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Sums and scalar multiples of totally differentiable maps are totally differentiable with the expected derivatives
Statement
If are totally differentiable at and , then and are totally differentiable at , with
Facts & Assumptions
Given: Total first-order expansions for and at .
The total-derivative definition supplies the normwise remainder estimate for every ; in positive source dimension this is equivalently the normalized remainder tending to zero as tends to zero (The total (Fréchet) derivative as the linear first-order approximation with remainder).
A norm satisfies the triangle inequality and (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms).
Proof
If the source dimension is zero, each derivative is the unique map from the zero space and the remainder condition holds on its sole increment. If , both functions and all candidate derivatives take only the zero value. In either case the displayed sum and scalar-multiple identities hold for these unique zero linear maps. For the remainder assume that both source and target dimensions are positive.
Add the two expansions and multiply the first by to obtain remainders for and for , with the displayed candidate linear maps.
By [L2], is bounded by the sum of two quantities tending to zero, and tends to zero (also when ).
Sums and scalar multiples of linear maps are linear, so step 2.1 verifies the definition with exactly the two stated derivatives.
Depends on
Used by
- Classical Neumann solutions differ by componentwise constants Corollary
- Uniqueness of classical Dirichlet and compatible Neumann solutions Corollary
- The real complex-squaring map is locally but not globally invertible off the origin Counterexample
- A regularized logistic finite sum with an exact minimizer Example
- The Euclidean norm and its square are convex, with a ball of subgradients at zero for the norm Example
- The graph of a Cᵏ Euclidean map is a regular level set Example
- The unit circle is locally a C¹ graph at every point Example
- Two equations implicitly determine two variables near the origin Example
- A bounded-domain Dirichlet Green function is unique and positive Lemma
- A policy score has zero conditional mean before the action Lemma
- Finite-response score mean and reward-gradient identities Lemma
- Gradient of a finite sum is the finite sum of gradients Lemma
- Newton maps are uniform contractions near a point with invertible derivative Lemma
- Pseudo-residuals are the negative gradient in sample-score space Lemma
- Smooth Euclidean hypersurface graphs and compact localization Lemma
- Smooth sections form a module over smooth functions Proposition
- The Euclidean implicit function theorem with derivative formula Theorem
Dependency tree · two levels
17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Lebl, Basic Analysis I, §8.3 (standard reference, not scraped)