Understanding Greens Theorem: Exploring P and Q in Vector Calculus

[help1please]

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?

 

[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.

 

[help1please]

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?

 

[help1please]

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

 

[HallsofIvy]

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".