The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$

Depends on

• The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\|h\|_2)$ remainder
• Total differentiability gives a local $O(\|h\|_2)$ increment bound and therefore continuity
• Every Euclidean linear map has a unique matrix and satisfies $\|Lh\|_2\le K\|h\|_2$ for some $K\ge0$
• A linear map $L:\mathbb{R}^m\to\mathbb{R}^n$ in Euclidean coordinates

Used by

• An everywhere-positive-definite Hessian implies strict convexity Corollary
• Cartesian and polar forms of the Cauchy–Riemann equations agree away from the origin Corollary
• A figure-eight curve is an immersed image but not an embedded submanifold Counterexample
• Analytic transport data Example
• The inverse-square field is divergence free, and its flux through the sphere bounding the translated unit ball vanishes Example
• The image of every immersion need not be an embedded submanifold False statement
• A C¹ map sends a compact set of content zero to a set of content zero Lemma
• A nondegenerate stationary envelope solves the Hamilton–Jacobi equation Lemma
• A quasilinear solution lifts to augmented characteristics Lemma
• A transport equation restricts to a linear ODE along each characteristic Lemma
• A vector line integral along an image arc is the parameter line integral of the pulled-back field Lemma
• Chart and partition independence of surface measure Lemma
• Choice-free smooth inverse function theorem in Euclidean space Lemma
• Compactly supported scaled Euclidean bumps Lemma
• Compatibility of a characteristic strip with Cauchy data Lemma
• Explicit compactly supported smooth cutoffs Lemma
• In source rank coordinates, the remaining components depend only on the rank coordinates Lemma
• Newton maps are uniform contractions near a point with invertible derivative Lemma
• Repeated derivatives along a line expand by the multinomial formula Lemma
• The Burgers slope obeys a Riccati law along characteristics Lemma
• The Charpit flow preserves the PDE constraint Lemma
• The curl flux integrand of a C² patch is a two-dimensional curl of the pulled-back field Lemma
• The inverse-projected characteristic graph satisfies the quasilinear PDE Lemma
• The oriented area vector transforms by the parameter Jacobian determinant Lemma
• The principal symbol depends only on the first derivative of a smooth coordinate change Lemma
• At interior base points, the graph faces of an adapted presentation induce the outward unit normal Proposition
• Chain rule for smooth half-space maps Proposition
• Compatibility of smooth atlases is an equivalence relation, and smooth Euclidean maps compose Proposition
• Identity maps and composites of smooth maps are smooth Proposition
• The tangent plane is invariant under regular reparametrization Proposition
• A C² function is convex exactly when its Hessian is positive semidefinite Theorem
• A constrained local extremum annihilates every velocity of a differentiable parametrization Theorem
• A differentiable map on a connected open Euclidean set has zero derivative exactly when it is constant Theorem
• A divergence-free C¹ field on a star-shaped open subset of ℝ³ has a vector potential Theorem
• A holomorphic function with zero derivative on a domain is constant Theorem
• Change of variables for an injective C¹ map on a compact Jordan set Theorem
• Cᵏ Euclidean maps are closed under componentwise algebra and composition Theorem
• Differentiable convex functions are characterized by the gradient inequality Theorem
• Homogeneous linear transport is solved by the inverse characteristic flow Theorem
• Local linear transport has a unique solution from noncharacteristic Cauchy data Theorem

…and 11 more results.

Dependency tree · two levels

13 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)