Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Directional derivatives and partial derivatives of a map
Definition
Let , , , and . If the line map is defined near , its derivative at is the directional derivative
For a standard basis vector (The standard list with and for is an ordered basis of ; hence , and is the zero space with basis and dimension ), is the th partial derivative, written . These are ordinary vector-valued one-variable derivatives in the sense of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral.
Depends on
- The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral
- Vector-valued functions $f : A \to \mathbb{R}^m$, their limits and continuity, with the dictionary to the metric notions
- The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$
Used by
- The map y(x²+y²)/x off the line x=0, extended by zero on that line, has every directional derivative zero at the origin but is discontinuous there Counterexample
- x²y/(x²+y²) has every directional derivative at the origin but is not totally differentiable there Counterexample
- xy/(x²+y²) has both partial derivatives at the origin but is discontinuous there Counterexample
- Cᵏ maps and multi-index derivative notation in Euclidean space Definition
- Divergence and curl of a C¹ vector field Definition
- Exact and closed C1 vector fields Definition
- The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case Definition
- The Wirtinger derivatives ∂_z f and ∂_z̄f, and antiholomorphic functions Definition
- Wirtinger operators in ℂᵐ Definition
- sin(xy) and its mixed partial derivatives Example
- xy sin(1/(x²+y²)) is differentiable at the origin with unbounded partial derivatives nearby Example
- FALSE: continuity alone makes the regular-patch surface-area formula applicable False statement
- A rectangular second difference equals a mixed partial times the side lengths Lemma
- Continuous second partials of a scalar potential commute Lemma
- A total derivative computes every directional derivative, and its matrix is the Jacobian Theorem
- For C¹ functions, holomorphy, complex linearity of the real derivative, and the Cauchy–Riemann system agree Theorem
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative Theorem
Dependency tree · two levels
52 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)