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- 07:47, 18 May 2024 Arc Length of Curve in Polar Coordinates/Function of Angle (hist | edit) [1,240 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $C$ be a curve embedded in a polar plane. <onlyinclude> Let the radial coordinate $r$ of $C$ be defined as a function of the angular coordinate $\theta$: :$r = \map f \theta$ The '''arc length''' $s$ of $C$ between $\theta = \alpha$ and $\theta = \beta$ is defined as: :$\ds s := \in...")
- 07:38, 18 May 2024 Arc Length of Curve in Polar Coordinates/Function of Radius (hist | edit) [1,178 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $C$ be a curve embedded in a polar plane. Let the angular coordinate $\theta$ of $C$ be defined as a function of the radial coordinate $r$: :$\theta = \map f r$ The '''arc length''' $s$ of $C$ between $r = u$ and $r = v$ is defined as: :$\ds s := \int_u^v \sqrt {1 + r^...") originally created as "Arc Length of Curve in Polar Coordinates"
- 10:32, 17 May 2024 Lemniscate of Bernoulli from Tangents to Rectangular Hyperbola (hist | edit) [1,133 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\KK$ be a rectangular hyperbola. Let $\LL$ be the locus of the foot of the perpendicular from the origin to the tangents to $\KK$. Then $\LL$ is the '''lemniscate of Bernoulli'''. </onlyinclude> == Proof == :File:Lemniscat...")
- 23:26, 16 May 2024 Legendre Symbol/Examples/2 over 7 (hist | edit) [962 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Use of Legendre Symbol == <onlyinclude> :$\paren {\frac 2 7} = 1$ </onlyinclude> == Proof == From Quadratic Residue Examples: $7$: {{:Quadratic Residue/Examples/7}} That is, $2$ is a quadratic residue of $7$: :$2^2 \equiv 4 \pmod 7$ Hence the result by definition of '''Legendre symbol'''. {{qed}} == Sources == * {{BookRefere...")
- 23:26, 16 May 2024 Legendre Symbol/Examples/3 over 7 (hist | edit) [941 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Use of Legendre Symbol == <onlyinclude> :$\paren {\frac 3 7} = -1$ </onlyinclude> == Proof == From Quadratic Residue Examples: $: {{:Quadratic Residue/Examples/7}} That is, $3$ is not a quadratic residue of $7$. Hence the result by definition of '''Legendre symbol'''. {{qed}} == Sources == * {{BookReference|The Penguin Dic...")
- 23:18, 16 May 2024 Legendre Symbol/Examples (hist | edit) [379 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Legendre Symbol == <onlyinclude> === Example: $\paren {\frac 2 7}$ === {{:Legendre Symbol/Examples/2 over 7}} === Example: $\paren {\frac 3 7}$ === {{:Legendre Symbol/Examples/3 over 7}}</onlyinclude> Category:Examples of Use of Legendre Symbol")
- 17:39, 16 May 2024 Legendre Polynomial/Examples/P4 (hist | edit) [1,101 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Legendre Polynomial == The $4$th '''Legendre polynomial''' is: <onlyinclude> :$\map {P_4} x = \dfrac 1 8 \paren {35 x^4 - 30 x^2 + 3}$ </onlyinclude> == Proof == {{ProofWanted|expand Generating Function for Legendre Polynomials}} Category:Examples of Legendre Polynomials")
- 17:39, 16 May 2024 Legendre Polynomial/Examples/P3 (hist | edit) [1,049 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Legendre Polynomial == The $3$rd '''Legendre polynomial''' is: <onlyinclude> :$\map {P_3} x = \dfrac 1 2 \paren {5 x^3 - 3 x}$ </onlyinclude> == Proof == {{ProofWanted|expand Generating Function for Legendre Polynomials}} Category:Examples of Legendre Polynomials")
- 17:38, 16 May 2024 Legendre Polynomial/Examples/P2 (hist | edit) [1,471 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Legendre Polynomial == The $2$nd '''Legendre polynomial''' is: <onlyinclude> :$\map {P_2} x = \dfrac 1 2 \paren {3 x^2 - 1}$ </onlyinclude> == Proof == {{ProofWanted|expand Generating Function for Legendre Polynomials}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Legendre Polynomial/Examples/P1|n...")
- 17:37, 16 May 2024 Legendre Polynomial/Examples/P1 (hist | edit) [2,278 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Legendre Polynomial == The $1$st '''Legendre polynomial''' is: <onlyinclude> :$\map {P_1} x = x$ </onlyinclude> == Proof == {{ProofWanted|expand Generating Function for Legendre Polynomials}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Legendre Polynomial/Examples/P0|next = Legendre Polynomial/Ex...")
- 17:36, 16 May 2024 Legendre Polynomial/Examples/P0 (hist | edit) [1,856 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Legendre Polynomial == The zeroth '''Legendre polynomial''' is: <onlyinclude> :$\map {P_0} x = 1$ </onlyinclude> == Proof == {{ProofWanted|expand Generating Function for Legendre Polynomials}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Legendre Polynomial/Examples|next = Le...")
- 17:24, 16 May 2024 Length of Legendre Polynomial (hist | edit) [3,158 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\map {P_n} x$ denote the '''Legendre polynomial of order $n$'''. Let $\norm {\map {P_n} x}$ denote the '''length''' of $\map {P_n} x$. Then: :$\norm {\map {P_n} x} := \sqrt {\frac 2 {2 n + 1} }$ </onlyinclude> == Proof == Applying Bonnet's Recursion Formula for $n - 1$: :$n \map {P_n} x = \paren {2 n - 1} x \map {P_{n - 1} } x - \paren {n - 1} \map {...")
- 16:31, 16 May 2024 Bonnet's Recursion Formula (hist | edit) [396 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\map {P_n} x$ denote the Legendre polynomial of order $n$. '''Bonnet's Recursion Formula''' states: :$\paren {n + 1} \map {P_{n + 1} } x = \paren {2 n + 1} x \map {P_n} x - n \map {P_{n - 1} } x$ </onlyinclude> == Proof == {{ProofWanted}} {{Namedfor|Pierre Ossian Bonnet|cat = Bonnet}} Category:Legendre Polynomials")
- 09:50, 15 May 2024 Method of Least Squares (Approximation Theory)/Examples/Arbitrary Example 1 (hist | edit) [1,703 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Method of Least Squares == <onlyinclude> Let $B$ be a false balance. $2$ items are weighed on $B$: first individually and then together. The recorded weights are: :$17 \, \mathrm g$ and $25 \, \mathrm g$ for the separate items :$40 \, \mathrm g$ for the combined weight. The '''Definition:Method of Least Square...")
- 21:05, 14 May 2024 Method of Least Squares (Approximation Theory)/Examples (hist | edit) [391 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Method of Least Squares == <onlyinclude> === Arbitrary Example === {{:Method of Least Squares (Approximation Theory)/Examples/Arbitrary Example 1}}</onlyinclude> Category:Examples of Method of Least Squares (Approximation Theory)")
- 20:50, 14 May 2024 Principle of Least Action/Historical Note (hist | edit) [744 bytes] Prime.mover (talk | contribs) (Created page with "== Historical Note on Principle of Least Action == <onlyinclude> The '''Principle of Least Action''' was first put forward by {{AuthorRef|Pierre Louis Moreau de Maupertuis}} in $1744$, and since modified. </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Principle of Least Action|next = Definition:Lowest Common Denominator/Also known as|entry = least action, principle o...")
- 20:49, 14 May 2024 Principle of Least Action (hist | edit) [1,434 bytes] Prime.mover (talk | contribs) (Created page with "== Physical Law == <onlyinclude> The '''Principle of Least Action''' is a physical law which states the following: Let $S$ be a dynamical system moving under conservative forces from point $A$ to point $B$. The motion of $S$ takes place in such a way that the action has a Def...")
- 09:38, 14 May 2024 Newton's Laws of Motion/Also known as (hist | edit) [430 bytes] Prime.mover (talk | contribs) (Created page with "== Newton's Laws of Motion: Also known as == <onlyinclude> '''Newton's Laws of Motion''' can also be referred to as just '''the Laws of Motion'''. </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Strong Law of Large Numbers|next = Symbols:Abbreviations/L/LCD|entry = laws of motion}} Category:Definitions/Newton's Laws of Motion")
- 16:18, 13 May 2024 Law of Species/Examples (hist | edit) [242 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Law of Species == <onlyinclude> === Arbitrary Example === {{:Law of Species/Examples/Arbitrary Example 1}}</onlyinclude> Category:Examples of Use of Law of Species")
- 16:18, 13 May 2024 Law of Species/Also known as (hist | edit) [1,028 bytes] Prime.mover (talk | contribs) (Created page with "== Law of Species: Also known as == <onlyinclude> The '''Law of Species''' is also known as the '''Law of Quadrants'''. </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Law of Species|next = Law of Species/Examples/Arbitrary Example 1|entry = species}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev =...")
- 16:12, 13 May 2024 Law of Species (hist | edit) [1,861 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $T$ be a right spherical triangle whose angles are $A$, $B$ and $C$ and whose respective sides opposite those angles are $a$, $b$ and $c$. Let $C$ be the right angle of $T$. Then: :$(1): $A$ and $a$ are of the same species, and $B$ and $B$ are of...")
- 07:32, 12 May 2024 Latin Square/Examples/Order 3 (hist | edit) [828 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Latin Square of order $3$ == <onlyinclude> This is an example of a '''Latin square of order $3$''': $\quad \begin {array} {|ccc|} \hline A & B & C \\ C & A & B \\ B & C & A \\ \hline \end {array}$ </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edi...")
- 08:23, 11 May 2024 Spherical Coordinate Form of Laplace's Equation (hist | edit) [970 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Laplace's equation can be expressed in spherical coordinates as: :$\dfrac 1 {r^2} \map {\dfrac \partial {\partial r} } {r^2 \dfrac {\partial V} {\partial r} } + \dfrac 1 {r^2 \sin^2 \theta} \dfrac {\partial^2 V} {\partial \phi^2} + \dfrac 1 {r^2 \sin \theta} \map {\dfrac \partial {\partial \theta} } {\sin \theta \dfrac {\partial V} {\partial \theta} } = 0$ </onlyinclude> =...")
- 20:49, 10 May 2024 Kakutani's Fixed Point Theorem (hist | edit) [680 bytes] Kennethw (talk | contribs) (Created page with "==Theorem== Let $S \subset \R^n$ be nonempty, compact, and convex and let $\Phi \to 2^{\R^n}$ be a correspondence. If the following conditions are satisfied: $\Phi(x)$ is nonempty and convex for all x. $\Phi(.)$ is upper hemi-continuous. $\Phi$ has a fixed point.") Tag: Visual edit: Switched
- 05:43, 10 May 2024 Beer-Lambert-Bouguer Law/Also known as (hist | edit) [599 bytes] Prime.mover (talk | contribs) (Created page with "== Beer-Lambert-Bouguer Law: Also known as == <onlyinclude> The '''Beer-Lambert-Bouguer Law''' is also known as: :the '''Lambert-Beer Law''' :'''Beer's Law''' :the '''Beer-Lambert Law''' :'''Lambert's Law of Absorption''' :'''Lambert's Law'''. </onlyinclude> == Sources == * {{BookReference|Differential Equations|1972|George F. Simmons|prev = Temperature of Body under Newton's Law of Cooling|next = Beer-Lambert-Bouguer Law}}: $1$: The Nature...")
- 19:04, 9 May 2024 N over 2 times Reciprocal of 1 Plus n Squared x Squared to the Power of 3/2 Delta Sequence (hist | edit) [6,665 bytes] Hbghlyj (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\sequence {\map {\delta_n} x}$ be a sequence such that: :$\ds \map {\delta_n} x := \frac n 2 \frac 1 {\paren{1 + n^2 x^2}^{3 / 2} }$ Then $\sequence {\map {\delta_n} x}_{n \mathop \in {\N_{>0} } }$ is a delta sequence. That is, in the distributional sense it holds that: :$\ds \lim_{n \mathop \to \infty} \map {\delta_n} x = \map \delta x$ or :$\ds \lim_{n \m...")
- 09:56, 9 May 2024 Euler-Lagrange Equation for Conservative System (hist | edit) [1,167 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $P$ be a conservative system composed of $n \in \N$ particles. The Euler-Lagrange equation for $P$ is given by: :$\map {\dfrac \d {\d t} } {\dfrac {\partial L} {\partial \dot q_j} } - \dfrac {\partial \L} {\partial q_j} = 0$ where: :$L$ denotes the Lagrangian of $P$ :$q_j$ denotes the...")
- 07:49, 9 May 2024 Lagrange's Four Square Theorem/Examples/59 (hist | edit) [964 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Use of Lagrange's Four Square Theorem == <onlyinclude> $59$ can be expressed as the sum of $4$ squares thus: {{begin-eqn}} {{eqn | l = 59 | r = 7^2 + 3^2 + 1^2 + 0^2 }} {{eqn | r = 5^2 + 5^2 + 3^2 + 0^2 }} {{eqn | r = 5^2 + 4^2 + 3^2 + 3^2 }} {{end-eqn}} </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition...")
- 07:44, 9 May 2024 Lagrange's Four Square Theorem/Examples/23 (hist | edit) [862 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Use of Lagrange's Four Square Theorem == <onlyinclude> $23$ can be expressed as the sum of $4$ squares thus: :$23 = 3^2 + 3^2 + 2^2 + 1^2$ </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Lagrange's Four Square Theorem/Examples/1|next = Lagrange's Four Square Theorem/Examples/59|entry = Lagrange's the...")
- 07:42, 9 May 2024 Lagrange's Four Square Theorem/Examples/1 (hist | edit) [875 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Use of Lagrange's Four Square Theorem == <onlyinclude> $1$ can be trivially expressed as the sum of $4$ squares: :$1 = 1^2 + 0^2 + 0^2 + 0^2$ </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Lagrange's Four Square Theorem/Historical Note/Mistake|next = Lagrange's Four Square Theorem/Examples/23|entry...")
- 07:27, 9 May 2024 Lagrange's Four Square Theorem/Examples (hist | edit) [501 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Lagrange's Four Square Theorem == <onlyinclude> === Example: Lagrange's Four Square Theorem/Examples/1|Example: $1$$ === {{:Lagrange's Four Square Theorem/Examples/1}} === Example: $ === {{:Lagrange's Four Square Theorem/Examples/23}} === Example: $ === {{:Lagrange's Four Square Theorem/Examples/59}}</onlyinclude> Category:Examples of Use of Lagran...")
- 06:49, 9 May 2024 Equivalence of Formulations of Lagrange Interpolation Formula (hist | edit) [870 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{TFAE}} === Formulation Lagrange Interpolation Formula/Formulation 1|Formulation $1$$ === {{:Lagrange Interpolation Formula/Formulation 1}} === Formulation $ === {{:Lagrange Interpolation Formula/Formulation 1}} == Proof == {{ProofWanted}} == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Lagrange Interpolation Formula/Formulation...")
- 06:41, 9 May 2024 Lagrange Interpolation Formula/Formulation 2 (hist | edit) [2,227 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $f \R \to \R$ be a real function. Let $f$ have known values $y_i = \map f {x_i}$ for $n \in \set {0, 1, \ldots, n}$. Let a value $y' = \map f {x'}$ be required to be estimated at some $x'$. Then: {{begin-eqn}} {{eqn | l = y' | o = \approx | r = \dfrac {y_1 \paren {x' - x_2} \paren {x' - x_3} \cdots \paren...")
- 17:53, 8 May 2024 Lagrange Interpolation Formula/Formulation 1 (hist | edit) [2,895 bytes] Prime.mover (talk | contribs) (Created page with "{{MissingLinks}} == Theorem == <onlyinclude> Let $\tuple {x_0, \ldots, x_n}$ and $\tuple {a_0, \ldots, a_n}$ be ordered tuples of real numbers such that $x_i \ne x_j$ for $i \ne j$. Then there exists a unique polynomial $P \in \R \sqbrk X$ of degree at most $n$ such that: :$\map P {x_i} = a_i$ for all $i \in \set {...")
- 17:31, 8 May 2024 Lagrange Interpolation Formula/Also known as (hist | edit) [850 bytes] Prime.mover (talk | contribs) (Created page with "== Lagrange Interpolation Formula: Also known as == <onlyinclude> The '''Lagrange interpolation formula''' can al;so be styled as '''Lagrange's interpolation formula'''. </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Lagrange's Method of Multipliers/Examples/Arbitrary Example 1|next = Lagrange Inter...")
- 15:14, 8 May 2024 Lagrange's Method of Multipliers/Examples/Arbitrary Example 1 (hist | edit) [2,606 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Lagrange's Method of Multipliers == <onlyinclude> Let it be required to find the maximum of $u = x y$ subject to the constraint $x + y = 1$. We write: :$L = x y + \lambda \paren {x + y - 1}$ Differentiation {{WRT|Differentiation}} $x$, $y$ and $\lambda$ and equating to zero gives: {{begin-eqn}} {{eqn | l...")
- 14:29, 8 May 2024 Lagrange's Method of Multipliers/Examples (hist | edit) [358 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Use of Lagrange's Method of Multipliers == <onlyinclude> === Arbitrary Example === {{:Lagrange's Method of Multipliers/Examples/Arbitrary Example 1}}</onlyinclude> Category:Examples of Use of Lagrange's Method of Multipliers")
- 22:43, 7 May 2024 Kruskal's Algorithm/Also known as (hist | edit) [472 bytes] Prime.mover (talk | contribs) (Created page with "== Kruskal's Algorithm: Also known as == <onlyinclude> It is clear that '''Kruskal's Algorithm''' is a greedy algorithm: at each stage the minimum possible weight is chosen, without any analysis as to whether there may be a combination of larger weights which may produce a smaller-weight spanning tree. For this reason, it is sometimes called '''Kruskal's Greedy Algorithm'''. </onlyinclude> Category:Kruskal's Algorithm")
- 10:30, 7 May 2024 Cauchy-Kovalevsky Theorem/Historical Note (hist | edit) [899 bytes] Prime.mover (talk | contribs) (Created page with "== Historical Note on Cauchy-Kovalevsky Theorem == <onlyinclude> The Cauchy-Kovalevsky theorem was a generalization of a theorem of {{AuthorRef|Augustin Louis Cauchy}}'s on partial differential equations. It was given by {{AuthorRef|Sofia Vasilyevna Kovalevskaya}} in $1975$. </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpag...")
- 10:29, 7 May 2024 Cauchy-Kovalevsky Theorem (hist | edit) [2,222 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\KK$ denote the field of either the real or complex numbers. Let $V = \KK^m$. Let $W = \KK^n$. Let $A_1, A_2, \ldots, A_{n − 1}$ be analytic functions defined on some neighborhood of $\tuple {0, 0}$ in $W \times V$, taking values in the $m \times m$ Definition:Squar...")
- 10:28, 7 May 2024 Cauchy-Kovalevsky Theorem/Also known as (hist | edit) [274 bytes] Prime.mover (talk | contribs) (Created page with "== Cauchy-Kovalevsky Theorem: Also known as == <onlyinclude> The '''Cauchy-Kovalevsky Theorem''' is also known as the '''Cauchy-Kovalevskaya Theorem'''. </onlyinclude> Category:Cauchy-Kovalevsky Theorem")
- 06:13, 7 May 2024 Planar Diagram/Examples/Trefoil Knot (hist | edit) [714 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Planar Diagram == <onlyinclude> This is the planar diagrams of the two types of right-hand trefoil knot: :300px </onlyinclude> == Sources == * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Definition:Planar Diagram|next = Definition:Reidemeister Move|entry = knot t...")
- 22:25, 6 May 2024 Knot (Knot Theory)/Examples/Right-Handed Trefoil (hist | edit) [216 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Knot == <onlyinclude> '''Right-handed trefoil''': :300px </onlyinclude> Category:Examples of Knots")
- 22:25, 6 May 2024 Knot (Knot Theory)/Examples/Left-Handed Trefoil (hist | edit) [213 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Knot == <onlyinclude> '''Left-handed trefoil''': :300px </onlyinclude> Category:Examples of Knots")
- 22:21, 6 May 2024 Knot (Knot Theory)/Examples/Trefoil Knot (hist | edit) [508 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Knot == <onlyinclude> These are the planar diagrams of the two types of trefoil knot: === Left-Handed Trefoil === {{:Definition:Left-Handed Trefoil Knot}} === Right-Handed Trefoil === {{:Definition:Right-Handed Trefoil Knot}}</onlyinclude> Category:Examples of Knots")
- 08:09, 6 May 2024 Knot (Knot Theory)/Examples (hist | edit) [229 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Knots == <onlyinclude> === Trefoil Knot === {{:Knot (Knot Theory)/Examples/Trefoil Knot}}</onlyinclude> Category:Examples of Knots")
- 08:02, 6 May 2024 Planar Diagram/Examples (hist | edit) [237 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Planar Diagrams == <onlyinclude> === Trefoil Knot === {{:Planar Diagram/Examples/Trefoil Knot}}</onlyinclude> Category:Examples of Planar Diagrams")
- 06:14, 6 May 2024 Outer Jordan Content of Right Triangle (hist | edit) [2,809 bytes] CircuitCraft (talk | contribs) (Created page with "== Theorem == Let $T \subseteq \R^2$ be defined as: :$T = \set {\tuple {x, y} \in \R^2 : x \ge 0 \land y \ge 0 \land x + y \le 1}$ Then: :$\map {m^*} T = \dfrac 1 2$ == Proof == Let $\epsilon > 0$ be arbitrary. By the Axiom of Archimedes, let $n \in \N$ such that: :$n > 2 \epsilon$ Define $C \subseteq \powerset {\R^2}$ as: :$C = \set {\closedint {\dfrac p n} {\dfrac {p + 1} n} \times \closedint {\dfrac q n} {\dfrac {q + 1} n} : p, q \in \set {0, 1, \dotsc, n...")
- 10:11, 4 May 2024 Kinetic Energy of Body at Constant Angular Speed (hist | edit) [996 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $B$ be a body rotating at an angular speed $\omega$ about some axis of rotation $R$. Let $I$ denote the moment of inertia of $B$ about $R$. Then the kinetic energy $T$ of $B$ brought about by this rotation is given by: :$T = \dfrac {I \omega^2} 2$...")
- 10:06, 4 May 2024 Kinetic Energy of Body at Constant Speed (hist | edit) [925 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $B$ be a body of mass $m$ moving at a speed of $v$. Let $v$ be considerably less than the speed of light. Then the kinetic energy $T$ of $B$ is given by: :$T \approx \dfrac {m v^2} 2$ {{expand|Add a subpage giving the relativistic KE of $B$}} == Proof == {{ProofWanted}} == Sources == * {{BookReference|...")