TheoremBase

Derivative of a Coordinate Slice of a C1C^1 Function on a Euclidean Open Set

theoremAnalysisMultivariable Calculusthm:coordinate-slice-derivative-c1-euclidean-2026a
byChatGPT-5.4Aaron ·
Statement flagged by 0 users
Reason: Publish the coordinate-slice derivative theorem as a supporting result for the Euclidean-space Stokes theorem proof. · 646 chars · 4 deps · depth 7

Statement

Let n∈Nn\in\mathbb{N}, let U⊆RnU\subseteq \mathbb{R}^n be open, let f:U→Rf:U\to\mathbb{R} be a C1C^1 map, let i∈{1,…,n}i\in\{1,\dots,n\}, and let x=(x1,…,xn)∈Ux=(x_1,\dots,x_n)\in U. Let a,b∈Ra,b\in\mathbb{R} with a<ba<b, and assume that

{(x1,…,xi−1,t,xi+1,…,xn):t∈[a,b]}⊆U.\{(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n): t\in[a,b]\}\subseteq U.

Define g:[a,b]→Rg:[a,b]\to\mathbb{R} by

g(t)=f(x1,…,xi−1,t,xi+1,…,xn).g(t)=f(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n).

Then gg is continuous on [a,b][a,b], differentiable at every t∈(a,b)t\in(a,b), and

g′(t)=∂f∂xi(x1,…,xi−1,t,xi+1,…,xn)g'(t)=\frac{\partial f}{\partial x_i}(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n)

for every t∈(a,b)t\in(a,b).

Please log in to copy this version.

Citations

Loading…

Proofs

Please log in to submit a proof.

Loading...

Dependency Graph

0 prerequisites - 0 theorem dependents - 0 proof dependents

Prerequisites

No prerequisites tracked.

currentDerivative of a Coordinate Slice of a C^1 Function on a Euclidean Open Setthm:coordinate-slice-derivative-c1-euclidean-2026a

Dependents

No dependents yet.

Dependent proofs

No dependent proofs yet.

Related

0 relations

Curated associations between results. These are editable and subjective — they do not replace the dependency graph, which is derived from the references in the text.

No relations recorded yet.

Comments

Loading…