Difference between revisions of "Total derivative"

Latest revision as of 01:27, 15 November 2016

Definition

Let [ilmath](U,\Vert\cdot\Vert_U)[/ilmath] and [ilmath](V,\Vert\cdot\Vert_V)[/ilmath] be normed spaces over the same field (either both real or complex), let [ilmath]A\in\mathcal{P}(U)[/ilmath] be an arbitrary subset of [ilmath]U[/ilmath] and let [ilmath]a\in A[/ilmath] be a point such that [ilmath]A[/ilmath] is a neighbourhood of [ilmath]a[/ilmath] in [ilmath]U[/ilmath], then a map, [ilmath]f:A\rightarrow V[/ilmath] is "differentiable at [ilmath]a[/ilmath]" if:^[1]:

• there exists a linear map, [ilmath]L:U\rightarrow V[/ilmath] such that
  • [math]\lim_{h\rightarrow 0}\left(\frac{\Vert f(a+h)-f(a)-L(v)\Vert_V}{\Vert h\Vert_U}\right)\eq 0[/math] (i.e. the limit exists for some [ilmath]L:U\rightarrow V[/ilmath])

The linear map, [ilmath]L[/ilmath], is called the derivative of [ilmath]f[/ilmath] at [ilmath]a[/ilmath] and [ilmath]f[/ilmath] is said to be differentiable at [ilmath]a[/ilmath].

• Claim 1: the derivative of [ilmath]f[/ilmath] at [ilmath]a[/ilmath] (if it exists) is unique.

We denote the derivative of [ilmath]f[/ilmath] at [ilmath]a[/ilmath] by: [ilmath]df\vert_a[/ilmath].

If the vector spaces [ilmath]U[/ilmath] and [ilmath]V[/ilmath] are finite dimensional then recall all norms on finite dimensional vector spaces are equivalent and thus the choice of norm doesn't matter.

• See: For finite dimensional vector spaces the derivative at a point is independent of the norm

Alternative definition

There is a "remainder" such that [ilmath]\lim_{h\rightarrow 0}\left(\frac{\Vert R(h)\Vert_V}{\Vert h\Vert_U}\right)\eq 0[/ilmath]

References

1. ↑ Introduction to Smooth Manifolds - John M. Lee