Sup

I've actually exhausted (downloaded and tried to comprehend) most of the internet sources regarding the Legendre transformation. Most of them say that it's used within a thermodynamic or mechanical context, and then proceed to apply it without explanation of how it works. Some of them, such as the one Devil Sunrise found, show the geometric interpretation in such an abstract way that I don't really understand. However, none of them show them within basic mathematical contexts.

 Old

I'm looking for a very simple mathematical definition, which just isn't available anywhere. (Or if it is, I've yet to find.) I feel like an actual, simple example will do much more for me than a complicated definition. Try to avoid the sup function if possible, because I know that it can be avoided. As a simple example, I attempted to transform:

df = 2x*e^(5y)*dx + 5x^2*e^(5y)*dy

Here is my work.
f = x^2*e^(5y)
df = w*dx + z*dy
w = 2x*e^(5y)
z = 5x^2*e^(5y)

Let g = f - w*x
g = x^2*e^(5y) - 2x^2*e^(5y)
g = -x^2*e^(5y)
dg = -2x*e^(5y)*dx - 5x^2*e^(5y)*dy

If this works properly:
dg = -x*dw + z*dy

dw = 2*e^(5y)*dx + 10*x*e^(5y)*dy
z*dy = 5x^2*e^(5y)*dy

dg = -2*e^(5y)*dx - 10*x*e^(5y)*dy + 5x^2*e^(5y)*dy
dg = -2x*e^(5y)*dx - 5x^2*e^(5y)*dy

Okay. They match, so my transform worked as it should. Now... what is the point of this procedure? You still need both variables. Also, I'm not sure how to transform it back. Supposedly, the Legendre transform is its own transform, but I haven't tried it yet. I'm not sure what's going on.

Also, I don't really understand the notation that most of these sources use:

(dy/dx)_y

I'm not sure what this means.

Edit: So I finally figured out how to use the Legendre Transform:

 Image

My new questions are:

1. How does this fit into the sup function? What's the point of writing the f*(p) = sup(px-f(x)), when the Legendre Transform takes f'(x,y) = wdx+zdy and turns it into: g'(x,y) = -xdw+zdy?

2. I read in one paper that the Legendre Transform of a parabola is a parabola, but a parabola exists in two-dimensional space and is a function of one variable. How do I apply the Legendre Transform to single variable functions?

3. What is the geometric interpretation of switching the differentials? What is convexity? Why is convexity a requirement for the Legendre Transform?

4. In my f(x,y) = x^2*e^(5y) example, how is g'(x,y) any more useful than f'(x,y)? Both g'(x,y) and f'(x,y) depend on x and y.

5. How is this process done in three variables?