Given a sequentially Cauchy complete Archimedean ordered field$\mathbb{R}$ of scalars and a type of indices $I$, one could define a real vector space$V \coloneqq \mathbb{R}^I$ with a basis vector function $e:I \hookrightarrow V$. Let $f:V \to \mathbb{R}$ be a differentiable scalar function, and given an index $i:I$, the partial derivative$\partial_{i}$ is pointwise defined as

$\partial_{i}(f)(v) \coloneqq \lim_{(x, y) \to (x, x)} \frac{f(v + x e_i) - f(v + y e_i)}{x - y}$