From c57fe104a57ad3d9bbf818be430881d92f00731f Mon Sep 17 00:00:00 2001 From: Luc Date: Mon, 23 Oct 2023 16:32:39 +0200 Subject: [PATCH] Added and restructed mathematics section. --- config/en/mkdocs.yaml | 19 ++- docs/en/chemistry/start.md | 3 + .../first-order-ode.md} | 2 +- .../second-order-ode.md | 133 ++++++++++++++++++ .../systems-of-linear-ode.md | 2 + .../sets-and-numbers.md | 0 docs/en/philosophy/start.md | 3 + docs/en/physics/start.md | 3 + 8 files changed, 163 insertions(+), 2 deletions(-) create mode 100644 docs/en/chemistry/start.md rename docs/en/mathematics/{calculus/first-order-differential-equations.md => ordinary-differential-equations/first-order-ode.md} (97%) create mode 100644 docs/en/mathematics/ordinary-differential-equations/second-order-ode.md create mode 100644 docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md rename docs/en/mathematics/{calculus => set-theory}/sets-and-numbers.md (100%) create mode 100644 docs/en/philosophy/start.md create mode 100644 docs/en/physics/start.md diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index fe5ff03..7b48f1f 100755 --- a/config/en/mkdocs.yaml +++ b/config/en/mkdocs.yaml @@ -40,7 +40,24 @@ extra_javascript: nav: - 'Welcome': index.md - 'Mathematics': - - 'Start': mathmematics/start.md + - 'Start': mathematics/start.md + - 'Calculus': + - 'Limits': mathematics/calculus/limits.md + - 'Continuity': mathematics/calculus/continuity.md + - 'Differentation': mathematics/calculus/differentation.md + - 'Transcendental functions': + - 'Inverse functions': mathematics/calculus/transcendental-functions/inverse-functions.md + - 'Exponential and logarithmic functions': mathematics/calculus/transcendental-functions/exponential-and-logarithmic-functions.md + - 'Extremes values': mathematics/calculus/extremes-values.md + - 'Concavity and inflections': mathematics/calculus/concavity-and-inflections.md + - 'Taylor polynomials': mathematics/calculus/taylor-polynomials.md + - 'Integration': mathematicse/calculus/integration.md + - 'Integration techniques': mathematics/calculus/integration-techniques.md + - 'Improper integrals': mathematics/calculus/improper-integrals.md + - 'Ordinary differential equations': + - 'First order differential equations': mathematics/ordinary-differential-equations/first-order-ode.md + - 'Second order differential equations': mathematics/ordinary-differential-equations/second-order-ode.md + - 'Systems of linear differential equations': mathematics/ordinary-differential-equations/systems-of-linear-ode.md - 'Physics': - 'Start': physics/start.md diff --git a/docs/en/chemistry/start.md b/docs/en/chemistry/start.md new file mode 100644 index 0000000..5691f52 --- /dev/null +++ b/docs/en/chemistry/start.md @@ -0,0 +1,3 @@ +# Mathematics + +Welcome to the chemistry page. \ No newline at end of file diff --git a/docs/en/mathematics/calculus/first-order-differential-equations.md b/docs/en/mathematics/ordinary-differential-equations/first-order-ode.md similarity index 97% rename from docs/en/mathematics/calculus/first-order-differential-equations.md rename to docs/en/mathematics/ordinary-differential-equations/first-order-ode.md index 3ac664c..06da3bc 100755 --- a/docs/en/mathematics/calculus/first-order-differential-equations.md +++ b/docs/en/mathematics/ordinary-differential-equations/first-order-ode.md @@ -1,6 +1,6 @@ # First-order differential equations -## First order linear differential equations +## First-order linear differential equations A first-order **linear** differential equation is one of the type diff --git a/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md b/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md new file mode 100644 index 0000000..8828373 --- /dev/null +++ b/docs/en/mathematics/ordinary-differential-equations/second-order-ode.md @@ -0,0 +1,133 @@ +# Second-order ordinary differential equations + +For simplicity, all definitions and statements are for complex values functions and vector spaces over $\mathbb{C}$. + +## Linear second-order ODEs with constant coefficients + +Let $L[y] = f$ be given by + +$$ +L[y] = \ddot y + p \dot y + qy = f \qquad (*), +$$ + +with $f,p,q \in \mathbb{R}$. + +*Definition*: the set of all solutions to $(*)$ is called the general solution. + +*Property*: if $y_1,y_2$ are both solutions to the homogeneous case $L[y]=0$ then $\forall c_1,c_2 \in \mathbb{R}$, $y=c_1y_1 + c_2y_2$ is a solution. + +$$ +L[y] = L[c_1y_1 + c_2y_2] = c_1L[y_1] + c_2L[y_2], +$$ + +Then the consequence is that the general solution is a linear space. + +$(*)$ is said to have **resonance** if $f$ can be split into linearly independent terms of which at least one lies in the solution space of $(*)$. + +
+ +### Solving homogeneous linear second-order ODEs with constant coefficients + +Therefore solving + +$$ +L[y] = \ddot y + p \dot y + qy = 0. +$$ + +Ansatz: let $y(t) = e^{\lambda t}$ with $\lambda \in \mathbb{C}$. Then + +$$ +L[y(t)] = \lambda^2 e^{\lambda t} + p \lambda e^{\lambda t} + q e^{\lambda t} = e^{\lambda t} (\lambda^2 + p \lambda + q) = 0, +$$ + +obtaining the characteristic equation $\Chi(\lambda) = \lambda^2 + p \lambda + q = 0$. If two roots $\lambda_1,\lambda_2 \in \mathbb{C}$ are found the solution space is + +$$ +y(t) = c_1 e^{\lambda_1 t} + c_2 e^{\lambda_2 t}, \quad c_1,c_2 \in \mathbb{C}, +$$ + +if instead one root $\lambda_1 \in \mathbb{C}$ is foundt the solution space is + +$$ +y(t) = (c_1 + c_2t) e^{\lambda_1 t}. +$$ + +*Proof*: will at some point be added. + +#### Example + +Let the homogeneous linear second-order ode be given by $\ddot y + 4 \dot y + 8y = 0$. Then the characteristic equation is given by $\Chi(\lambda) = \lambda^2 + 4\lambda + 8 = 0$ with solutions $\lambda_1 = -2 + 2i$ and $\lambda_2 = -2 - 2i$. Then the general solution is given by + +$$ +y(t) = c_1 e^{(-2 + 2i)1 t} + c_2 e^{(-2 - 2i) t}, \quad c_1,c_2 \in \mathbb{C}, +$$ + +and we can write the real solution as + +$$ +y(t) = e^{-2t}\big(d_1\cos 2t + d_2 \sin 2t \big), \quad d_1,d_2 \in \mathbb{R}. +$$ + +
+ +### Solving inhomogeneous linear second-order ODEs with constant coefficients + +*Theorem*: let $y_p$ be a particular solution to $(*)$. Then the general solution to $(*)$ is given by + +$$ +y = y_H + y_p, +$$ + +with $y_H$ the solution to the homegeneous case. + +*Proof*: let $y$ be a solution to $(*)$, then $L[y - y_p] = L[y] - L[y_p] = f - f = 0$. Therefore $y = (y - y_p) + y_p = y_H + y_p$. + +#### Method of variation of parameters + +We need the general solution to the homogeneous case + +$$ +y_H(t) = c_1 y_1(t) + c_2 y_2(t), \qquad c_1,c_2 \in \mathbb{C}. +$$ + +Ansatz: let $y_p(t) = c_1(t) y_2(t) + c_2(t) y_2(t)$, then taking the derivative of $y_p(t)$ + +$$ +\dot y_p(t) = \dot c_1(t) y_2(t) + \dot c_2(t) y_2(t) + c_1(t) \dot y_2(t) + c_2(t) \dot y_2(t), +$$ + +we demand that $\dot c_1(t) y_2(t) + \dot c_2(t) y_2(t) = 0$. Then taking the second derivative of $y_p(t)$ + +$$ +\ddot y_p(t) = \dot c_1(t) \dot y_2(t) + \dot c_2(t) \dot y_2(t) + c_1(t) \ddot y_2(t) + c_2(t) \ddot y_2(t), +$$ + +then we have for $(*)$ + +$$ +\ddot y_p(t) + p \dot y_p(t) + q = c_1\big(\ddot y_1 + p \dot y_1 + q y_1\big) + c_2\big(\ddot y_2 + p \dot y_2 + q y_2\big) + \dot c_1 \dot y_1 + \dot c_2 \dot y_2 = f +$$ + +we demand that $\dot c_1 \dot y_1 + \dot c_2 \dot y_2 = f$. Then we can create a linear system of demands + +$$ +\begin{pmatrix} y_1 && y_2 \\ \dot y_1 && \dot y_2\end{pmatrix} \begin{pmatrix} \dot c_1 \\ \dot c_2 \end{pmatrix} = \begin{pmatrix} 0 \\ f \end{pmatrix}, +$$ + +named the Wronskian and we can solve for $c_1(t)$ and $c_2(t)$ by integration. + +#### Ansatz method + +Let $f(t) = p(t)e^{\lambda t}$, rule of thumb: $y_p$ is of related type to inhomogeneity $f$. Then for $A_n, B_n and P_n$ polynomials of degree $\leq n$ and $\alpha \in \R$ + +| Inhomogeneity | Particular solution | +| ------ | --------------- | +| $L[y] = P_n$ | $t^m A_n$ | +| $L[y] = P_n e^{\alpha t}$ | $t^m A_n e^{\alpha t}$ | +| $L[y] = P_n \cos \omega t$ | $t^m \big(A_n \cos \omega t + B_n \sin \omega t \big)$ | +| $L[y] = P_n \sin \omega t$ | $t^m \big(A_n \cos \omega t + B_n \sin \omega t \big)$ | +| $L[y] = P_n e^{\alpha t} \cos \omega t$ | $t^m e^{\alpha t} \big(A_n \cos \omega t + B_n \sin \omega t \big)$ | +| $L[y] = P_n e^{\alpha t} \sin \omega t$ | $t^m e^{\alpha t} \big(A_n \cos \omega t + B_n \sin \omega t \big)$ | + +Choose $m \in \N \cup \{0\}$ as small as possible such that no term in the ansatz solves the homogeneous equation $L[y] = 0$. + diff --git a/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md b/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md new file mode 100644 index 0000000..b31ad71 --- /dev/null +++ b/docs/en/mathematics/ordinary-differential-equations/systems-of-linear-ode.md @@ -0,0 +1,2 @@ +# Systems of linear ordinary differential equations + diff --git a/docs/en/mathematics/calculus/sets-and-numbers.md b/docs/en/mathematics/set-theory/sets-and-numbers.md similarity index 100% rename from docs/en/mathematics/calculus/sets-and-numbers.md rename to docs/en/mathematics/set-theory/sets-and-numbers.md diff --git a/docs/en/philosophy/start.md b/docs/en/philosophy/start.md new file mode 100644 index 0000000..af3dc90 --- /dev/null +++ b/docs/en/philosophy/start.md @@ -0,0 +1,3 @@ +# Philosophy + +Welcome to the pilosophy page. \ No newline at end of file diff --git a/docs/en/physics/start.md b/docs/en/physics/start.md new file mode 100644 index 0000000..2928a39 --- /dev/null +++ b/docs/en/physics/start.md @@ -0,0 +1,3 @@ +# Physics + +Welcome to the physics page. \ No newline at end of file