Theorem: partial^2 f / partial-x partial-y - partial^2 f / partial-y partial-x = 0 (d^2 = 0 via Schwarz)

Theorem (d-squared-zero canonical): for a smooth function f(x, y), the equality of mixed partial derivatives partial^2 f / partial-x partial-y = partial^2 f / partial-y partial-x (Schwarz/Clairaut symmetry) is equivalent to d circ d = 0 on…