A differentiable functional f on a convex set S n is increasing if and only f(x) 0 for every x S, that

A differentiable functional f on a convex set S ⊆ ℜn is increasing if and only ∇f(x) ≥ 0 for every x ∈ S, that is, if every partial derivative Dxi f[x] is nonnegative.