Question

Legendre @ Osprey

Answer 1

Re-Legendre:


For \(f = f(x,y)\), we know that:

\(df = f_x dx +f_y dy \)

we re-write that as:

\(df = p dx +q dy \)

Explicitly: \(p = f_x, q = f_y\)

And so the indie variables are initially \((x,y)\)

But we suggest that there is a function \(g = g(p,x)\) that simply has to follow this rule:

\(g(p,x) = f - p x\)

And that means that:

\(dg = d(g(p,x) )= df - d(p x)\)

i think this sucks

\(= (p dx +q dy) - (x dp + p dx)\)

\(= - x dp + q dy \)

And then we can conclude that \(g(p,x)\) lines up with \(f(x,y)\), as simple as that ??