A total derivative computes every directional derivative, and its matrix is the Jacobian

Depends on

• The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\|h\|_2)$ remainder
• 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
• Every Euclidean linear map has a unique matrix and satisfies $\|Lh\|_2\le K\|h\|_2$ for some $K\ge0$

Used by

• A rank drop at one point need not persist locally Counterexample
• x²y/(x²+y²) has every directional derivative at the origin but is not totally differentiable there Counterexample
• The Jacobian determinant of a square-dimensional C¹ map is the determinant of its Jacobian matrix Definition
• Wirtinger operators in ℂᵐ Definition
• Coordinate inclusions are immersions and coordinate projections are submersions Example
• r² sin(1/r) is differentiable at the origin with a discontinuous gradient Example
• The orthogonal group is a regular level set of dimension n(n-1)/2 Example
• The special linear group is a codimension-one embedded submanifold Example
• Rank at one point need not determine nearby rank 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
• 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
• The divergence and curl of a cross product Lemma
• The oriented area vector transforms by the parameter Jacobian determinant 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
• The tangent plane is invariant under regular reparametrization Proposition
• 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
• Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ∂_z̄f=0, or with the Cauchy–Riemann equations Theorem
• For a differentiable scalar field, Dᵥf(a)=⟨∇ f(a),v⟩ and the unit direction of steepest ascent is the normalized gradient Theorem
• Regular level surfaces have local regular parametrizations with the same tangent plane Theorem

Dependency tree · two levels

14 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)