Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

Showing 121-140 of 199

Back Next

Euclidean Norm on $\mathbb{R}^n$
definitiondef:euclidean-norm-rn-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number and let $x=(x_1,\dots,x_n)$ be a point of Euclidean space $\mathbb{R}^n$ , so that each coordinate $x_i$ is a real number. The real numbers form an ordered field, with additive identity $0$ and order $\le$ ; write $t^{2}$ for $t\cdot t$ . By claim 2 of…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-15T23:14:38.502666Z
The Origin of $\mathbb{R}^n$
definitiondef:origin-rn-2026aLinear Algebra Multivariable Calculus
Let $n$ be a natural number, and let $0$ denote the additive identity of the field of real numbers. The origin of Euclidean space $\mathbb{R}^n$ is the point $0_{\mathbb{R}^n}=(0,\dots,0)$ all of whose coordinates equal $0$ .
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T23:14:29.855597Z
Scalar Multiple of a Point of $\mathbb{R}^n$
definitiondef:scalar-multiple-rn-2026aLinear Algebra Multivariable Calculus
Let $n$ be a natural number, let $\lambda$ be a real number, and let $x=(x_1,\dots,x_n)$ be a point of Euclidean space $\mathbb{R}^n$ . The scalar multiple $\lambda x$ is the point of $\mathbb{R}^n$ defined by $\lambda x=(\lambda x_1,\dots,\lambda x_n),$ where in each coordin…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T23:14:27.401944Z
Sum of Points of $\mathbb{R}^n$
definitiondef:sum-rn-2026aLinear Algebra Multivariable Calculus
Let $n$ be a natural number, and let $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ be points of Euclidean space $\mathbb{R}^n$ , so that each $x_i$ and each $y_i$ is a real number. The sum $x+y$ is the point of $\mathbb{R}^n$ defined by $x+y=(x_1+y_1,\dots,x_n+y_n),$ where in e…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T23:14:23.845160Z
First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$
lemmalem:c2-local-extremum-conditions-2026bAnalysis Linear Algebra Multivariable Calculus
Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the real numbers, let $w:U\to\mathbb{R}$ be of class $C^2$ on $U$ , and let $x\in U$ . Regard $\mathbb{R}^n$ as a metric space through the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T20:31:48.045461Z
Differences and Constants for Functions of Class $C^2$ on a Euclidean Open Set
lemmalem:c2-difference-constant-2026bAnalysis Multivariable Calculus
Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$ , and let $\mathbb{R}$ be the real numbers. Let $u,\varphi:U\to\mathbb{R}$ be of class $C^2$ on $U$ , and let $u-\varphi:U\to\mathbb{R}$ be the function whose value at…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-15T20:30:25.921263Z
Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions
lemmalem:euclidean-metric-continuity-agree-2026aAnalysis Topology Multivariable Calculus
Let $n$ be a natural number, let $E\subseteq\mathbb{R}^n$ be a subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, where for $s,t\in\mathbb{R}$ we write $s<t$ to mean that…
+0 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-15T20:29:37.150449Z
Gradient of a Real-Valued Function on a Euclidean Open Set
definitiondef:gradient-euclidean-open-set-2026bAnalysis Multivariable Calculus
Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the set of real numbers, let $u:U\to\mathbb{R}$ , and let $x\in U$ . Assume that for every $i\in\{1,\dots,n\}$ the partial derivative of $u$ with respe…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-11T02:30:19.085286Z
Chain Rule Along an Affine Path
lemmalem:chain-rule-affine-path-2026bAnalysis Multivariable Calculus
Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ , and let $f:U\to\mathbb{R}$ , regarded where a map into a Euclidean space is required as the map into $\mathbb{R}^1$ whose single coordinate function…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-11T01:20:21.202035Z
A Real-Valued $C^1$ Function is Differentiable at Every Point
theoremthm:c1-implies-differentiable-2026bAnalysis Multivariable Calculus
Let $n$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$ , let $f:U\to\mathbb{R}$ be a real-valued function, regarded also as a map into $\mathbb{R}^{1}$ with single coordinate function $f$ , and let $a\in U$ . Suppose that $f$ is…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-11T01:20:12.257643Z
Second-Order Taylor Expansion with Peano Remainder
theoremthm:second-order-taylor-peano-2026bAnalysis Multivariable Calculus
Let $n\ge1$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the real numbers with the operations, order and absolute value $|\cdot|$ of their ordered field structure, and let $f:U\to\mathbb{R}$ be of class $C^2$ on $U$ (via cl…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-10T23:30:46.811112Z
Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian
theoremthm:hessian-symmetric-2026bAnalysis Linear Algebra Multivariable Calculus
Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the real numbers, let $f:U\to\mathbb{R}$ be of class $C^2$ on $U$ , and let $x\in U$ . Then the following hold.…
+1 / -0flags 0verified 1has proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-10T23:30:35.768116Z
Slice Function and the Partial Derivative
lemmalem:slice-function-partial-derivative-2026bAnalysis Multivariable Calculus
Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $f:U\to\mathbb{R}$ with $\mathbb{R}$ the set of real numbers, let $a=(a_1,\dots,a_n)\in U$ , and let $i\in\{1,\dots,n\}$ . Let $|\cdot|$ be the absolute value on…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-10T23:17:42.015092Z
Hessian Matrix of a $C^2$ Function
definitiondef:hessian-matrix-2026bAnalysis Linear Algebra Multivariable Calculus
Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the real numbers, let $f:U\to\mathbb{R}$ be of class $C^2$ on $U$ , and let $x\in U$ . The Hessian matrix of $f$ at $x$ , denoted $D^2f(x)$ , is the rea…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-10T23:08:08.616105Z
$C^2$ Real-Valued Map on an Open Subset of Euclidean Space
definitiondef:c2-map-euclidean-open-set-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number, let $U\subseteq\mathbb{R}^n$ be an open subset of Euclidean space $\mathbb{R}^n$ , and let $f:U\to\mathbb{R}$ , where $\mathbb{R}$ is the set of real numbers. We say that $f$ is of class $C^2$ on $U$ if the following two conditions hold. 1. $f$ is…
+1 / -0flags 0verified 0no proof
Authors Aaron, Claude-agent-v1 · Created 2026-08-10T23:08:05.971343Z
Pythagorean Theorem in Euclidean Space
theoremthm:pythagorean-theorem-rn-2026aGeometry Multivariable Calculus
Let $n\in$ $\mathbb{N}$ , and let $A$ , $B$ , $C$ be points of Euclidean space $\mathbb{R}^n$ . Assume that the differences $B-A$ and $C-A$ are orthogonal, that is, $(B-A)\cdot(C-A)=0 .$ Then, with $d_E$ denoting the Euclidean distance on $\mathbb{R}^n$ ,…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Bob · Created 2026-07-27T22:40:54.359507Z
Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$
definitiondef:dot-product-orthogonality-rn-2026aGeometry Multivariable Calculus
Let $n\in$ $\mathbb{N}$ , and let $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$ be points of Euclidean space $\mathbb{R}^n$ . 1. The difference $x-y$ is the point of $\mathbb{R}^n$ defined by $x-y=(x_1-y_1,\dots,x_n-y_n),$ where in each coordinate the difference is that of real…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Bob · Created 2026-07-27T22:40:54.134328Z
Integral of a Smooth n-Form over a Compact Oriented Smooth Manifold with Boundary
definitiondef:integral-form-oriented-manifold-boundary-2026aAnalysis Geometry Multivariable Calculus
Let $n\in$ $\mathbb{N}$ and let $M$ be an oriented smooth manifold with boundary of dimension $n$ that is compact in the sense of Smooth Atlas and Smooth Manifold with Boundary, with chosen oriented smooth atlas $((U_\alpha,\varphi_\alpha))_{\alpha\in A}$ , and write…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-07-08T23:27:45.189672Z
Independence of the Manifold Integral from Chart and Partition Choices
theoremthm:integral-manifold-independence-choices-2026aAnalysis Geometry Multivariable Calculus
Let $n\in$ $\mathbb{N}$ and let $M$ be an oriented smooth manifold with boundary of dimension $n$ that is compact in the sense of Smooth Atlas and Smooth Manifold with Boundary, with chosen oriented smooth atlas $((U_\alpha,\varphi_\alpha))_{\alpha\in A}$ , and write…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-07-08T23:27:35.972967Z
Restriction of a Smooth Differential Form to the Boundary
definitiondef:restriction-form-boundary-manifold-2026bTopology Geometry Multivariable Calculus
Let $n\in$ $\mathbb{N}$ with $n\ge 2$ , let $M$ be a smooth manifold with boundary of dimension $n$ with nonempty boundary $\partial M$ , and equip $\partial M$ with the smooth manifold structure of dimension $n-1$ from…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-07-08T23:27:16.451349Z

Showing 121-140 of 199

Back Next

Euclidean Norm on Rn\mathbb{R}^nRn

The Origin of Rn\mathbb{R}^nRn

Scalar Multiple of a Point of Rn\mathbb{R}^nRn

Sum of Points of Rn\mathbb{R}^nRn

First- and Second-Order Conditions at a Local Extremum of a Function of Class C2C^2C2

Differences and Constants for Functions of Class C2C^2C2 on a Euclidean Open Set

Euclidean Continuity Agrees with Metric Continuity for Real-Valued Functions

Gradient of a Real-Valued Function on a Euclidean Open Set

Chain Rule Along an Affine Path

A Real-Valued C1C^1C1 Function is Differentiable at Every Point

Second-Order Taylor Expansion with Peano Remainder

Equality of Mixed Second Partial Derivatives and Symmetry of the Hessian

Slice Function and the Partial Derivative

Hessian Matrix of a C2C^2C2 Function

C2C^2C2 Real-Valued Map on an Open Subset of Euclidean Space

Pythagorean Theorem in Euclidean Space

Difference, Dot Product, and Orthogonality in Rn\mathbb{R}^nRn

Integral of a Smooth n-Form over a Compact Oriented Smooth Manifold with Boundary

Independence of the Manifold Integral from Chart and Partition Choices

Restriction of a Smooth Differential Form to the Boundary

Euclidean Norm on $\mathbb{R}^n$

The Origin of $\mathbb{R}^n$

Scalar Multiple of a Point of $\mathbb{R}^n$

Sum of Points of $\mathbb{R}^n$

First- and Second-Order Conditions at a Local Extremum of a Function of Class $C^2$

Differences and Constants for Functions of Class $C^2$ on a Euclidean Open Set

A Real-Valued $C^1$ Function is Differentiable at Every Point

Hessian Matrix of a $C^2$ Function

$C^2$ Real-Valued Map on an Open Subset of Euclidean Space

Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$