Homotopy Type Theory directional derivative > history (Rev #4, changes)

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Definition

Given a sequentially Cauchy complete Archimedean ordered field \mathbb{R} of scalars, let VV be a \mathbb{R}-real vector space VV, and let f:Vf:V \to \mathbb{R} be a directionally differentiable function valued in scalars. Given a vector w:Vw:V, the directional derivative w\Del_{w} is pointwise defined as

w(f)(v)lim (x,y)(x,x)f(v+xw)f(v+yw)xy\Del_{w}(f)(v) \coloneqq \lim_{(x, y) \to (x, x)} \frac{f(v + x w) - f(v + y w)}{x - y}

See also

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