The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case

Depends on

• Directional derivatives and partial derivatives of a map $U\subseteq\mathbb{R}^m\to\mathbb{R}^n$
• The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$
• 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

• For one regular constraint, the objective gradient is a scalar multiple of the constraint gradient Corollary
• Green's first identity on a glued elementary solid region Corollary
• Green's second identity on a glued elementary solid region Corollary
• Lagrange multipliers for a regular graph constraint y=ψ(x) Corollary
• The curl of a curl is the gradient of the divergence minus the Laplacian Corollary
• The Jacobian determinant of a holomorphic map is |f'|² and is positive exactly where f'≠0 Corollary
• The subdifferential of a differentiable convex function is its gradient singleton Corollary
• Vector forms: the boundary integrals of fn and of n× F Corollary
• A critical value can have a smooth level set Counterexample
• A curl-free C¹ field on the complement of a line that is not conservative 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 degenerate constraint x²+y²=0 defeats the multiplier conclusion Counterexample
• Continuously differentiable maps, local inverses, and local diffeomorphisms Definition
• Divergence and curl of a C¹ vector field Definition
• Exact and closed C1 vector fields Definition
• Piecewise-C1 path-connected domains, potential functions, conservative fields, and path independence Definition
• The Hessian matrix and critical points of a scalar field Definition
• The Jacobian determinant of a square-dimensional C¹ map is the determinant of its Jacobian matrix Definition
• The Laplacian of a C² function and of a C² vector field Definition
• The rank of a derivative and constant-rank Euclidean maps Definition
• The variational equation along an ODE solution Definition
• Wirtinger operators in ℂᵐ Definition
• A Euclidean sphere is a regular level set with tangent hyperplanes Example
• A function with vanishing Laplacian has zero boundary flux of its gradient on the unit box Example
• A positive-definite quadratic ellipsoid is a regular level set Example
• r² sin(1/r) is differentiable at the origin with a discontinuous gradient Example
• The inverse-square field is divergence free, and its flux through the sphere bounding the translated unit ball vanishes 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 polynomial map (x,y)↦(1+x+2y+x², 2x+3y+xy) and its Jacobian 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: 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 cyclic permutation of the coordinates of ℝ³ preserves Jordan measurability and integrals Lemma
• A vector line integral along an image arc is the parameter line integral of the pulled-back field Lemma
• Divergence and curl are linear and satisfy the scalar product rules Lemma
• Each coordinate of the oriented area vector is the Jacobian determinant of the matching cyclic projection Lemma
• The curl flux integrand of a C² patch is a two-dimensional curl of the pulled-back field Lemma

…and 16 more results.

Dependency tree · two levels

31 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)