Greens Theorem

1. Jun 11, 2012

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

I understand Greens Theorem, been doing a bit of it recently, but I have perhaps a very... novice question.

What is P and Q exactly? They showed us on the board, but I am unsure what they are. Are they vectors? Or are they functions of a vector?

2. Jun 11, 2012

algebrat

They are x and y components of a vector, (P,Q), and if you're curious or have seen enough machinery, you might like to know that Pdx+Qdy=(P,Q).(dx,dy), that is, the dot product F.dr.

3. Jun 11, 2012

I often see it written as $$P_{(x,y)}$$ and $$P_{(x,y)}$$ and I am told that they are functions of x and y?

4. Jun 11, 2012

I knew the thingy about the dot product... that became obvious when translating it to vector form.

5. Jun 12, 2012

HallsofIvy

Staff Emeritus
Well, what exactly is your question? Yes, the notation P(x,y) means that P is a function of the two variables x and y. You, for some reason, write that twice. Did you mean Q(x, y) for the second? It also means that Q is a function of the two variables x and y.

In Green's theorem, in the form
$$\oint P(x,y)dx+ Q(x,y)dy= \int\int\left(\frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}\right)dxdy$$

P and Q are simply two functions of x and y. They can be pretty much any functions as long as they satisfy the hypotheses: they must have continuous partial derivatives inside and on the closed curve.

You can think of P and Q as the x and y components of a vector valued function, as algebrat suggests. Taking the z-component to be 0, the integrand on the right can be thought of as the "curl",
$$\nabla\times \vec{F}(x,y)= \left(\frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}\right)\vec{k}$$
where $\vec{F}(x,y)$ is the vector valued function $\vec{F}(x,y)= P(x,y)\vec{i}+ Q(x,y)\vec{j}$.

In vector form, Green's theorem can written
$$\oint \vec{F}\cdot d\vec{\sigma}= \int\int \nabla\times \vec{F}\cdot d\vec{S}$$
a special form of the "generalized Stoke's theorem".

Last edited: Jun 12, 2012