)/2$ and $F=\nabla f$. So calculate $\nabla f$ to get $F$.

 

[HallsofIvy]

Perhaps the problem is that you don't know what $\nabla f$ means? If f is as function of two variables, x and y, the $\nabla f$ is the vector function
$$\frac{\partial f}{\partial x}\vec{i}+ \frac{\partial f}{\partial y}\vec{j}$$

That should be easy to calculate for this f.