If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative

Depends on

• Small coordinate-by-coordinate increments stay inside a Euclidean ball and telescope the total increment
• Directional derivatives and partial derivatives of a map $U\subseteq\mathbb{R}^m\to\mathbb{R}^n$
• The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case
• The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\|h\|_2)$ remainder
• The mean value inequality: if $f : [a,b] \to \mathbb{R}^m$ is continuous and differentiable on $(a,b)$ with $\lVert f'\rVert_2 \le M$, then $\lVert f(b)-f(a)\rVert_2 \le M(b-a)$
• A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions
• Cauchy-Schwarz $\lvert\langle x,y\rangle\rvert \le \lVert x\rVert_2\lVert y\rVert_2$ with its equality case, the triangle inequality for $\lVert\cdot\rVert_2$, the parallelogram law and polarisation

Used by

• A critical value can have a smooth level set Counterexample
• The cone x²+y²=z² has a rank drop at its apex Counterexample
• The cusp y²=x³ has a rank drop at the origin Counterexample
• The real complex-squaring map is locally but not globally invertible off the origin Counterexample
• Cᵏ Euclidean maps and diffeomorphisms Definition
• A Euclidean sphere is a regular level set with tangent hyperplanes Example
• A positive-definite quadratic ellipsoid is a regular level set Example
• Brownian finite-dimensional density Example
• The map (x,y)↦(x,xy) has nonconstant rank on every neighbourhood of the origin Example
• The one-sheeted hyperboloid is a regular surface of revolution Example
• The orthogonal group is a regular level set of dimension n(n-1)/2 Example
• The polynomial map (x,y)↦(1+x+2y+x², 2x+3y+xy) and its Jacobian Example
• The unit circle is locally a C¹ graph at every point Example
• Two constraints on a sphere-plane circle, where one multiplier solution is only a local maximum Example
• FALSE: a critical value must have a singular level set False statement
• FALSE: an everywhere-invertible derivative gives a global inverse False statement
• FALSE: every injective real-differentiable planar map has nonzero Jacobian False statement
• FALSE: every level set of a smooth map is locally a graph False statement
• A C¹ map sends a compact set of content zero to a set of content zero Lemma
• A vector line integral along an image arc is the parameter line integral of the pulled-back field Lemma
• Planar Brownian annular exit probability Lemma
• Repeated derivatives along a line expand by the multinomial formula Lemma
• The curl flux integrand of a C² patch is a two-dimensional curl of the pulled-back field Lemma
• The curl measures the antisymmetric part of the total derivative Lemma
• At interior base points, the graph faces of an adapted presentation induce the outward unit normal Proposition
• Compatibility of smooth atlases is an equivalence relation, and smooth Euclidean maps compose Proposition
• Smooth maps are continuous Proposition
• A continuous path-independent field has a potential constructed by line integrals Theorem
• A divergence-free C¹ field on a star-shaped open subset of ℝ³ has a vector potential Theorem
• Cᵏ Euclidean maps are closed under componentwise algebra and composition Theorem
• Continuous first partial derivatives and the Cauchy–Riemann equations imply complex differentiability pointwise, and holomorphy when they hold throughout an open set Theorem
• Every smooth manifold embeds in some finite-dimensional Euclidean space Theorem
• For C¹ functions, holomorphy, complex linearity of the real derivative, and the Cauchy–Riemann system agree Theorem
• Poincare's lemma on a star-shaped domain: every closed C1 field is exact Theorem

Dependency tree · two levels

48 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, §8.4 (standard reference, not scraped)