Given a calculus field?sequentially Cauchy complete Archimedean ordered field$\mathrm{F\mathbb{R}}$ F \mathbb{R} of scalars and a type of indices $I$, one could define a calculus vector space?real vector space$V\u2254{\mathrm{F\mathbb{R}}}^{I}$ V \coloneqq F^I \mathbb{R}^I with a basis vector function $e:I \hookrightarrow V$. Let $f:V\to \mathrm{F\mathbb{R}}$ f:V \to F \mathbb{R} be a differentiable scalar function, and given an index $i:I$, the partial derivative$\partial_{i}$ is pointwise defined as

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