Second Order ODE/x^2 y'' = 2 x y' + (y')^2

The second order ODE: :$x^2 y'' = 2 x y' + \paren {y'}^2$ has the general solution: :$y = -\dfrac {x^2} 2 - C_1 x - {C_1}^2 \, \map \ln {x - C_1} + C_2$


Second Order ODE/y y'' = y^2 y' + (y')^2

The second order ODE: :$(1): \quad y y'' = y^2 y' + \paren {y'}^2$ subject to the initial conditions: :$y = -\dfrac 1 2$ and $y' = 1$ when $x = 0$ has the particular solution: :$2 y - 3 = 8 y \, \map \exp {\dfrac {3 x} 2}$


Second Order ODE/y'' = 1 + (y')^2

The second order ODE: :$(1): \quad y'' = 1 + \paren {y'}^2$ has the general solution: :$y = \map {\ln \sec} {x + C_1} + C_2$


Velocity of Bead on Brachistochrone

Consider a wire bent into the shape of an arc of a cycloid $C$ and inverted so that its cusps are uppermost and on the same horizontal line. Let $C$ be defined by Equation of Cycloid embedded in a cartesian plane: :$x = a \left({\theta - \sin \theta}\right)$ :$y = -a \left({1 - \cos \theta}\right)$ Let a bead $B$ be released from some point on the wire. Let $B$ slide without friction under the influence of a constant gravitational field exerting an acceleration $g$. Let $s_0$ be the arc length along the cycloid. Let $s$ be the arc length along the cycloid at any subsequent point in time. Then the speed $v$ of $B$ relative to $C$ is defined by the equation: :$4 a v^2 = g \left({ {s_0}^2 - s^2}\right)$


Open Ball is Subset of Open Ball

Let $M = \struct {A, d}$ be a metric space. Let $x, y$ be points of $A$. Then: : $\epsilon-\delta \ge \map d {x, y} \implies \map {B_\delta} y \subseteq \map {B_\epsilon} x$ where $\map {B_\epsilon} x$ denotes the open $\epsilon$-ball in $M = \struct {A, d}$.


Pursuit Curve of Boat in River

Consider a straight river $R$ whose parallel banks are aligned with the $y$-axis and the line $x = c$ of a cartesian plane. Let the current of $R$ have a constant and uniform speed $a$ in the negative $y$ direction. Let a boat $B$ be launched from the point $\left({c, 0}\right)$ and headed directly towards the origin with speed $b$ relative to the water. The path of $B$ is defined by the equation: :$c^k \left({y + \sqrt {x^2 + x^2} }\right) = x^{k + 1}$


Current in Electric Circuit/L, R, C in Series

Consider the electrical circuit $K$ consisting of: : a resistance $R$ : an inductance $L$ : a capacitance $C$ in series with a source of electromotive force $E$ which is a function of time $t$. :File:CircuitRLCseries.png The electric current $I$ in $K$ is given by the linear second order ODE: :$L \dfrac {\mathrm d^2 I} {\mathrm d t^2} + R \dfrac {\mathrm d I} {\mathrm d t} + \dfrac 1 C I = \dfrac {\mathrm d E} {\mathrm d t}$


Electric Charge in Electric Circuit/L, R, C in Series

Consider the electrical circuit $K$ consisting of: : a resistance $R$ : an inductance $L$ : a capacitance $C$ in series with a source of electromotive force $E$ which is a function of time $t$. :File:CircuitRLCseries.png The electric charge $Q$ in $K$ is given by the linear second order ODE: :$L \dfrac {\mathrm d^2 Q} {\mathrm d t^2} + R \dfrac {\mathrm d Q} {\mathrm d t} + \dfrac 1 C Q = E$


Current in Electric Circuit/L, R in Series

Consider the electrical circuit $K$ consisting of: :a resistance $R$ :an inductance $L$ in series with a source of electromotive force $E$ which is a function of time $t$. :File:CircuitRLseries.png The electric current $I$ in $K$ is given by the linear first order ODE: :$L \dfrac {\d I} {\d t} + R I = E$


Current in Electric Circuit/L, R in Series/Constant EMF at t = 0

Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$. The electric current $I$ in $K$ is given by the equation: :$I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$


Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 1

Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$. After a sufficiently long time, the electric current $I$ in $K$ is given by the equation: :$E_0  = R I$


Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 2

Let the electric current flowing in $K$ at time $t = 0$ be $0$. Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$. The electric current $I$ in $K$ is given by the equation: :$I = \dfrac {E_0} R \paren {1 - e^{-R t / L} }$


Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 3

Let the electric current flowing in $K$ at time $t = 0$ be $I_0$. Let  EMF imposed upon $K$ be zero. The electric current $I$ in $K$ is given by the equation: :$I = I_0 e^{-R t / L}$


Current in Electric Circuit/L, R in Series/Condition for Ohm's Law

Ohm's Law is satisfied by $K$ whenever the current $I$ is at a maximum or a minimum.


Current in Electric Circuit/L, R in Series/Minimum Current implies Increasing EMF

Let the current $I$ be at a minimum. Then the EMF $E$ is increasing.


Current in Electric Circuit/L, R in Series/Maximum Current implies Decreasing EMF

Let the current $I$ be at a maximum. Then the EMF $E$ is decreasing.


Second Order ODE/y y'' = (y')^2

The second order ODE: :$(1): \quad y y'' = \paren {y'}^2$ has the general solution: :$y = C_2 e^{C_1 x}$


First Order ODE/(1 - x y) y' = y^2

The first order ODE: :$(1): \quad \paren {1 - x y} y' = y^2$ has the general solution: :$x y = \ln y + C$


First Order ODE/(2 x + 3 y + 1) dx + (2 y - 3 x + 5) dy = 0

The first order ODE: :$(1): \quad \paren {2 x + 3 y + 1} \rd x + \paren {2 y - 3 x + 5} \rd y = 0$ has the general solution: :$3 \, \map \arctan {\dfrac {y + 1} {x - 1} } = \map \ln {\paren {y + 1}^2 + \paren {x - 1}^2} + C$


First Order ODE/x y' = Root of (x^2 + y^2)

is a homogeneous differential equation with solution: :$3 x^2 \ln x = y \sqrt {x^2 + y^2} + x^2 \, \map \ln {y + \sqrt {x^2 + y^2} } + y^2 + C x^2$


Bernoulli's Equation/y^2 dx = (x^3 - x y) dy

The first order ODE: :$(1): \quad y^2 \rd x = \paren {x^3 - x y} \rd y$ has the general solution: :$3 y = 2 x^2 + C x^2 y^2$


First Order ODE/(x^2 y^3 + y) dx = (x^3 y^2 - x) dy

The first order ODE: :$(1): \quad \paren {x^2 y^3 + y} \rd x = \paren {x^3 y^2 - x} \rd y$ has the general solution: :$-\dfrac 1 {2 x^2 y^2} = \ln \dfrac y x + C$


Second Order ODE/y y'' + (y')^2 - 2 y y' = 0

The second order ODE: :$y y'' + \paren {y'}^2 - 2 y y' = 0$ has the general solution: :$y^2 = C_2 e^{2 x} + C_1$


Rationals are Everywhere Dense in Sorgenfrey Line

$\Q$ is everywhere dense in the Sorgenfrey line.


Linear First Order ODE/x dy + y dx = x cosine x dx

The linear first order ODE: :$x \rd y + y \rd x = x \cos x \rd x$ has the general solution: :$x y = x \sin x + \cos x + C$


First Order ODE/x y dy = x^2 dy + y^2 dx

The first order ODE: :$(1): \quad x y \rd y = x^2 \rd y + y^2 \rd x$ has the general solution: :$y = x \ln y + C x$


Second Order ODE/y'' + 2 x (y')^2 = 0

The second order ODE: :$(1): \quad y'' + 2 x \paren {y'}^2 = 0$ has the general solution: :$C_1 \map \arctan {C_1 x} = y + C_2$


First Order ODE/(y over x^2) dx + (y - 1 over x) dy = 0

The first order ordinary differential equation: :$(1): \quad \dfrac y {x^2} \rd x + \paren {y - \dfrac 1 x} \rd y = 0$ is an exact differential equation with solution: :$\dfrac {y^2} 2 - \dfrac y x = C$


Focal Property of Parabola


First Order ODE/(3 x^2 over y^4 - 1 over y^2) dy - 2 x over y^3 dx = 0

The first order ODE: :$(1): \quad \paren {\dfrac {3 x^2} {y^4} - \dfrac 1 {y^2} } \rd y - \dfrac {2 x} {y^3} \rd x = 0$ is an exact differential equation with general solution: :$\dfrac 1 y - \dfrac {x^2} {y^3} = C$


First Order ODE/(y - 1 over x) dx + (x - y) dy = 0

The first order ODE: :$(1): \quad \paren {y - \dfrac 1 x} \rd x + \paren {x - y} \rd y = 0$ is an exact differential equation with solution: :$x y - \ln x - \dfrac {y^2} 2 + C$


First Order ODE/1 over x^3 y^2 dx + (1 over x^2 y^3 + 3 y) dy = 0

The first order ODE: :$(1): \quad \dfrac 1 {x^3 y^2} \rd x + \paren {\dfrac 1 {x^2 y^3} + 3 y} \rd y = 0$ is an exact differential equation with solution: :$-\dfrac 1 {2 x^2 y^2} + \dfrac {3 y^2} 2 = C$


Linear First Order ODE/y' - (y over x) = 3 x

The linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} - \dfrac y x = 3 x$ has the general solution: :$\dfrac y x = 3 x + C$ or: :$y = 3 x^2 + C x$


Characterization of Closure by Basis

Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathcal B \subseteq \tau$ be a basis. Let $A$ be a subset of $T$. Let $x$ be a point of $T$. Then $x \in A^-$ {{iff}}: :for every $U \in \mathcal B$: ::if $x \in U$ then $A \cap U \ne \varnothing$ where: :$A^-$ denotes the closure of $A$


Sorgenfrey Line is Separable

The Sorgenfrey line is separable.


First Order ODE/y dx + x dy = 0

The first order ODE: :$(1): \quad y \rd x + x \rd y = 0$ has the general solution: :$x y = C$


First Order ODE/y dy = k dx

Let $k \in \R$ be a real number. The first order ODE: :$y \, \dfrac {\d y} {\d x} = k$ has the general solution: :$y^2 = 2 k x + C$


First Order ODE/x dy = (y + y^3) dx

The first order ODE: :$(1): \quad x \dfrac {\d y} {\d x} = y + y^3$ has the general solution: :$y = \dfrac x {\sqrt {C^2 - x^2} }$


First Order ODE/y dy = k x dx

Let $k \in \R$ be a real number. The first order ODE: :$y \rd y = k x \rd x$ has the general solution: :$y^2 = k x^2 + C$


Primitive of Reciprocal of Root of x squared plus k

Let $k \in \R$. Then: :$\displaystyle \int \frac {\d x} {\sqrt {\size {x^2 + k} } } = \map \ln {x + \sqrt {\size {x^2 + k} } } + C$


Bernoulli's Equation/x^2 dy = (2 x y + y^2) dx

The first order ODE: :$(1): \quad x^2 \rd y = \paren {2 x y + y^2} \rd x$ has the general solution: :$y = - \dfrac {x^2} {x + C}$


Bernoulli's Equation/2 x y dx + (x^2 + 2 y) dy = 0

The first order ODE: :$(1): \quad 2 x y \rd x + \paren {x^2 + 2 y} \rd y = 0$ has the solution: :$y \paren {x^2 + y} = C$


Linear First Order ODE/y' - (y over x) = k x

Let $k \in \R$ be a real number. The linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} - \dfrac y x = k x$ has the general solution: :$\dfrac y x = k x + C$ or: :$y = k x^2 + C x$


First Order ODE/x dy = k y dx

The first order ODE: :$(1): \quad x \rd y = k y \rd x$ has the general solution: :$y = C x^k$


First Order ODE/dy = k y dx

Let $k \in \R$ be a real number. The first order ODE: :$\dfrac {\d y} {\d x} = k y$ has the general solution: :$y = C e^{k x}$


First Order ODE/(1 over x^3 y^2 + 1 over x) dx + (1 over x^2 y^3 - 1 over y) dy = 0

The first order ordinary differential equation: :$(1): \quad \paren {\dfrac 1 {x^3 y^2} + \dfrac 1 x} \rd x + \paren {\dfrac 1 {x^2 y^3} - \dfrac 1 y} \rd y = 0$ is an exact differential equation with solution: :$-\dfrac 1 {2 x^2 y^2} = \ln \dfrac y x + C$


Linear First Order ODE/y' + (y over x) = k x^n

Let $k, n \in \R$ be real numbers. The linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} + \dfrac y x = k x^n$ has the general solution: :$y = \begin{cases} \dfrac {k x^{n + 1} } {n + 2} + \dfrac C x & : n \ne -2 \\ & \\ \dfrac {k \ln x} x + \dfrac C x & : n = -2 \end{cases}$


Linear First Order ODE/(x^2 + y) dx = x dy

The linear first order ODE: :$(1): \quad \paren {x^2 + y} \rd x = x \rd y$ has the general solution: :$y = x^2 + C x$


Linear First Order ODE/x y' + y = x^2 cosine x

The linear first order ODE: :$x \, \dfrac {\d y} {\d x} + y = x^2 \cos x$ has the general solution: :$y = 2 \cos x + x \sin x - \dfrac 2 x \sin x + \dfrac C x$


First Order ODE/(6x + 4y + 3) dx + (3x + 2y + 2) dy = 0

The first order ODE: :$(1): \quad \paren {6 x + 4 y + 3} \rd x + \paren {3 x + 2 y + 2} \rd y = 0$ has the general solution: :$3 x + 2 y + \map \ln {\paren {3 x + 2 y}^2} + x = C$


First Order ODE/Cosine (x + y) dx = sine (x + y) dx + x sine (x + y) dy

The first order ordinary differential equation: :$\map \cos {x + y} \rd x = \map \sin {x + y} \rd x + x \map \sin {x + y} \rd y$ is an exact differential equation with solution: :$x \map \cos {x + y} = C$


Linear First Order ODE/x y' + y = f (x)

The linear first order ODE: :$(1): \quad x \, \dfrac {\d y} {\d x} + y = \map f x$ has the general solution: :$\displaystyle x y = \int \map f x \rd x + C$


Second Order ODE/x^2 y'' + x y' = 1

The second order ODE: :$x^2 y'' + x y' = 1$ has the general solution: :$y = \dfrac {\paren {\ln x}^2} 2 + C_1 \ln x + C_2$


Sorgenfrey Line is Lindelöf

The Sorgenfrey line is Lindelöf.


First Order ODE/y' ln (x - y) = 1 + ln (x - y)

The first order ordinary differential equation: :$(1): \quad \dfrac {\d y} {\d x} \map \ln {x - y} \d x = 1 + \map \ln {x - y}$ is an exact differential equation with solution: :$\paren {x - y} \map \ln {x - y} = C - y$


Linear First Order ODE/y' + 2 x y = exp -x^2

The linear first order ODE: :$\dfrac {\d y} {\d x} + 2 x y = \map \exp {-x^2}$ has the general solution: :$y = \paren {x + C} \map \exp {-x^2}$


Linear First Order ODE/(1 + x^2) y' + 2 x y = 4 x^3

The linear first order ODE: :$\paren {1 + x^2} \dfrac {\d y} {\d x} + 2 x y = 4 x^3$ has the general solution: :$y = \dfrac {x^4} {1 + x^2} + \dfrac C {1 + x^2}$


First Order ODE/(1 + x^2) y' + x y = 0

The first order ODE: :$\paren {1 + x^2} y' + x y = 0$ has the general solution: :$y = \dfrac C {\sqrt {1 + x^2} }$


First Order ODE/(x exp y + y - x^2) dy = (2 x y - exp y - x) dx

The first order ordinary differential equation: :$(1): \quad \paren {x e^y + y - x^2} \rd y = \paren {2 x y - e^y - x} \rd x$ is an exact differential equation with solution: :$2 x e^y + x^2 + y^2 - 2 x^2 y = C$


First Order ODE/exp x (1 + x) dx = (x exp x - y exp y) dy

The first order ordinary differential equation: :$(1): \quad e^x \paren {1 + x} \rd x = \paren {x e^x - y e^y} \rd y$ has the solution: :$2 x e^x e^{-y} + y^2 + C$


Two Stage Radioactive Decay

Let $R_A$ be a radioactive isotope which decays into another radioactive isotope $R_B$ with the rate constant $k_1$. Let $R_B$ decay into a third element $R_C$ with the rate constant $k_2$. Let the initial quantity of $R_A$ be $x_0$. Let the amounts of $R_A$ and $R_B$ present at time $t$ be $x$ and $y$ respectively. Then: :$y = \begin{cases} \dfrac {k_1 x_0} {k_2 - k_1} \left({e^{-k_1 t} - e^{-k_2 t} }\right) & : k_1 \ne k_2 \\ & \\ k_1 x_0 t e^{-k_1 t} & : k_1 = k_2 \end{cases}$


General Solution of Riccati Equation from Particular Solution

Consider the Riccati equation: :$(1): \quad y' = \map p x + \map q x y + \map r x y^2$ Let $\map {y_1} x$ be a particular solution to $(1)$. Then the general solution to $(1)$ has the form: :$\map y x = \map {y_1} x + \map z x$ where $\map z x$ is the general solution to the Bernoulli equation: :$z' - \paren {q - 2 r y_1} z = r z^2$


Riccati Equation/y' = (y over x) + x^3 y^2 - x^5

The Riccati equation: :$(1): \quad y' = \dfrac y x + x^3 y^2 - x^5$ has the general solution: :$C \exp \dfrac {2 x^5} 5 = \dfrac {y - x} {y + x}$


Bernoulli's Equation/y' - (1 over x + 2 x^4) y = x^3 y^2

The first order ODE: :$(1): \quad y' - \paren {\dfrac 1 x + 2 x^4} y = x^3 y^2$ has the general solution: :$y = \dfrac {2 x} {C \, \map \exp {-\dfrac {2 x^5} 5} - 1}$


Motion of Body with Variable Mass

Let $B$ be a body undergoing a force $\mathbf F$. Let $B$ be travelling at a velocity $\mathbf v$ at time $t$. Let mass travelling at a velocity $\mathbf v + \mathbf w$ be added to $B$ at a rate of $\dfrac {\d m} {\d t}$. Let $m$ be the mass of $B$ at time $t$. Then the equation of motion of $B$ is given by: :$\mathbf w \dfrac {\d m} {\d t} + \mathbf F = m \dfrac {\d \mathbf v} {\d t}$


Burnout Velocity of Upward Rocket under Constant Gravity

Let $R$ be a rocket whose structural mass is $m_1$. Let $R$ contain fuel of initial mass $m_2$. Let $R$ be fired straight up from the surface of a planet whose gravitational field exerts an Acceleration Due to Gravity of $g$, assumed constant. Let $R$ burn fuel at a constant rate $a$, producing a constant exhaust velocity $b$ relative to $R$. Let all forces on $R$ except that due to the gravitational field be neglected. Then the burnout velocity of $R$ is given by: :$v_b = b \ln \left({1 + \dfrac {m_1} {m_2} }\right) - \dfrac {g m_2} a$


Burnout Height of Upward Rocket under Constant Gravity

Let $R$ be a rocket whose structural mass is $m_1$. Let $R$ contain fuel of initial mass $m_2$. Let $R$ be fired straight up from the surface of a planet whose gravitational field exerts an Acceleration Due to Gravity of $g$, assumed constant. Let $R$ burn fuel at a constant rate $a$, with a constant exhaust velocity $b$ relative to $R$. Let all forces on $R$ except that due to  the gravitational field be neglected. Then the burnout height of $R$ is given by: :$h_b = -\dfrac {g m_2^2} {2 a^2} + \dfrac {b m_2} a + \dfrac {b m_1} a \ln \dfrac {m_1} {m_1 + m_2}$


Linear First Order ODE/y' = x + y

The linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = x + y$ has the general solution: :$y = C e^x - x - 1$


Linear First Order ODE/y' = x + y/y(0) = 1

The linear first order ODE: :$(1): \quad \dfrac {\d y} {\d x} = x + y$ with initial condition: :$\map y 0 = 1$ has the particular solution: :$y = 2 e^x - x - 1$


Existence and Uniqueness of Solution for Linear Second Order ODE with two Initial Conditions

Let $\map P x$, $\map Q x$ and $\map R x$ be continuous real functions on a closed real interval $\closedint a b$. Let $x_0$ be any point in $\closedint a b$. Let $y_0$ and ${y_0}'$ be real numbers. Then the linear second order ordinary differential equation: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = \map R x$ has a unique particular solution $\map y x$ on $\closedint a b$ such that: :$\map y {x_0} = y_0$ and: :$\map {y'} {x_0} = {y_0}'$


General Solution of Linear 2nd Order ODE from Homogeneous 2nd Order ODE and Particular Solution

Consider the nonhomogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = \map R x$ Let $\map {y_g} x$ be the general solution of the homogeneous linear second order ODE: :$(2): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ Let $\map {y_p} x$ be a particular solution of $(1)$. Then $\map {y_g} x + \map {y_p} x$ is the general solution of $(1)$.


Trivial Solution of Homogeneous Linear 2nd Order ODE

The homogeneous linear second order ODE: :$\dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ has the particular solution: :$\map y x = 0$ that is, the zero constant function. This particular solution is referred to as the '''trivial solution'''.


Linear Combination of Solutions to Homogeneous Linear 2nd Order ODE

Let $c_1$ and $c_2$ be real numbers. Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ Then: :$c_1 \, \map {y_1} x + c_2 \, \map {y_2} x$ is also a particular solution to $(1)$. That is, a linear combination of particular solutions to a homogeneous linear second order ODE is also a particular solution to that ODE.


Solutions of Linear 2nd Order ODE have Common Zero iff Linearly Dependent

Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ on a closed interval $\closedint a b$. Let $y_1$ and $y_2$ both have a zero for the same value of $x$ in $\closedint a b$. Then $y_1$ and $y_2$ are constant multiples of each other. That is, $y_1$ and $y_2$ are linearly dependent.


Real Function is Linearly Dependent with Zero Function

Let $f \left({x}\right)$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$. Let $g \left({x}\right)$ be the constant zero function on $\left[{a \,.\,.\, b}\right]$: :$\forall x \in \left[{a \,.\,.\, b}\right]: g \left({x}\right) = 0$ Then $f$ and $g$ are linearly dependent on $\left[{a \,.\,.\, b}\right]$.


Two Linearly Independent Solutions of Homogeneous Linear Second Order ODE generate General Solution

Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ on a closed interval $\closedint a b$. Let $y_1$ and $y_2$ be linearly independent. Then the general solution to $(1)$ is: :$y = C_1 \map {y_1} x + C_2 \map {y_2} x$ where $C_1 \in \R$ and $C_2 \in \R$ are arbitrary constants.


Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE

Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ on a closed interval $\closedint a b$. Let $y_1$ and $y_2$ be linearly independent. Then their Wronskian is either never zero, or zero everywhere on $\closedint a b$.


Real Number Line is Lindelöf

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is Lindelöf.


Zero Wronskian of Solutions of Homogeneous Linear Second Order ODE iff Linearly Dependent

Let $\map {y_1} x$ and $\map {y_2} x$ be particular solutions to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ on a closed interval $\closedint a b$. Then: :$y_1$ and $y_2$ are linearly dependent {{iff}}: :the Wronskian $\map W {y_1, y_2}$ of $y_1$ and $y_2$ is zero everywhere on $\closedint a b$.


Linear Second Order ODE/y'' + y = 0

The second order ODE: :$(1): \quad y'' + y = 0$ has the general solution: :$y = C_1 \sin x + C_2 \cos x$


Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0

The second order ODE: :$(1): \quad x^2 y'' - 2 x y' + 2 y = 0$ has the general solution: :$y = C_1 x + C_2 x^2$ on any closed real interval which does not contain $0$.


Linear Second Order ODE/x^2 y'' - 2 x y' + 2 y = 0/y(1) = 3, y'(1) = 5

The second order ODE: :$(1): \quad x^2 y'' - 2 x y' + 2 y = 0$ with initial conditions: :$\map y 1 = 3$ :$\map {y'} 1 = 5$ has the particular solution: :$y = x + 2 x^2$


Linear Second Order ODE/y'' - 3 y' + 2 y = 0

The second order ODE: :$(1): \quad y'' - 3 y' + 2 y = 0$ has the general solution: :$y = C_1 e^x + C_2 e^{2 x}$


Linear Second Order ODE/y'' - 3 y' + 2 y = 0/y(0) = -1, y'(0) = 1

The second order ODE: :$(1): \quad y'' - 3 y' + 2 y = 0$ with initial conditions: :$\map y 0 = -1$ :$\map {y'} 0 = 1$ has the particular solution: :$y = -3 e^x + 2 e^{2 x}$


Linear Second Order ODE/y'' - 4 y' + 4 y = 0

The second order ODE: :$(1): \quad y'' - 4 y' + 4 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 x e^{2 x}$


Particular Solution to Homogeneous Linear Second Order ODE gives rise to Another

Let $\map {y_1} x$ be a particular solution to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ such that $y_1$ is not the trivial solution. Then there exists a standard procedure to determine another particular solution $\map {y_2} x$ of $(1)$ such that $y_1$ and $y_2$ are linearly independent.


Induced Solution to Homogeneous Linear Second Order ODE is Linearly Independent with Inducing Solution

Let $\map {y_1} x$ be a particular solution to the homogeneous linear second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + \map P x \dfrac {\d y} {\d x} + \map Q x y = 0$ such that $y_1$ is not the trivial solution. Let $\map {y_2} x$ be the real function defined as: :$\map {y_2} x = \map v x \, \map {y_1} x$ where: :$\displaystyle v = \int \dfrac 1 { {y_1}^2} e^{-\int P \rd x} \rd x$ Then $y_2$ and $y_1$ are linearly independent.


Linear Second Order ODE/x^2 y'' + x y' - y = 0

The second order ODE: :$(1): \quad x^2 y'' + x y' - y = 0$ has the general solution: :$y = C_1 x + \dfrac {C_2} x$


Linear Second Order ODE/y'' - y = 0

The second order ODE: :$(1): \quad y'' - y = 0$ has the general solution: :$y = C_1 e^x + C_2 e^{-x}$


Second Order ODE/x y'' + 3 y' = 0

The second order ODE: :$(1): \quad x y'' + 3 y' = 0$ has the general solution: :$y = C_1 + \dfrac {C_2} {x^2}$


Linear Second Order ODE/x^2 y'' + x y' - 4 y = 0

The second order ODE: :$(1): \quad x^2 y'' + x y' - 4 y = 0$ has the general solution: :$y = C_1 x^2 + \dfrac {C_2} {x^2}$


Topological Subspace of Real Number Line is Lindelöf

Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $W$ be a non-empty subset of $\R$. Then $R_W$ is Lindelöf where $R_W$ denotes the topological subspace of $R$ on $W$.


Bessel's Equation/x^2 y'' + x y' + (x^2 - (1 over 4)) y = 0/Particular Solution

The special case of Bessel's equation: :$(1): \quad x^2 y'' + x y' + \left({x^2 - \dfrac 1 4}\right) y = 0$ has a particular solution: :$y = \dfrac {\sin x} {\sqrt x}$


Linear Second Order ODE/(x - 1) y'' - y' + y = 0

The second order ODE: :$(1): \quad \paren {x - 1} y'' - x y' + y = 0$ has the general solution: :$y = C_1 x + C_2 e^x$


Linear Second Order ODE/x^2 y'' + 2 x y' - 2 y = 0

The second order ODE: :$(1): \quad x^2 y'' + 2 x y' - 2 y = 0$ has the general solution: :$y = C_1 x + \dfrac {C_2} {x^2}$


Second Order ODE/x y'' - (2 x + 1) y' + (x + 1) y = 0

The second order ODE: :$(1): \quad x y'' - \paren {2 x + 1} y' + \paren {x + 1} y = 0$ has the general solution: :$y = C_1 e^x + C_2 x^2 e^x$


Exponential Function is Solution of Constant Coefficient Homogeneous LSOODE iff Index is Root of Auxiliary Equation

Let: :$(1): \quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. Then: :$y = e^{m_1 x}$ is a solution to $(1)$ {{iff}} :$m_1$ is a root of the auxiliary equation $m^2 + p m + q = 0$


Solution of Constant Coefficient Homogeneous LSOODE

Let: :$(1): \quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. Let $m_1$ and $m_2$ be the roots of the auxiliary equation $m^2 + p m + q = 0$.


Solution of Constant Coefficient Homogeneous LSOODE/Real Roots of Auxiliary Equation

{{:Solution of Constant Coefficient Homogeneous LSOODE}} Let $p^2 > 4 q$. Then $(1)$ has the general solution: :$y = C_1 e^{m_1 x} + C_2 e^{m_2 x}$


Solution of Constant Coefficient Homogeneous LSOODE/Complex Roots of Auxiliary Equation

{{:Solution of Constant Coefficient Homogeneous LSOODE}} Let $p^2 < 4 q$. Then $(1)$ has the general solution: :$y = e^{a x} \paren {C_1 \cos b x + C_2 \sin b x}$ where: :$m_1 = a + i b$ :$m_2 = a - i b$


Solution of Constant Coefficient Homogeneous LSOODE/Equal Real Roots of Auxiliary Equation

{{:Solution of Constant Coefficient Homogeneous LSOODE}} Let $p^2 = 4 q$. Then $(1)$ has the general solution: :$y = C_1 e^{m_1 x} + C_2 x e^{m_1 x}$


Linear Second Order ODE/y'' + y' - 6 y = 0

The second order ODE: :$(1): \quad y'' + y' - 6 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-3 x}$


Linear Second Order ODE/y'' + 2 y' + y = 0

The second order ODE: :$(1): \quad y'' + 2 y' + y = 0$ has the general solution: :$y = C_1 e^{-x} + C_2 x e^{-x}$


Linear Second Order ODE/2 y'' - 4 y' + 8 y = 0

The second order ODE: :$(1): \quad 2 y'' - 4 y + 8 y = 0$ has the general solution: :$y = e^x \paren {C_1 \cos \sqrt 3 x + C_2 \sin \sqrt 3 x}$


Linear Second Order ODE/y'' - 9 y' + 20 y = 0

The second order ODE: :$(1): \quad y'' - 9 y' + 20 y = 0$ has the general solution: :$y = C_1 e^{4 x} + C_2 e^{5 x}$


Condition for Solutions to Constant Coefficient Homogeneous LSOODE to tend to Zero

Let: :$(1): \quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. Let the general solution to $(1)$ be $\map y {x, C_1, C_2}$. Then: :$\displaystyle \lim_{x \mathop \to \infty} \map y {x, C_1, C_2} = 0$ {{iff}} :$p$ and $q$ are both strictly positive.


Derivative of Solution to Constant Coefficient Homogeneous LSOODE is also Solution

Let: :$(1): \quad y'' + p y' + q y = 0$ be a constant coefficient homogeneous linear second order ODE. Let $\map y x$ be a particular solution of $(1)$. Then its derivative $\map {y'} x$ is also a particular solution of $(1)$.


Conversion of Cauchy-Euler Equation to Constant Coefficient Linear ODE

Consider the Cauchy-Euler equation: :$(1): \quad x^2 \dfrac {\d^2 y} {\d x^2} + p x \dfrac {\d y} {\d x} + q y = 0$ By making the substitution: :$x = e^t$ it is possible to convert $(1)$ into a constant coefficient homogeneous linear second order ODE: :$\dfrac {\d^2 y} {\d t^2} + \paren {p - 1} \dfrac {\d y} {\d t} + q y = 0$


Linear Second Order ODE/x^2 y'' + 2 x y' - 12 y = 0

The second order ODE: :$(1): \quad x^2 y'' + 2 x y' - 12 y = 0$ has the general solution: :$y = C_1 x^3 + C_2 x^{-4}$


Cardinality of Image of Set not greater than Cardinality of Set

Let $X, Y$ be sets. Let $f:X \to Y$ be a mapping. Let $A$ be a subset of $X$. Then $\left\vert{f^\to\left({A}\right)}\right\vert \le \left\vert{A}\right\vert$ where $\left\vert{A}\right\vert$ denotes the cardinality of $A$.


Projector has Norm 1

An idempotent operator $P$ is a projector on the Hilbert Space $H$ {{iff}} $P$ has norm $1$: :$\displaystyle \norm P \equiv \sup_{x \mathop \in H} \frac {\norm P} {\norm x} = 1$


Linear Second Order ODE/y'' + 3 y' - 10 y = 0

The second order ODE: :$(1): \quad y'' + 3 y' - 10 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-5 x}$


Linear Second Order ODE/y'' + 4 y = 0

The second order ODE: :$(1): \quad y'' + 4 y = 0$ has the general solution: :$y = C_1 \cos 2 x + C_2 \sin 2 x$


Linear Second Order ODE/y'' + 4 y = 3 sine x

The second order ODE: :$(1): \quad y'' + 4 y = 3 \sin x$ has the general solution: :$y = C_1 \cos 2 x + C_2 \sin 2 x + \sin x$


Linear Second Order ODE/y'' + 10 y' + 25 y = 0

The second order ODE: :$(1): \quad y'' + 10 y' + 25 y = 0$ has the general solution: :$y = C_1 e^{-5 x} + C_2 x e^{-5 x}$


Linear Second Order ODE/y'' + 10 y' + 25 y = 14 exp -5 x

The second order ODE: :$(1): \quad y'' + 10 y' + 25 y = 14 e^{-5 x}$ has the general solution: :$y = C_1 \cos 2 x + C_2 \sin 2 x + \sin x$


Linear Second Order ODE/y'' - 2 y' + 5 y = 0

The second order ODE: :$(1): \quad y'' - 2 y' + 5 y = 0$ has the general solution: :$y = e^x \paren {C_1 \cos 2 x + C_2 \sin 2 x}$


Linear Second Order ODE/y'' + y' - 12 y = 0

The second order ODE: :$(1): \quad y'' + y' - 12 y = 0$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-4 x}$


Linear Second Order ODE/y'' + 2 y' + 10 y = 0

The second order ODE: :$(1): \quad y'' + 2 y' + 10 y = 0$ has the general solution: :$y = e^{-x} \paren {C_1 \cos 3 x + C_2 \sin 3 x}$


Linear Second Order ODE/y'' + 4 y' + 4 y = 0

The second order ODE: :$(1): \quad y'' + 4 y' + 4 y = 0$ has the general solution: :$y = \paren {C_1 + C_2 x} e^{-2 x}$


Linear Second Order ODE/y'' + y' - 2 y = 0

The second order ODE: :$(1): \quad y'' + y' - 2 y = 0$ has the general solution: :$y = C_1 e^x + C_2 e^{-2 x}$


Linear Second Order ODE/y'' - 4 y = 0

The second order ODE: :$(1): \quad y'' - 4 y = 0$ has the general solution: :$y = C_1 e^{2 x} + C_2 e^{-2 x}$


Linear Second Order ODE/y'' - y' - 6 y = 0

The second order ODE: :$(1): \quad y'' - y' - 6 y = 0$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x}$


Linear Second Order ODE/y'' - y' - 6 y = 20 exp -2 x

The second order ODE: :$(1): \quad y'' - y' - 6 y = 20 e^{-2 x}$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x} - 4 x e^{-2 x}$


Linear Second Order ODE/y'' - 2 y' = 0

The second order ODE: :$(1): \quad y'' - 2 y' = 0$ has the general solution: :$y = C_1 + C_2 e^{2 x}$


Directed iff Finite Subsets have Upper Bounds

Let $\left({S, \precsim}\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is directed {{iff}} :for every a finite subset $A$ of $H$ ::$\exists h \in H: \forall a \in A: a \precsim h$


Linear Second Order ODE/y'' + k^2 y = sine b x

The second order ODE: :$(1): \quad y'' + k^2 y = \sin b x$ has the general solution: :$y = \begin{cases} C_1 \sin k x + C_2 \cos k x + \dfrac {\sin b x} {k^2 - b^2} & : b \ne k \\ C_1 \sin k x + C_2 \cos k x - \dfrac {x \cos k x} {2 k} & : b = k  \end{cases}$


Combination of Solutions to Non-Homogeneous LSOODE with same Homogeneous Part

Let $\map {y_1} x$ be a particular solution of the linear second order ODE: :$(1): \quad y'' + \map P x y' + \map Q x y = \map {R_1} x$ Let $\map {y_2} x$ be a particular solution of the linear second order ODE: :$(2): \quad y'' + \map P x y' + \map Q x y = \map {R_2} x$ Then $\map y x = \map {y_1} x + \map {y_2} x$ is a particular solution of the linear second order ODE: :$(3): \quad y'' + \map P x y' + \map Q x y = \map {R_1} x + \map {R_2} x$


Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x + 6 cosine x + 8 x^2 - 4 x

The second order ODE: :$(1): \quad y'' + 4 y = 4 \cos 2 x + 6 \cos x + 8 x^2 - 4 x$ has the general solution: :$y = C_1 \sin 2 x + C_2 \cos 2 x + x \sin 2 x + 2 \cos x - 1 - x + 2 x^2$


Linear Second Order ODE/y'' + 4 y = 4 cosine 2 x

The second order ODE: :$(1): \quad y'' + 4 y = 4 \cos 2 x$ has the general solution: :$y = C_1 \sin 2 x + C_2 \cos 2 x + x \sin 2 x$


Linear Second Order ODE/y'' + 4 y = 6 cosine x

The second order ODE: :$(1): \quad y'' + 4 y = 6 \cos x$ has the general solution: :$y = C_1 \sin 2 x + C_2 \cos 2 x + 2 \cos x$


Linear Second Order ODE/y'' + 4 y = 8 x^2 - 4 x

The second order ODE: :$(1): \quad y'' + 4 y = 8 x^2 - 4 x$ has the general solution: :$y = C_1 \sin 2 x + C_2 \cos 2 x - 1 - x + 2 x^2$


Linear Second Order ODE/y'' - 2 y' + y = 0

The second order ODE: :$(1): \quad y'' - 2 y' + y = 0$ has the general solution: :$y = C_1 e^x + C_2 x e^x$


Linear Second Order ODE/y'' - 2 y' + y = 2 x

The second order ODE: :$(1): \quad y'' - 2 y' + y = 2 x$ has the general solution: :$y = C_1 e^x + C_2 x e^x + 2 x + 4$


Linear Second Order ODE/y'' - y' - 6 y = exp -x

The second order ODE: :$(1): \quad y'' - y' - 6 y = e^{-x}$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-2 x} - \dfrac {e^{-x} } 4$


Filtered iff Finite Subsets have Lower Bounds

Let $\struct {S, \precsim}$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is filtered {{iff}}: :for every a finite subset $A$ of $H$ ::$\exists h \in H: \forall a \in A: h \precsim a$


Tangent of Half Angle plus Quarter Pi

:$\map \tan {\dfrac x 2 + \dfrac \pi 4} = \tan x + \sec x$


Linear Second Order ODE/y'' + 4 y = tangent 2 x

The second order ODE: :$(1): \quad y'' + 4 y = \tan 2 x$ has the general solution: :$y = C_1 \cos 2 x + C_2 \sin 2 x - \dfrac 1 4 \cos 2 x \, \map \ln {\sec 2 x + \tan 2 x}$


Linear Second Order ODE/y'' + 2 y' + y = exp -x log x

The second order ODE: :$(1): \quad y'' + 2 y' + y = e^{-x} \ln x$ has the general solution: :$y = C_1 e^{-x} + C_2 x e^{-x} - \dfrac {x^2 e^{-x} \ln x} 2 - \dfrac 3 4 x^2 e^{-x}$


Linear Second Order ODE/y'' - 2 y' - 3 y = 0

The second order ODE: :$(1): \quad y'' - 2 y' - 3 y = 0$ has the general solution: :$y = C_1 e^{3 x} + C_2 e^{-x}$


Linear Second Order ODE/y'' + 2 y' + 5 y = 0

The second order ODE: :$(1): \quad y'' + 2 y' + 5 y = 0$ has the general solution: :$y = e^{-x} \paren {C_1 \cos 2 x + C_2 \sin 2 x}$


Linear Second Order ODE/y'' + 2 y' + 5 y = exp -x secant 2 x

The second order ODE: :$(1): \quad y'' + 2 y' + 5 y = e^{-x} \sec 2 x$ has the general solution: :$y = e^{-x} \paren {C_1 \cos 2 x + C_2 \sin 2 x} + \dfrac {x e^{-x} \sin 2 x} 2 + \dfrac {e^{-x} \cos 2 x \ln \cos 2 x} 4$


Singleton is Directed and Filtered Subset

Let $\struct {S, \precsim}$ be a preordered set. Let $x$ be an element of $S$. Then $\set x$ is directed and filtered subset of $S$.


Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = 0

The second order ODE: :$(1): \quad \paren {x^2 - 1} y'' - 2 x y' + 2 y = 0$ has the general solution: :$y = C_1 x + C_2 \paren {x^2 + 1}$


Linear Second Order ODE/(x^2 - 1) y'' - 2 x y' + 2 y = (x^2 - 1)^2

The second order ODE: :$(1): \quad \paren {x^2 - 1} y'' - 2 x y' + 2 y = \paren {x^2 - 1}^2$ has the general solution: :$y = C_1 x + C_2 \paren {x^2 + 1} + \dfrac {x^4} 6 - \dfrac {x^2} 2$


Directed iff Lower Closure Directed

Let $\left({S, \precsim}\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is directed {{iff}} $H^\precsim$ is directed where $H^\precsim$ denotes the lower closure of set.


Filtered iff Upper Closure Filtered

Let $\left({S, \precsim}\right)$ be a preordered set. Let $H$ be a non-empty subset of $S$. Then $H$ is filtered {{iff}} $H^\succsim$ is filtered where $H^\succsim$ denotes the upper closure of set.


Linear Second Order ODE/(x^2 + x) y'' + (2 - x^2) y' - (2 + x) y = 0

The second order ODE: :$(1): \quad \paren {x^2 + x} y'' + \paren {2 - x^2} y' - \paren {2 + x} y = 0$ has the general solution: :$y = C_1 e^x + \dfrac {C_2} x$


Position of Cart attached to Wall by Spring/x = x0 at t = 0

Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = x_0 \cos \alpha t$


Directed in Join Semilattice

Let $\left({S, \preceq}\right)$ be a join semilattice. Let $H$ be a non-empty lower subset of $S$. Then $H$ is directed {{iff}} :$\forall x, y \in H: x \vee y \in H$


Position of Cart attached to Wall by Spring under Damping/Underdamped/x = x0 at t = 0

Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the horizontal position of $C$ at time $t$ can be expressed as: :$x = \dfrac {x_0} \alpha e^{-b t} \left({\alpha \cos \alpha t + b \sin \alpha t}\right)$


Canonical Form of Underdamped Oscillatory System

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the value of $x$ can be expressed in the form: :$x = \dfrac {x_0 \, a} \alpha e^{-b t} \, \map \cos {\alpha t - \theta}$ where: :$\alpha = \sqrt {a^2 - b^2}$ :$\theta = \map \arctan {\dfrac b \alpha}$ This can be referred to as the '''canonical form''' of the solution of $(1)$.


Period of Oscillation of Underdamped System is Regular

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the period of its movement is well-defined, in the sense that its zeroes are regularly spaced, and given by: :$T = \dfrac {2 \pi} {\sqrt {a^2 - b^2} }$


Period of Oscillation of Underdamped Cart attached to Wall by Spring

Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the period of oscillation of $C$ can be expressed as: :$T = \dfrac {2 \pi} {\sqrt {\dfrac k m - \dfrac {c^2} {4 m^2} } }$


Natural Frequency of Underdamped System

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$\dfrac {\d^2 x} {\d t^2} + 2 b \dfrac {\d x} {\d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. Then the natural frequency of $S$ is given by: :$\nu = \dfrac {\sqrt {a^2 - b^2} } {2 \pi}$


Natural Frequency of Underdamped Cart attached to Wall by Spring

Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$. Then the natural frequency of $C$ can be expressed as: :$\nu = \dfrac 1 {2 \pi} \sqrt {\dfrac k m - \dfrac {c^2} {4 m^2} }$


Linear Second Order ODE/y'' + 2 b y' + a^2 y = 0/b less than a

The second order ODE: :$(1): \quad y'' + 2 b y' + a^2 y = 0$ where $b^2 < a^2$ has the general solution: :$y = e^{-b x} \paren {C_1 \, \map \cos {\sqrt {a^2 - b^2} } x + C_2 \, \map \sin {\sqrt {a^2 - b^2} } x}$


Linear Second Order ODE/y'' + 2 b y' + a^2 y = K cosine omega x/b less than a

The second order ODE: :$(1): \quad y'' + 2 b y' + a^2 y = K \cos \omega x$ where $b^2 < a^2$ has the general solution: :$y = e^{-b x} \paren {C_1 \cos \alpha x + C_2 \sin \alpha x} + \dfrac K {\sqrt {4 b^2 \omega^2 + \paren {a^2 - \omega^2}^2} } \map \cos {\omega x - \phi}$ where: :$\alpha = \sqrt {a^2 - b^2}$ :$\phi = \map \arctan {\dfrac {2 b \omega} {a^2 - \omega^2} }$


Condition for Resonance in Forced Vibration of Underdamped System

Consider a physical system $S$ whose behaviour is defined by the second order ODE: :$(1): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$ where: :$K \in \R: k > 0$ :$a, b \in \R_{>0}: b < a$ Then $S$ is in resonance when: :$\omega = \sqrt {a^2 - 2 b^2}$ and thus the resonance frequency is: :$\nu_R = \dfrac {\sqrt {a^2 - 2 b^2} } {2 \pi}$ This resonance frequency exists only when $a^2 - 2 b^2 > 0$.


Filtered in Meet Semilattice

Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $H$ be a non-empty upper subset of $S$. Then $H$ is filtered {{iff}} :$\forall x, y \in H: x \wedge y \in H$


Directed in Join Semilattice with Finite Suprema

Let $\left({S, \preceq}\right)$ be a join semilattice. Let $H$ be a non-empty lower subset of $S$. Then $H$ is directed {{iff}} :for every non-empty finite subset $A$ of $H$, $\sup A \in H$


Filtered in Meet Semilattice with Finite Infima

Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $H$ be a non-empty lower subset of $S$. Then $H$ is filtered {{iff}} :for every non-empty finite subset $A$ of $H$, $\inf A \in H$


Interval between Local Maxima for Underdamped Free Vibration

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px Let $T$ be the period of oscillation of $S$. Then the successive local maxima of $x$ occur for $t = 0, T, 2T, \ldots$


Ratio of Successive Local Maxima for Underdamped Free Vibration

Consider a physical system $S$ whose behaviour can be described with the second order ODE in the form: :$(1): \quad \dfrac {\mathrm d^2 x} {\mathrm d t^2} + 2 b \dfrac {\mathrm d x} {\mathrm d t} + a^2 x = 0$ for $a, b \in \R_{>0}$. Let $b < a$, so as to make $S$ underdamped. :600px Let $T$ be the period of oscillation of $S$. Let $x_1$ and $x_2$ be successive local maxima of $x$. Then: :$\dfrac {x_1} {x_2} = e^{b T}$


Motion of Particle in Polar Coordinates

Consider a particle $p$ of mass $m$ moving in the plane under the influence of a force $\mathbf F$. Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$. Let $\mathbf F$ be expressed as: :$\mathbf F = F_r \mathbf u_r + F_\theta \mathbf u_\theta$ where: :$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$ :$F_r$ and $F_\theta$ are the magnitudes of the components of $\mathbf F$ in the directions of $\mathbf u_r$ and $\mathbf u_\theta$ respectively. Then the second order ordinary differential equations governing the motion of $m$ under the force $\mathbf F$ are: {{begin-eqn}} {{eqn | l = F_\theta       | r = m \left({r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} + 2 \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} }\right)       | c =  }} {{eqn | l = F_r       | r = m \left({\dfrac {\mathrm d^2 r} {\mathrm d t^2} - r \left({\dfrac {\mathrm d \theta} {\mathrm d t} }\right)^2}\right)       | c =  }} {{end-eqn}}


Derivative of Angular Component under Central Force

Let a point mass $p$ of mass $m$ be under the influence of a central force $\mathbf F$. Let the position of $p$ at time $t$ be given in polar coordinates as $\polar {r, \theta}$. Let $\mathbf r$ be the radius vector from the origin to $p$. Then the rate of change of the angular coordinate of $p$ is inversely proportional to the square of the radial coordinate of $p$.


Linear Second Order ODE/y'' + y = K

The second order ODE: :$(1): \quad y'' + y = K$ has the general solution: :$y = C_1 \sin x + C_2 \cos x + K$


Velocity Vector in Polar Coordinates

Consider a particle $p$ moving in the plane. Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$. Then the velocity $\mathbf v$ of $p$ can be expressed as: :$\mathbf v = r \dfrac {\mathrm d \theta} {\mathrm d t} \mathbf u_\theta + \dfrac {\mathrm d r} {\mathrm d t} \mathbf u_r$ where: :$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$


Derivatives of Unit Vectors in Polar Coordinates

Consider a particle $p$ moving in the plane. Let the position of $p$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$. Let: :$\mathbf u_r$ be the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ be the unit vector in the direction of the angular coordinate of $p$ Then the derivative of $\mathbf u_r$ and $\mathbf u_\theta$ {{WRT}} $\theta$ can be expressed as: {{begin-eqn}} {{eqn | l = \dfrac {\mathrm d \mathbf u_r} {\mathrm d \theta}       | r = \mathbf u_\theta }} {{eqn | l = \dfrac {\mathrm d \mathbf u_\theta} {\mathrm d \theta}       | r = -\mathbf u_r }} {{end-eqn}}


Acceleration Vector in Polar Coordinates

Consider a particle $p$ moving in the plane. Let the position of $p$ at time $t$ be given in polar coordinates as $\left\langle{r, \theta}\right\rangle$. Then the acceleration $\mathbf a$ of $p$ can be expressed as: :$\mathbf a = \left({r \dfrac {\mathrm d^2 \theta} {\mathrm d t^2} + 2 \dfrac {\mathrm d r} {\mathrm d t} \dfrac {\mathrm d \theta} {\mathrm d t} }\right) \mathbf u_\theta + \left({\dfrac {\mathrm d^2 r} {\mathrm d t^2} - r \left({\dfrac {\mathrm d \theta} {\mathrm d t} }\right)^2}\right) \mathbf u_r$ where: :$\mathbf u_r$ is the unit vector in the direction of the radial coordinate of $p$ :$\mathbf u_\theta$ is the unit vector in the direction of the angular coordinate of $p$


Existence of Non-Empty Finite Suprema in Join Semilattice

Let $\left({S, \preceq}\right)$ be a join semilattice. Let $A$ be a non-empty finite subset of $S$. Then $A$ admits a supremum in $\left({S, \preceq}\right)$.


Euler's Product form of Riemann Zeta Function

Let $s \in \R: s > 1$. Then: :$\displaystyle \sum_{k \mathop \in \N_{>0} } \dfrac 1 {k^s} = \prod_{p \mathop \in \Bbb P} \dfrac 1 {1 - 1 / p^s}$ where $\Bbb P$ denotes the set of all prime numbers.


Variance of Exponential Distribution

Let $X$ be a continuous random variable with the exponential distribution with parameter $\beta$. Then the variance of $X$ is: :$\var X = \beta^2$


Cardinality of Singleton

Let $A$ be a set. Then $\left\vert{A}\right\vert = 1$ {{iff}} $\exists a: A = \left\{ {a}\right\}$ where $\left\vert{A}\right\vert$ denotes the cardinality of $A$.


Existence of Non-Empty Finite Infima in Meet Semilattice

Let $\struct {S, \preceq}$ be a meet semilattice. Let $A$ be a non-empty finite subset of $S$. Then $A$ admits a infimum in $\struct {S, \preceq}$.


Thales' Theorem/Historical Note

This result was known by the ancient Babylonians as early as $2000$ BCE. The proof of what is now known as Thales' Theorem that {{AuthorRef|Thales of Miletus}} actually used is unknown.  Whether he was aware of the Interior Angles Theorem and used it in his proof cannot be decided. The Interior Angles Theorem is traditionally ascribed to {{AuthorRef|Pythagoras of Samos|Pythagoras}}, who lived near {{AuthorRef|Thales of Miletus|Thales}}, and may have met him. Thus it is possible that {{AuthorRef|Pythagoras of Samos|Pythagoras}} learned of the Interior Angles Theorem either directly or indirectly from {{AuthorRef|Thales of Miletus|Thales}} himself.


External Angle of Triangle equals Sum of other Internal Angles

The external angle of a triangle equals the sum of the other two internal angles. {{:Euclid:Proposition/I/32}}


Pappus's Centroid Theorem

=== Pappus's Centroid Theorem for Surface Area === {{:Pappus's Centroid Theorem/Surface Area}} === Pappus's Centroid Theorem for Volume === {{:Pappus's Centroid Theorem/Volume}}


Pappus's Centroid Theorem/Volume

Let $C$ be a plane figure that lies entirely on one side of a straight line $L$. Let $S$ be the solid of revolution generated by $C$ around $L$. Then the volume of $S$ is equal to the area of $C$ multiplied by the distance travelled by the centroid of $C$ around $L$ when generating $S$.


Pappus's Centroid Theorem/Surface Area

Let $C$ be a plane figure that lies entirely on one side of a straight line $L$. Let $S$ be the solid of revolution generated by $C$ around $L$. Then the surface area of $S$ is equal to the perimeter length of $C$ multiplied by the distance travelled by the centroid of $C$ around $L$ when generating $S$.


Pythagoras's Theorem for Parallelograms

Let $\triangle ABC$ be a triangle. Let $ACDE$ and $BCFG$ be parallelograms constructed on the sides $AC$ and $BC$ of $\triangle ABC$. Let $DE$ and $FG$ be produced to intersect at $H$. Let $AJ$ and $BI$ be constructed on $A$ and $B$ parallel to and equal to $HC$. Then the area of the parallelogram $ABIJ$ equals the sum of the areas of the parallelograms $ACDE$ and $BCFG$.


Mapping Preserves Finite and Filtered Infima

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be meet semilattices. Let $f: S_1 \to S_2$ be a mapping. Let $f$ preserve finite infima and preserve filtered infima. Then $f$ also preserves all infima


Volume of Sphere from Surface Area

The volume $V$ of a sphere of radius $r$ is given by: :$V = \dfrac {r A} 3$ where $A$ is the surface area of the sphere.


Volume of Solid of Revolution/Historical Note

The technique of finding the Volume of Solid of Revolution by dividing up the solid of revolution into many thin disks and approximating them to cylinders was devised by {{AuthorRef|Johannes Kepler}} sometime around or after $1612$, reportedly on the occasion of his wedding in $1613$. His inspiration was in the problem of finding the volume of wine barrels accurately. He published his technique in his $1615$ work {{BookLink|Nova Stereometria Doliorum Vinariorum|Johannes Kepler}} (New Stereometry of Wine Barrels). {{AuthorRef|Gottfried Wilhelm von Leibniz}} redefined the problem by applying the techniques of integral calculus around $1680$.


Factors of Mersenne Number M67

The Mersenne number $M_{67}$ has the factors: :$193 \, 707 \, 721$ :$761 \, 838 \, 257 \, 287$


Lower Bound is Lower Bound for Subset

Let $\struct {S, \preceq}$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$B \subseteq A$ Let $L$ be an element of $S$. Let $L$ be a lower bound for $A$. Then $L$ is a lower bound for $B$.


Upper Bound is Upper Bound for Subset

Let $\left({S, \preceq}\right)$ be a preordered set. Let $A, B$ be subsets of $S$ such that :$B \subseteq A$ Let $U$ be an upper bound for $A$. Then $U$ is an upper bound for $B$.


Largest Rectangle with Given Perimeter is Square

Let $\SS$ be the set of all rectangles with a given perimeter $L$. The element of $\SS$ with the largest area is the square with length of side $\dfrac L 4$.


Mapping Preserves Finite and Directed Suprema

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be join semilattices. Let $f: S_1 \to S_2$ be a mapping. Let $f$ preserve finite suprema and preserve directed suprema. Then $f$ also preserves all suprema


Integral of Power/Historical Note

The conventional proof of Integral of Power of course holds for all real $n \ne -1$, not just where $n$ is a strictly positive rational. However, the real point of this page is Fermat's proof, which demonstrates how integration was achieved before the full machinery of calculus had been thoroughly constructed. {{AuthorRef|Bonaventura Francesco Cavalieri}} had previously made progress with this problem, proving it for integral $1 \le n \le 9$ but the algebra for the proof of each power was more difficult than the previous one, and he found $10$ too much hard work. The clear beauty of {{AuthorRef|Pierre de Fermat|Fermat}}'s approach was that it works for ''all'' $n$, rational as well as integral.


Integer as Sum of Polygonal Numbers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then $n$ is: :$(1): \quad$ Either triangular or the sum of $2$ or $3$ triangular numbers :$(2): \quad$ Either square or the sum of $2$, $3$ or $4$ square numbers :$(3): \quad$ Either pentagonal or the sum of $2$, $3$, $4$ or $5$ pentagonal numbers :and so on. That is: :for all $k \ge 3$, $n$ is the sum of no more than $k$ polygonal numbers of order $k$.


Infima Preserving Mapping on Filters is Increasing

Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $f: S \to T$ be a mapping. For every filter $F$ in $\struct {S, \preceq}$, let $f$ preserve the infimum on $F$. Then $f$ is increasing.


Infimum of Upper Closure of Set

Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$. Let $U = T^\succeq$ be the upper closure of $T$ in $S$. Let $s \in S$. Then $s$ is the infimum of $T$ {{iff}} it is the infimum of $U$.


Cavalieri's Principle/Extension

Let two solid figures $S_1$ and $S_2$ have equal height. Let the areas of the sections made by planes parallel to their bases and at equal distances from the bases always have the same ratio. Then the volumes of $S_1$ and $S_2$ are in that same ratio.


Cavalieri's Principle

Let two solid figures $S_1$ and $S_2$ have equal height. Let sections made by planes parallel to their bases and at equal distances from the bases always have equal area. Then the volumes of $S_1$ and $S_2$ are equal.


Volume of Cone

Let $K$ be a cone whose base is of area $A$ and whose height is $h$. Then the volume of $K$ is given by: :$V_K = \dfrac {A h} 3$


Volume of Cylinder

The volume $V_C$ of a cylinder whose bases are circles of radius $r$ and whose height is $h$ is given by the formula: :$V_C = \pi r^2 h$


Slope of Tangent to Cycloid

Let $C$ be a cycloid generated by the equations: :$x = a \paren {\theta - \sin \theta}$ :$y = a \paren {1 - \cos \theta}$ The slope of the tangent to $C$ at the point $\tuple {x, y}$ is given by: :$\dfrac {\d y} {\d x} = \cot \dfrac \theta 2$


Upper Closure of Singleton

Let $\left({S, \preceq}\right)$ be an ordered set. Let $s$ be an element of $S$. Then: :$\left\{ {s}\right\}^\succeq = s^\succeq$ where: :$\left\{ {s}\right\}^\succeq$ denotes the upper closure of $\left\{ {s}\right\}$ :$s^\succeq$ denotes the upper closure of $s$


Lower Closure of Singleton

Let $\left({S, \preceq}\right)$ be an ordered set. Let $s$ be an element of $S$. Then: : $\left\{ {s}\right\}^\preceq = s^\preceq$ where: :$\left\{ {s}\right\}^\preceq$ denotes the lower closure of $\left\{ {s}\right\}$ :$s^\preceq$ denotes the lower closure of $s$.


Upper Closure of Element is Filter

Let $\struct {S, \preceq}$ be an ordered set. Let $s$ be an element of $S$. Then: :$s^\succeq$ is a filter in $\struct {S, \preceq}$ where $s^\succeq$ denotes the upper closure of $s$.


Length of Logarithmic Spiral

Consider a logarithmic spiral $S$ given by the equation: :$r = a e^{b \theta}$ Construct a tangent to $S$ at the point $Q = \left({a, 0}\right)$. Let the tangent cross the $y$-axis at $P$. Then the length of $PQ$ equals the total length of $S$ from $P$ inwards to the origin.


Length of Logarithmic Spiral/Historical Note

The length of a logarithmic spiral was first found by {{AuthorRef|Evangelista Torricelli}} in $1645$. This was the first time anybody had found the length of a non-straight-line curve for anything other than a circle. Before this had been done, few people could accept that this was possible to do. For example, {{AuthorRef|René Descartes}} had stated in his {{BookLink|La Géométrie|René Descartes}} in $1637$: :''Geometry should not include lines that are like strings, in that they are sometimes straight and sometimes curved, since the ratios between straight and curved lines are not known, and I believe cannot be discovered by human minds.'' {{AuthorRef|Galileo}}'s response was: :'' Who is so blind as not to see that, if there are two equal straight lines, one of which is then bent into a curve, that curve will be equal to the straight line?''


Infimum of Upper Closure of Element

Let $\struct {S, \preceq}$ be an ordered set. Let $s$ be an element of $S$. Then: :$\map \inf {s^\succeq} = s$ where $s^\succeq$ denotes the upper closure of $s$.


Volume of Gabriel's Horn

Consider Gabriel's horn, the solid of revolution formed by rotating about the $x$-axis the curve: :$y = \dfrac 1 x$ Consider the volume $V$ of the space enclosed by the planes $x = 1$, $x = a$ and the portion of Gabriel's horn where $1 \le x \le a$. Then: :$V = \pi \left({1 - \dfrac 1 a}\right)$


Pascal's Theorem

Let $ABCDEF$ be a hexagon whose $6$ vertices lie on a conic section and whose opposite sides are not parallel. Then the points of intersection of the opposite sides, when produced as necessary, all lie on a straight line.


Lower Closure of Element is Ideal

Let $\left({S, \preceq}\right)$ ne an ordered set. Let $s$ be an element of $S$. Then $s^\preceq$ is ideal in $\left({S, \preceq}\right)$ where $s^\preceq$ denotes the lower closure of $s$.


Supremum of Lower Closure of Element

Let $\left({S. \preceq}\right)$ be an ordered set. Let $s$ be an element of $S$. Then: :$\sup \left({s^\preceq}\right) = s$ where $s^\preceq$ denotes the lower closure of $s$.


Upper Closure is Decreasing

Let $\struct {S, \preceq}$ be an ordered set. Let $x, y$ be elements of $S$ such that :$x \preceq y$ then $y^\succeq \subseteq x^\succeq$ where $y^\succeq$ denotes the upper closure of $y$.


Lower Closure is Increasing

Let $\struct {S, \preceq}$ be an ordered set. Let $x, y$ be elements of $S$ such that :$x \preceq y$ then $x^\preceq \subseteq y^\preceq$ where $y^\preceq$ denotes the lower closure of $y$.


Suprema Preserving Mapping on Ideals is Increasing

Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $f: S \to T$ be a mapping. For every ideal $I$ in $\left({S, \preceq}\right)$, let $f$ preserve the supremum on $I$. Then $f$ is increasing.


Imaginary Unit to Power of Itself

:$i^i = e^{-\pi / 2}$ where $i$ is the imaginary unit. Its decimal expansion is: :$0 \cdotp 20787 \, 95763 \, 50761 \, 90854 \, 6955 \ldots$ {{OEIS|A049006}}


Imaginary Unit to Power of Itself/Complete

:$i^i = \set {\exp \paren {\dfrac {4 k + 3} 2 \pi}: k \in \Z}$ where $i$ is the imaginary unit.


Riemann Zeta Function of 4

The Riemann zeta function of $4$ is given by: {{begin-eqn}} {{eqn | l = \map \zeta 4       | r = \dfrac 1 {1^4} +  \dfrac 1 {2^4} +  \dfrac 1 {3^4} +  \dfrac 1 {4^4} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^4} {90}       | c =  }} {{eqn | o = \approx       | r = 1 \cdotp 08232 \, 3 \ldots       | c =  }} {{end-eqn}}


Riemann Zeta Function of 6

The Riemann zeta function of $6$ is given by: {{begin-eqn}} {{eqn | l = \map \zeta 6       | r = \dfrac 1 {1^6} +  \dfrac 1 {2^6} +  \dfrac 1 {3^6} +  \dfrac 1 {4^6} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^6} {945}       | c =  }} {{eqn | o = \approx       | r = 1 \cdotp 01734 \, 3 \ldots       | c =  }} {{end-eqn}}


Infima Preserving Mapping on Filters Preserves Filtered Infima

Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $f: S \to T$ be a mapping. For every filter $F$ in $\struct {S, \preceq}$, let $f$ preserve the infimum on $F$. Then $f$ preserves filtered infima.


Approximate Formula for Number of Partitions

The number of partitions $\map p n$ of a (strictly) positive integer $n$ is given by the approximation: :$\map p n \approx \dfrac {e^{\pi \sqrt {2 n / 3} } } {4 n \sqrt 3}$


Number of Partitions as Coefficient of Power Series

The number of partitions $\map p n$ of a (strictly) positive integer $n$ is equal to the coefficient of $x^n$ when the expression: :$\map f n = \dfrac 1 {\paren {1 - x} \paren {1 - x^2} \paren {1 - x^3} \cdots}$ is expanded into a power series. That is: :$\map f n = 1 + \map p 1 x + \map p 2 x^2 + \map p 3 x^3 + \cdots$ or: :$\displaystyle \sum_{n \mathop \in \Z_{\ge 0} } \map p n q^n = \prod_{j \mathop \in \Z_{>0} } \dfrac 1 {1 - q^j}$ where $\map p 0 := 1$.


Suprema Preserving Mapping on Ideals Preserves Directed Suprema

Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $f: S \to T$ be a mapping. Let every filter $F$ in $\left({S, \preceq}\right)$, $f$ preserve the infimum on $F$. Then $f$ preserves directed suprema.


Euler's Equations of Motion for Rotation of Rigid Body

Let a rigid body $B$ rotate about an axis $A$ which is fixed in relation to $B$ and parallel to the principal axis of inertia of $B$. Then the rotation of $B$ about $A$ is described by: :$\mathbf I \cdot \dot {\boldsymbol \omega} + \boldsymbol \omega \times \left({\mathbf I \cdot \boldsymbol\omega}\right) = \mathbf M$ where: :$\mathbf M$ is the applied torque applied to $B$ about $A$ :$\mathbf I$ is the moment of inertia of $B$ with respect to $A$ :$\boldsymbol \omega$ is the angular velocity about $A$.


Euler's Hydrodynamical Equation for Flow of Ideal Incompressible Fluid

:$\begin{cases} \dfrac {D \mathbf u} {D t} & = & - \nabla w + \mathbf g \\ \nabla \cdot \mathbf u & = & 0 \end{cases}$ where: :$\mathbf u$ denotes the flow velocity vector, with components in an $N$-dimensional space $u_1, u_2, \dots, u_N$ :$\dfrac D {D t}$ denotes the material derivative in time :$\cdot$ denotes the dot product :$\nabla$ denotes the nabla operator, used to represent the specific thermodynamic work gradient (first equation), and the flow velocity divergence (second equation) :$\mathbf u \cdot \nabla$ is the convective derivative :$w$ is the thermodynamic work per unit mass, the internal source term :$\mathbf g$ denotes body acceleration per unit mass acting on the continuum. {{stub|Absolutely no idea as to how best to present this. Leaving it as a stub for now.}} {{Namedfor|Leonhard Paul Euler|cat = Euler}}


Euler Buckling Formula

:$F = \dfrac {\pi^2 E I} {\paren {K L}^2}$ where: :$F$ = maximum or critical force (vertical load on column) :$E$ = modulus of elasticity :$I$ = area moment of inertia of the cross section of the rod :$L$ = unsupported length of column :$K$ = column effective length factor, whose value depends on the conditions of end support of the column, as follows: ::For both ends pinned (hinged, free to rotate), $K = 1.0$ ::For both ends fixed, $K = 0.50$ ::For one end fixed and the other end pinned, $K \approx 0.699$ ::For one end fixed and the other end free to move laterally, $K = 2.0$ :$K L$ is the effective length of the column


Mellin Transform of Heaviside Step Function/Lemma

Let $t \in \R$. Let $s \in \C$ with $\map \Re s < 0$. Then: :$\displaystyle \lim_{t \mathop \to +\infty} t^s = 0$


Up-Complete Lower Bounded Join Semilattice is Complete

Let $\left({S, \preceq}\right)$ be an up-complete lower bounded join semillattice. Then $\left({S, \preceq}\right)$ is complete.


Gregory Series

For $-\dfrac \pi 4 \le \theta \le \dfrac \pi 4$: :$\theta = \tan \theta - \dfrac 1 3 \tan^3 \theta + \dfrac 1 5 \tan^5 \theta - \ldots$


Differential Form of Arc Length on Curved Surface

The first fundamental form for the element of arc length on a curved surface is given by: :$\d s^2 = E \rd u^2 + 2 F \d u \rd v + G \rd v^2$ where $E$, $F$ and $G$ are the coefficients of the first fundamental form.


Lattice is Complete iff it Admits All Suprema

Let $\left({S, \preceq}\right)$ be an ordered set. Then $\left({S, \preceq}\right)$ is a complete lattice {{iff}} :$\forall X \subseteq S: X$ admits a supremum.


Theorema Egregium

The Gaussian curvature of a surface does not change if one bends the surface without stretching it. That is, Gaussian curvature can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient $3$-dimensional Euclidean space. Thus the Gaussian curvature is an intrinsic invariant of a surface.


Gauss-Bonnet Theorem

Let $M$ be a compact $2$-dimensional Riemannian manifold with boundary $\partial M$. Let $\Kappa$ be the Gaussian curvature of $M$. Let $k_g$ be the geodesic curvature of $\partial M$. Then : :$\displaystyle \int_M \kappa \, \mathrm d A + \int_{\partial M} k_g \, \mathrm d s = 2 \pi \chi\left({M}\right)$ where: :$\mathrm d A$ is the element of area of the surface :$\mathrm d s$ is the line element along $\partial M$ :$\chi\left({M}\right)$ is the Euler characteristic of $M$.


Unique Factorization Theorem for Gaussian Integers

For every Gaussian integer $n$ such that $\left\vert{n}\right\vert > 1$, $n$ can be expressed as the product of one or more Gaussian primes, uniquely up to the order in which they appear.


Dirichlet's Principle (Harmonic Functions)

Let the function $\map u x$ be the particular solution to Poisson's equation: :$\Delta u + f = 0$ on a domain $\Omega$ of $\R^n$ with boundary condition: :$u = g$ on $\partial \Omega$ Then $u$ can be obtained as the minimizer of the Dirichlet's energy: :$\displaystyle E \sqbrk {\map v x} = \int_\Omega \paren {\frac 1 2 \cmod {\nabla v}^2 - v f} \rd x$ amongst all twice differentiable functions $v$ such that $v = g$ on $\partial \Omega$ . This result holds provided that there exists at least one function which makes the Dirichlet integral finite.


Galois Connection is Expressed by Minimum

Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $g: S \to T$, $d: T \to S$ be mappings. Then $\tuple {g, d}$ is a Galois connection {{iff}}: :$g$ is an increasing mapping and ::$\forall t \in T: \map d t = \min \set {g^{-1} \sqbrk {t^\succsim} }$ where :$\min$ denotes the minimum :$g^{-1} \sqbrk {t^\succsim}$ denotes the image of $t^\succsim$ under relation $g^{-1}$ :$t^\succsim$ denotes the upper closure of $t$


Closed Form for Pentagonal Numbers

The closed-form expression for the $n$th pentagonal number is: :$P_n = \dfrac {n \paren {3 n - 1} } 2$


Closed Form for Hexagonal Numbers

The closed-form expression for the $n$th hexagonal number is: :$H_n = n \paren {2 n - 1}$


Sum of Even Integers is Even

The sum of any finite number of even integers is itself even.


Galois Connection is Expressed by Maximum

Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $g: S \to T$, $d: T \to S$ be mappings. Then $\tuple {g, d}$ is a Galois connection {{iff}} :$d$ is an increasing mapping and: ::$\forall s \in S: \map g s = \map \max {d^{-1} \sqbrk {s^\preceq} }$ where: :$\max$ denotes the maximum :$d^{-1} \sqbrk {s^\preceq}$ denotes the image of $s^\preceq$ under relation $d^{-1}$ :$s^\preceq$ denotes the lower closure of $t$


Compass and Straightedge Construction for Regular Heptagon does not exist

There exists no compass and straightedge construction for the regular heptagon.


Integer to Rational Power is Irrational iff not Integer or Reciprocal

Let $m$ be a rational number. Let $n$ be a positive integer. Then $n^m$ is an irrational number {{iff}} $n^{\size m}$ is not an integer.


Squaring the Circle by Archimedean Spiral

Let $C$ be a circle. It is possible to construct a square of the same area as $C$ using an Archimedean spiral.


Surface Area of Sphere

The surface area $A$ of a sphere whose radius $a$ is given by: :$A = 4 \pi a^2$


Upper Bound for Number of Grains of Sand to fill Universe

The number of grains of sand that would fill the Universe, to the extent it was believed to exist in the days of {{AuthorRef|Archimedes of Syracuse|Archimedes}}, is bounded above by $10^{63}$.


Lower Adjoint Preserves All Suprema

Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $d: T \to S$ be an lower adjoint of Galois connection. Then $d$ preserves all suprema.


Oscillation at Point (Infimum) equals Oscillation at Point (Limit)

Let $f: D \to \R$ be a real function where $D \subseteq \R$. Let $x$ be a point in $D$. Let $N_x$ be the set of open subset neighborhoods of $x$. Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\map {\omega_f} x = \inf \set {\map {\omega_f} I: I \in N_x}$ where $\map {\omega_f} I$ is the oscillation of $f$ on a real set $I$: :$\map {\omega_f} I = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$ Let $\map {\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by: :$\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$ Then: :$\map {\omega_f} x \in \R$ {{iff}} $\map {\omega^L_f} x \in \R$ and, if $\map {\omega_f} x$ and $\map {\omega^L_f} x$ exist as real numbers: :$\map {\omega_f} x = \map {\omega^L_f} x$


All Infima Preserving Mapping is Upper Adjoint of Galois Connection

Let $\left({S, \preceq}\right)$ be a complete lattice. Let $\left({T, \precsim}\right)$ be an ordered set. Let $g: S \to T$ be all infima preserving mapping. Then there exists a mapping $d: T \to S$ such that $\left({g, d}\right)$ is Galois connection and :$\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$ where :$\min$ denotes the minimum :$g^{-1}\left[{t^\succsim}\right]$ denotes the image of $t^\succsim$ under relation $g^{-1}$ :$t^\succsim$ denotes the upper closure of $t$


Binomial Theorem/General Binomial Theorem/Historical Note

The General Binomial Theorem was first conceived by {{AuthorRef|Isaac Newton}} during the years $1665$ to $1667$ when he was living in his home in Woolsthorpe. He announced the result formally, in letters to [https://en.wikipedia.org/wiki/Henry_Oldenburg Henry Oldenburg] on $13$th June $1676$ and $24$th October $1676$ but did not provide a proper proof (at that time the need for the appropriate level of rigor had not been recognised). {{AuthorRef|Leonhard Paul Euler}} made an incomplete attempt in $1774$, but the full proof had to wait for {{AuthorRef|Carl Friedrich Gauss}} to provide it in $1812$. This was, in fact, the first time anything about infinite summations was proved adequately.


Image of Inverse Image

Let $S, T$ be sets. Let $f: S \to T$ be a mapping. Let $X$ be a subset of $T$. Then: :$f\left[{f^{-1}\left[{X}\right]}\right] \subseteq X$ where: :$f^{-1}\left[{X}\right]$ denotes the image of $X$ under the relation $f^{-1}$.


All Suprema Preserving Mapping is Lower Adjoint of Galois Connection

Let $\struct {S, \preceq}$ be an ordered set. Let $\struct {T, \precsim}$ be a complete lattice. Let $d: T \to S$ be all suprema preserving mapping. Then there exists a mapping $g: S \to T$ such that $\left({g, d}\right)$ is Galois connection and :$\forall s \in S: \map g s = \map \max {d^{-1} \sqbrk {s^\preceq} }$ where :$\min$ denotes the maximum :$d^{-1} \sqbrk {s^\preceq}$ denotes the image of $s^\preceq$ under relation $d^{-1}$ :$s^\preceq$ denotes the lower closure of $t$


Galois Connection Implies Order on Mappings

Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be ordered sets. Let $g: S \to T$ and $d: T \to S$ be mappings such that :$\tuple {g, d}$ is Galois connection. Then $d \circ g \preceq I_S$ and $I_T \precsim g \circ d$ where :$\preceq, \precsim$ denote the orderings on mappings, :$I_S$ denotes the identity mapping of $S$


Primitive of exp (-x^2) has no Solution in Elementary Functions

The primitive: :$\displaystyle \int \map \exp {-x^2} \rd x$ cannot be expressed in terms of a finite number of elementary functions.


Primitive of exp x over x has no Solution in Elementary Functions

The primitive: :$\displaystyle \int \frac {e^{a x} \, \mathrm d x} x$ cannot be expressed in terms of a finite number of elementary functions.


Primitive of Sine of x over x has no Solution in Elementary Functions

The primitive: :$\displaystyle \int \frac {\sin x \, \mathrm d x} x$ cannot be expressed in terms of a finite number of elementary functions.


Primitive of Reciprocal of Logarithm of x has no Solution in Elementary Functions

The primitive: :$\displaystyle \int \frac {\mathrm d x} {\ln x}$ cannot be expressed in terms of a finite number of elementary functions.


Pi is Transcendental

$\pi$ (pi) is transcendental.


Solution of Qunitic Equation using Elliptic Functions

A quintic equation can be solved using elliptic functions.


Riemann Surface is Path-Connected

A Riemann surface is path-connected.


Properties of Riemann Surface

This page gathers some results about Riemann surfaces.


Radó's Theorem (Riemann Surfaces)

A Riemann surface is second countable.


Conformal Isomorphism of Universal Cover of Riemann Surface

The universal cover of a Riemann surface is conformally isomorphic to either: :the Riemann sphere :the complex plane or :the unit disk.


Riemann Surface is Metrizable

A Riemann surface is metrizable.


Riemann Surface admits Metric of Constant Curvature

A Riemann surface admits a metric of constant curvature.


Riemann Sphere is only Elliptic Riemann Surface

The Riemann sphere is the only elliptic Riemann surface (up to conformal isomorphism).


Parabolic Riemann Surface is Plane, Punctured Plane or Torus

A parabolic Riemann surface is conformally isomorphic to either: :the complex plane :the punctured complex plane $\C \setminus \set 0$ or: :a torus.


Riemann's Rearrangement Theorem

Let $S$ be an infinite series which is conditionally convergent. Then its terms can be arranged in a permutation so that the new series converges to any given value, or diverges.


Riemann-Christoffel Tensor in Two Dimensions is Gaussian Curvature

Let $M$ be a Riemannian manifold of dimension $2$. Then the Riemann-Christoffel tensor on $M$ reduces to the Gaussian curvature on $M$.


Riemannian Manifold has Zero Gaussian Curvature iff Euclidean

Let $M$ be a Riemannian manifold of dimension $2$. The Gaussian curvature on $M$ is zero {{iff}} the Riemannian metric on $\mathcal M$ is the same as the Euclidean metric.


Particle on Curved Surface under no Force moves along Geodesic

Consider a particle $P$ which is constrained to move on a curved surface $C$. Let $P$ be such that no force acts upon it. Then $P$ moves along a geodesic.


Sum of Reciprocals of Powers as Euler Product

Let $\zeta$ be the Riemann zeta function. Let $s \in \C$ be a complex number with real part $\sigma > 1$. Then: :$\displaystyle \map \zeta s = \prod_{\text {$p$ prime} } \frac 1 {1 - p^{-s} }$ where the infinite product runs over the prime numbers.


Shift Mapping is Lower Adjoint iff Appropriate Maxima Exist

Let $\left({S, \preceq}\right)$ be a meet semilattice. Then the following two conditions are equivalent: $(1): \quad \forall x \in S, f:S \to S: \left({\forall s \in S: f\left({s}\right) = x \wedge s}\right) \implies f$ is lower adjoint $(2): \quad \forall x, t \in S: \max \left\{ {s \in S: x \wedge s \preceq t}\right\}$ exists.


Brouwerian Lattice iff Shift Mapping is Lower Adjoint

Let $\left({S, \preceq}\right)$ be a lattice. Then $\left({S, \preceq}\right)$ is a Brouwerian lattice {{iff}} :$\forall x \in S, f: S \to S: \left({\forall s \in S: f \left({s}\right) = x \wedge s}\right) \implies f$ is a lower adjoint


Brouwerian Lattice is Distributive

Let $\left({S, \preceq}\right)$ be a Brouwerian lattice. Then $\left({S, \preceq}\right)$ is a distributive lattice


Definite Integral of Reciprocal of Root of a Squared minus x Squared

:$\displaystyle \int_0^x \frac {\d t} {\sqrt{1 - t^2} } = \arcsin x$


Meromorphic Function is Elliptic iff Doubly Periodic

Let $f: \C \to \C$ be a meromorphic function.  Then $\map f z$ is an elliptic function {{iff}} it is also doubly periodic.


Entire Function is Transcendental iff Power Series Expansion is Infinite

Let $f$ be an entire function. Then $f$ is '''transcendental''' {{iff}} the power series expansion of $f$ has an infinite number of coefficients which are non-zero.


Entire Function is Transcendental iff not Complex Polynomial Function

Let $f$ be an entire function. Then $f$ is '''transcendental''' {{iff}} it is not a complex polynomial function.


Complex Function is Entire iff it has Everywhere Convergent Power Series

Let $f: \C \to \C$ be a complex function. Then $f$ is an '''entire function''' {{iff}} $f$ can be given by an everywhere convergent power series: :$\displaystyle \map f z = \sum_{n \mathop = 0}^\infty a_n z^n; \quad \lim_{n \mathop \to \infty} \sqrt [n] {\size {a_n} } = 0$


Brouwerian Lattice is Upper Bounded

Let $\struct {S, \vee, \wedge, \preceq}$ be a Brouwerian lattice. Then $S$ is upper bounded.


Extreme Value Theorem/Real Function

Let $f$ be continuous in a closed real interval $\left[{a \,.\,.\, b}\right]$. Then: :$\forall x \in \left[{a \,.\,.\, b}\right]: \exists x_s \in \left[{a \,.\,.\, b}\right]: f \left({x_s}\right) \le f \left({x}\right)$ :$\forall x \in \left[{a \,.\,.\, b}\right]: \exists x_n \in \left[{a \,.\,.\, b}\right]: f \left({x_n}\right) \ge f \left({x}\right)$


Galileo's Paradox

The natural numbers $\N$ are in one-to-one correspondence with their squares: That is, the mapping: :$\forall n \in \N: f: n \mapsto n^2$ is a bijection. Hence the set of square numbers is equinumerous to the set of natural numbers. That is, a set is equinumerous to one of its proper subsets.


Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice

Let $\left({S, \vee, \wedge, \preceq}\right)$ be a Brouwerian lattice. Let $a$ be an element of $S$. Let $g, d: S \to S$ be mappings such that :$\forall s \in S: g\left({s}\right) = a \to s$ and :$\forall s \in S: d\left({s}\right) = a \wedge s$ Then $\left({g, d}\right)$ is Galois connection.


Inequality with Meet Operation is Equivalent to Inequality with Relative Pseudocomplement in Brouwerian Lattice

Let $\struct {S, \vee, \wedge, \preceq}$ be s Brouwerian lattice. Let $a, x, y \in S$. Then :$a \wedge x \preceq y$ {{iff}} $x \preceq a \to y$


Weierstrass Approximation Theorem

Let $f$ be a real function which is continuous on the closed interval $\Bbb I$. Then $f$ can be uniformly approximated on $\Bbb I$ by a polynomial function to any given degree of accuracy.


Construction of Right Angle by Stretched Rope

Let a rope be knotted at regular intervals so that there are $12$ equal segments separated by knots. Tie the rope in a loop consisting of those $12$ segments. Fix one of the knots to the ground at the point you want the right angle to be placed. Stretch the rope tightly in the direction of one of the legs of the right angle you want to create and fix the fourth knot to the ground. Stretch the rope tightly in the direction of the other leg of the right angle you want to create and fix the third knot in the ground, at the same time making sure the remaining segment of $5$ knots is also tight. The point where the $3$-knot section and the $4$-knot section meet, the rope will be bent at a right angle. :400px


Up-Complete Product

Let $\struct {S, \preceq_1}$, $\struct {T, \preceq_2}$ be non-empty ordered sets. Let $\struct {S \times T, \preceq}$ be Cartesian product of $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$.


Cartesian Product of Ordered Sets is Ordered Set

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets. Let $\left({S_1 \times S_2, \preceq}\right)$ be the Cartesian product of $\left({S_1, \preceq_1}\right)$ and $\left({S_2, \preceq_2}\right)$. Then $\left({S_1 \times S_2, \preceq}\right)$ is also an ordered set.


Motion of Rocket in Outer Space

Let $B$ be a rocket travelling in outer space. Let the velocity of $B$ at time $t$ be $\mathbf v$. Let the mass of $B$ at time $t$ be $m$. Let the exhaust velocity of $B$ be constant at $\mathbf b$. Then the equation of motion of $B$ is given by: :$m \dfrac {\mathrm d \mathbf v} {\mathrm d t} = - \mathbf b \dfrac {\mathrm d m} {\mathrm d t}$


Velocity of Rocket in Outer Space

Let $B$ be a rocket travelling in outer space. Let the velocity of $B$ at time $t$ be $\mathbf v$. Let the mass of $B$ at time $t$ be $m$. Let the exhaust velocity of $B$ be constant at $\mathbf b$. Then the velocity of $B$ at time $t$ is given by: :$\map {\mathbf v} t = \map {\mathbf v} 0 + \mathbf b \ln \dfrac {\map m 0} {\map m t}$ where $\map {\mathbf v} 0$ and $\map m 0$ are the velocity and mass of $B$ at time $t = 0$.


Acceleration of Rocket in Outer Space

Let $B$ be a rocket travelling in outer space. Let the velocity of $B$ at time $t$ be $\mathbf v$. Let the mass of $B$ at time $t$ be $m$. Let the exhaust velocity of $B$ be constant at $\mathbf b$. Then the acceleration of $B$ at time $t$ is given by: :$\mathbf a = \dfrac 1 m \left({- \mathbf b \dfrac {\mathrm d m} {\mathrm d t} }\right)$


Up-Complete Product/Lemma 2

Let $X$ be a directed subset of $S \times T$. Then :$\map {\pr_1^\to} X$ and $\map {\pr_2^\to} X$ are directed where  :$\pr_1$ denotes the first projection on $S \times T$ :$\pr_2$ denotes the second projection on $S \times T$ :$\map {\pr_1^\to} X$ denotes the image of $X$ under $\pr_1$


Sum of Cosines of Multiples of Angle

{{begin-eqn}} {{eqn | l = \frac 1 2 + \sum_{k \mathop = 1}^n \cos \left({k x}\right)       | r = \frac 1 2 + \cos x + \cos 2 x + \cos 3 x + \cdots + \cos n x       | c =  }} {{eqn| r = \frac {\sin \left({\left({2 n + 1}\right) x / 2}\right)} {2 \sin \left({x / 2}\right)}       | c =  }} {{end-eqn}} where $x$ is not an integer multiple of $2 \pi$.


Eratosthenes' Measurement of Earth

The circumference of Earth was measured by {{AuthorRef|Eratosthenes of Cyrene}} to be of the order of $45 \, 000 \, \mathrm{km}$.


Sum of Reciprocals of Twin Primes

The sum of the reciprocals of all the twin primes: :$\dfrac 1 3 + \dfrac 1 5 + \dfrac 1 7 + \dfrac 1 {11} + \dfrac 1 {13} + \dfrac 1 {17} + \dfrac 1 {19} + \dfrac 1 {29} + \dfrac 1 {31} + \cdots$ is either finite or convergent.


Prime Number Theorem in Eulerian Logarithmic Integral Form

The Prime Number Theorem is equivalent to: :$\displaystyle \lim_{x \mathop \to \infty} \frac {\map \pi x} {\map \Li x} = 1$ where: :$\map \pi x$ is the prime-counting function :$\map \Li x$ is the Eulerian logarithmic integral: ::$\displaystyle \map \Li x := \int_2^x \dfrac {\d t} {\ln t}$


Exponential of Rational Number is Irrational

Let $r$ be a rational number such that $r \ne 0$. Then: :$e^r$ is irrational where $e$ is Euler's number.


Rational Points on Graph of Exponential Function

Consider the graph $f$ of the exponential function in the real Cartesian plane $\R^2$: :$f := \set {\tuple {x, y} \in \R^2: y = e^x}$ The only rational point of $f$ is $\tuple {0, 1}$.


Rational Points on Graph of Logarithm Function

Consider the graph of the logarithm function in the real Cartesian plane $\R^2$: :$f := \set {\tuple {x, y} \in \R^2: y = \ln x}$ The only rational point of $f$ is $\tuple {1, 0}$.


Rational Points on Graph of Sine Function

Consider the graph of the sine function in the real Cartesian plane $\R^2$: :$f := \set {\tuple {x, y} \in \R^2: y = \sin x}$ The only rational point of $f$ is $\tuple {0, 0}$.


Rational Points on Graph of Cosine Function

Consider the graph of the cosine function in the real Cartesian plane $\R^2$: :$f := \left\{ {\left({x, y}\right) \in \R^2: y = \cos x}\right\}$ The only rational point of $f$ is $\left({0, 1}\right)$.


Meet-Continuous iff Ideal Supremum is Meet Preserving

Let $\mathscr S = \struct {S, \vee, \wedge, \preceq}$ be an up-complete lattice. Let $f: \map {\it Ids} {\mathscr S} \to S$ be a mapping such that: :$\forall I \in \map {\it Ids} {\mathscr S}: \map f I = \sup_{\mathscr S} I$ where  :$\map {\it Ids} {\mathscr S}$ denotes the set of all ideals in $\mathscr S$ Then :$\mathscr S$ is meet-continuous {{iff}} :$f$ preserves meet as a mapping from $\struct {\map {\it Ids} {\mathscr S}, \subseteq}$ into $\mathscr S$


Rational Number is Algebraic

Let $r \in \Q$ be a rational number. Then $r$ is also an algebraic number.


Square Root of 2 is Algebraic of Degree 2

The square root of $2$ is an algebraic number of degree $2$.


Constructible Length with Compass and Straightedge

Let $L$ be a line segment in a Eucldiean space. Let the length of $L$ be $d$. Let $L'$ be a line segment of length $d'$ constructed from $L$ using a compass and straightedge construction. Then: :$d' = q d$ where $q$ is an algebraic number whose degree is at most $2$.


Gelfond-Schneider Constant is Transcendental

The Gelfond-Schneider constant: :$2^{\sqrt 2}$ is transcendental.


Sum of Bernoulli Numbers by Binomial Coefficients Vanishes

:$\forall n \in \Z_{>1}: \displaystyle \sum_{k \mathop = 0}^{n - 1} \binom n k B_k = 0$ where $B_k$ denotes the $k$th Bernoulli number.


Bernoulli Numbers are Rational

The Bernoulli numbers are rational.


Power Series Expansion for Tangent Function/Proof of Convergence

The radius of convergence of the Power Series Expansion for Tangent Function: :$\displaystyle \tan x = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} } {\paren {2 n}!} x^{2 n - 1}$ where $B_{2 n}$ denotes the Bernoulli numbers, is given as: :$\size x < \dfrac \pi 2$


Power Series Expansion for Tangent Function/Sequence

The Power Series Expansion for Tangent Function begins: :$\tan x = x + \dfrac 1 3 x^3 + \dfrac 2 {15} x^5 + \dfrac {17} {315} x^7 + \dfrac {62} {2835} x^9 + \cdots$


Partial Fractions Expansion of Cotangent

Let $x \in \R - \Z$ , that is such that $x$ is a real number that is not an integer. Then: :$\pi \cot \pi x = \dfrac 1 2 + 2 x \displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {x^2 - n^2}$


Tangent to Cycloid is Vertical at Cusps

The tangent to the cycloid whose locus is given by: : $x = a \paren {\theta - \sin \theta}$ : $y = a \paren {1 - \cos \theta}$ is vertical at the cusps.


Tangent to Cycloid passes through Top of Generating Circle

Let $C$ be a cycloid generated by the equations: :$x = a \paren {\theta - \sin \theta}$ :$y = a \paren {1 - \cos \theta}$ Then the tangent to $C$ at a point $P$ on $C$ passes through the top of the generating circle of $C$.


Tangent to Cycloid

Let $C$ be a cycloid generated by the equations: : $x = a \left({\theta - \sin \theta}\right)$ : $y = a \left({1 - \cos \theta}\right)$ Then the tangent to $C$ at a point $\left({x, y}\right)$ on $C$ is given by the equation: :$y - a \left({1 - \cos \theta}\right) = \dfrac {\sin \theta} {1 - \cos \theta} \left({x - a \theta + a \sin \theta}\right)$


Equation of Hypocycloid

Let a circle $C_1$ of radius $b$ roll without slipping around the inside of a circle $C_2$ of (larger) radius $a$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis. Let $H$ be the hypocycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \tuple {x, y}$ is described by the equations: :$x = \paren {a - b} \cos \theta + b \map \cos {\paren {\dfrac {a - b} b} \theta}$ :$y = \paren {a - b} \sin \theta - b \map \sin {\paren {\dfrac {a - b} b} \theta}$


Meet in Set of Ideals

Let $\mathscr S = \left({S, \preceq}\right)$ be a meet semilattice. Let ${\it Ids}\left({\mathscr S}\right)$ be the set of all ideals in $\mathscr S$. Let $I_1, I_2$ be ideals in $\mathscr S$. Then :$I_1 \wedge I_2 = \left\{ {i_1 \wedge i_2: i_1 \in I_1, i_2 \in I_2}\right\}$ where :$I_1 \wedge I_2$ denotes the meet in $\left({ {\it Ids}\left({\mathscr S}\right), \subseteq}\right)$


Intersection of Semilattice Ideals is Ideal

Let $\struct {S, \preceq}$ be a meet semilattice. Let $I_1, I_2$ be ideals in $\struct {S, \preceq}$. Then $I_1 \cap I_2$ is ideal in $\struct {S, \preceq}$


Number of Cusps of Hypocycloid from Integral Ratio of Circle Radii

Let $H$ be a hypocycloid $H$ generated by a circle $C_1$ of radius $b$ rolling within a circle $C_2$ of (larger) radius $a$. Let $a = n b$ where $n$ is an integer. Then $H$ has $n$ cusps.


Number of Cusps of Hypocycloid from Rational Ratio of Circle Radii

Consider the hypocycloid $H$ generated by a circle $C_1$ of radius $b$ rolling within a circle $C_2$ of (larger) radius $a$. Let $k = \dfrac a b$ be a rational number. Let $k$ be expressed in canonical form: :$k = \dfrac p q$ where $p$ and $q$ are coprime. Then $H$ has $p$ cusps.


Cusps of Hypocycloid from Irrational Ratio of Circle Radii

Consider the hypocycloid $H$ generated by a rotor $C_1$ of radius $b$ rolling within a stator $C_2$ of (larger) radius $a$. Let $k = \dfrac a b$ be an irrational number. Then $H$ has an infinite number of cusps, which are evenly and densely distributed around the circumference of $C_2$.


Equation of Astroid

Let $H$ be the astroid generated by the rotor $C_1$ of radius $b$ rolling without slipping around the inside of a stator $C_2$ of radius $a = 4 b$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. === Parametric Form === {{:Equation of Astroid/Parametric Form}} === Cartesian Form === {{:Equation of Astroid/Cartesian Form}}


Tangent to Astroid between Coordinate Axes has Constant Length

Let $C_1$ be a circle of radius $b$ roll without slipping around the inside of a circle $C_2$ of radius $a = 4 b$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis. Let $H$ be the astroid formed by the locus of $P$. The segment of the tangent to $H$ between the $x$-axis and the $y$-axis is constant, immaterial of the point of tangency.


Pendulum Contained by Cycloid moves along Cycloidal Path

Let a pendulum with a flexible rod be suspended from a point $P$. Let the rod be contained by a pair of bodies shaped as the arcs of a cycloid such that $P$ is the cusp between those two arcs. :400px Then the bob is constrained to move such that its path traces the arc of a cycloid.


Lower Closure of Meet of Lower Closures

Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $D_1, D_2$ be subsets of $S$. Then :$\left\{ {i_1 \wedge i_2: i_1 \in D_1^\preceq, i_2 \in D_2^\preceq}\right\}^\preceq = \left\{ {i_1 \wedge i_2: i_1 \in D_1, i_2 \in D_2}\right\}^\preceq$ where :$D_1^\preceq$ denotes the lower closure of $D_1$.


Equation of Cycloid in Cartesian Coordinates

Consider a circle of radius $a$ rolling without slipping along the $x$-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \tuple {x, y}$ is described by the equation: :$a \sin^{-1} \paren {\dfrac {\sqrt {2 a y - y^2} } a} = \sqrt {2 a y - y^2} + x$


Second Derivative of Locus of Cycloid

Consider a circle of radius $a$ rolling without slipping along the x-axis of a cartesian plane. Consider the point $P$ on the circumference of this circle which is at the origin when its center is on the y-axis. Consider the cycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The second derivative of the locus of $P$ is given by: :$y'' = -\dfrac a {y^2}$


Rate of Change of Cartesian Coordinates of Cycloid

Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$. Consider a point $P$ on the circumference of this circle. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. Then the rate of change of $x$ and $y$ can be expresssed as: {{begin-eqn}} {{eqn | l = \frac {\d x} {\d t}       | r = \mathbf v_0 \paren {1 - \cos \theta}       | c =  }} {{eqn | l = \frac {\d y} {\d t}       | r = \mathbf v_0 \sin \theta       | c =  }} {{end-eqn}} where $\theta$ is the angle turned by $C$ after time $t$.


Maximum Rate of Change of X Coordinate of Cycloid

Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$. Consider a point $P$ on the circumference of this circle. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The maximum rate of change of $x$ is $2 \mathbf v_0$, which happens when $P$ is at the top of the circle $C$.


Lower Closure is Closure Operator

Let $\left({S, \preceq}\right)$ be an ordered set. Then :lower closure of set is a closure operator.


Area inside Astroid

The area inside an astroid $H$ constructed within a circle of radius $a$ is given by: :$\AA = \dfrac {3 \pi a^2} 8$


Length of Arc of Astroid

The total length of the arcs of an astroid constructed within a stator of radius $a$ is given by: :$\LL = 6 a$


Area of Surface of Revolution from Astroid

Let $H$ be the astroid constructed within a circle of radius $a$. The surface of revolution formed by rotating $H$ around the $x$-axis: :400px evaluates to: :$\mathcal S = \dfrac {12 \pi a^2} 5$


Tusi Couple is Diameter of Stator

A Tusi couple is a degenerate case of the hypocycloid whose form is a straight line that forms a diameter of the stator.


Meet Preserves Directed Suprema

Let $\mathscr S = \struct {S, \preceq}$ be an up-complete meet semilattice such that :$\forall x \in S$, a directed subset $D$ of $S$: $x \preceq \sup D \implies x \preceq \sup \set {x \wedge y: y \in D}$


Equation of Astroid/Parametric Form

The point $P = \tuple {x, y}$ is described by the parametric equation: :$\begin{cases} x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$ where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$.


Equation of Astroid/Cartesian Form

The point $P = \tuple {x, y}$ is described by the equation: :$x^{2/3} + y^{2/3} = a^{2/3}$


Preceding iff Meet equals Less Operand

Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $x, y \in S$. Then :$x \preceq y$ {{iff}} $x \wedge y = x$


Simple Harmonic Motion of Point on Tusi Couple

Let $C_1$ and $C_2$ be the rotor and stator respectively of a Tusi couple $H$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let the center of $C_1$ move at a constant angular velocity $\omega$ around the center of $C_2$. Let $P$ be the point on the circumference of $C_1$ whose locus is $H$. Let $C_1$ be initially positioned so that $P$ its point of tangency to $C_2$, located on the $x$-axis. Then $P$ moves back and forward on the $x$-axis with simple harmonic motion with period $\dfrac {2 \pi} \omega$ and maximum speed $a \omega$.


Maximum Rate of Change of Y Coordinate of Astroid

Let $C_1$ and $C_2$ be the rotor and stator respectively of an astroid $H$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let the center $C$ of $C_1$ move at a constant angular velocity $\omega$ around the center of $C_2$. Let $P$ be the point on the circumference of $C_1$ whose locus is $H$. Let $C_1$ be positioned at time $t = 0$ so that $P$ its point of tangency to $C_2$, located on the $x$-axis. Let $\theta$ be the angle made by $OC$ to the $x$-axis at time $t$. Then the maximum rate of change of the $y$ coordinate of $P$ in the first quadrant occurs when $P$ is at the point where: :$x = a \paren {\dfrac 1 3}^{3/2}$ :$y = a \paren {\dfrac 2 3}^{3/2}$


Equation of Deltoid

The point $P = \tuple {x, y}$ is described by the parametric equation: :$\begin{cases} x & = 2 b \cos \theta + b \cos 2 \theta \\ y & = 2 b \sin \theta - b \sin 2 \theta \end{cases}$ where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$.


Length of Arc of Deltoid

The total length of the arcs of a deltoid constructed within a stator of radius $a$ is given by: :$\LL = \dfrac {16 a} 3$


Equation of Epicycloid

Let a circle $C_1$ of radius $b$ roll without slipping around the outside of a circle $C_2$ of radius $a$. Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin. Let $P$ be a point on the circumference of $C_1$. Let $C_1$ be initially positioned so that $P$ is its point of tangency to $C_2$, located at point $A = \tuple {a, 0}$ on the $x$-axis. Let $H$ be the epicycloid traced out by the point $P$. Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane. The point $P = \tuple {x, y}$ is described by the equations: :$x = \paren {a + b} \cos \theta - b \map \cos {\paren {\dfrac {a + b} b} \theta}$ :$y = \paren {a + b} \sin \theta - b \map \sin {\paren {\dfrac {a + b} b} \theta}$


Equation of Nephroid

The point $P = \tuple {x, y}$ is described by the parametric equation: :$\begin{cases} x & = 3 b \cos \theta - b \cos 3 \theta \\ y & = 3 b \sin \theta - b \sin 3 \theta \end{cases}$ where $\theta$ is the angle between the $x$-axis and the line joining the origin to the center of $C_1$.


Length of Arc of Nephroid

The total length of the arcs of a nephroid constructed around a stator of radius $a$ is given by: :$\LL = 12 a$


Supremum of Meet Image of Directed Set

Let $\struct {S, \preceq}$ be an up-complete meet semilattice. Let $f: S \times S \to S$ be a mapping such that :$\forall \tuple {x, y} \in S \times S: \map f {x, y} = x \wedge y$ Let $D$ be directed subset of $S \times S$ in Cartesian product $\struct {S \times S, \precsim}$ of $\struct {S, \preceq}$ and $\struct {S, \preceq}$. Then: :$\map \sup {\map {f^\to} D} = \sup \set {x \wedge y: x \in \map {\pr_1^\to} D, y \in \map {\pr_2^\to} D}$ where :$\pr_1$ denotes the first projection on $S \times S$ :$\pr_2$ denotes the second projection on $S \times S$ :$\map {\pr_1^\to} D$ denotes the image of $D$ under $\pr_1$


Meet is Increasing

Let $\left({S, \preceq}\right)$ be a meet semilattice. Let $f: S \times S \to S$ be a mapping such that :$\forall s, t \in S: f\left({s, t}\right) = s \wedge t$ Then: :$f$ is increasing as a mapping from Cartesian product $\left({S\times S, \precsim}\right)$ of $\left({S, \preceq}\right)$ and $\left({S, \preceq}\right)$ into $\left({S, \preceq}\right)$.


Meet Preserves Directed Suprema/Lemma 2

Let $x$ be an element of $S$, $D$ be a directed subset of $S$. Then: :$\paren {\sup D} \wedge x = \sup \set {d \wedge x: d \in D}$


Supremum by Suprema of Directed Set in Cartesian Product

Let $\struct {S, \preceq}$ be an up-complete meet semilattice. Let $\struct {S \times S, \precsim}$ be the Cartesian product of $\struct {S, \preceq}$ and $\struct {S, \preceq}$. Let $D$ be a directed subset of $S \times S$. Then: :$\sup D = \tuple {\map \sup {\map {\pr_1^\to} D}, \map \sup {\map {\pr_2^\to} D} }$


Evolute of Circle is its Center

The evolute of a circle is a single point: its center.