I'm trying to understand the proof for Green's Theorem and I've stumbled upon a few problems.
In my notes, it says that:
If $E$ is a simple (flat?) surface in $\mathbb{R}^2$ (I've been trying to find the right English terminology but I'm having trouble as it's not my first language, so I'm not sure if it's correct.), then $\partial E$ is its edge which is a piecewise smooth curve.
I don't really understand this conclusion.
Another thing I'm struggling with is the following:
If we have a scalar field Q (i.e. a function from $\mathbb{R}^2$ to $\mathbb{R}$) then $\frac{\partial Q}{\partial x}$ is a surface in 3-D space. (Again, not sure about the terminology, sorry.)
I've watched these videos: https://www.khanacademy.org/math/calculus/line_integrals_topic/greens_theorem/v/green-s-theorem-proof-part-1 hoping they would clarify the proof, which it did, but I don't understand it as intuitively as I want to.
If anyone could help me understand these concepts intuitively it would be much appreciated!
This isn't rigorous but the only reason I am posting this is "...could help me understand these concepts intuitively...". So here goes.
$\partial E$ would be a 'small change in E'. What elementary object is it, that if you keep adding side by side, will produce a surface? A line of course. It could be a curved one too, but a line nevertheless. And therefore the edge of the surface, since you are only incrementing(or decrementing) the surface at the edge.
The requirement of $\partial$ here is, that it could be differentiated with respect any of the two axes(assuming a planar surface) and you get a choice of selecting along which direction you want to increment. Hope that helps.
I cannot help with the second question, since I have never actually come across this before. I just answered with whatever physical intuition I had about differentiation, and its relation to change.