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Definition

Given a calculus field?sequentially Cauchy complete Archimedean ordered field $\mathrm{<span><del\; class="diffmod">\; F</del><ins\; class="diffmod">\; ℝ</ins></span>}$ F \mathbb{R} of scalars and a type of indices $I$, one could define a calculus vector space?real vector space $V≔{\mathrm{<span><del\; class="diffmod">\; F</del><ins\; class="diffmod">\; ℝ</ins></span>}}^I$ V \coloneqq F^I \mathbb{R}^I with a basis vector function $e:I \hookrightarrow V$. Let $f:V\to \mathrm{<span><del\; class="diffmod">\; F</del><ins\; class="diffmod">\; ℝ</ins></span>}$ 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

See also

• calculus field?sequentially Cauchy complete Archimedean ordered field

• calculus vector space?real vector space

• Newton-Leibniz operator

• directional derivative