Homotopy Type Theory partial derivative > history (Rev #2)

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Definition

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

i(f)(v)lim (x,y)(x,x)f(v+xe i)f(v+ye i)xy\partial_{i}(f)(v) \coloneqq \lim_{(x, y) \to (x, x)} \frac{f(v + x e_i) - f(v + y e_i)}{x - y}

See also

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