piefoot

The solution to the Euler equation gives the minimal surface with is obtained when y(x) is a catenoid, y=c cosh(x/c) here c satisfies 2=c cosh(1/c). The Euler equation gives the minimal distance when the function F is the length of the arc.

Solving 3.1 I think the solution is binomial(n1+r-1,r-1) where n1=n-r, but I obtained that formula by calculating the formula for r=1, then r=2, and the relation f(r,n) = sum(f(r-1,n-i),i,0,n) and using maxima with simpsum and factor to obtain a simplified expression whose form suggest the general rule, but I don't see a simpler way.

I think that the difference is from historical reasons, anyway when if you introduce in school partial derivatives with functions of only one variable that is confusing because here partial is not partial but global.

if a function depends only on multiple variables then those are partial derivatives, if the function depends only on one variable then there is only one derivative d/dx