From 561bac75579391c14e47eaccfabdf9eda98855da Mon Sep 17 00:00:00 2001 From: Thomas Albers Date: Wed, 27 Jul 2022 18:13:20 +0200 Subject: Initial commit --- math/cylindrical_coordinates.org | 37 ++++++++++++++++ math/derivatives.org | 31 ++++++++++++++ math/index.org | 13 ++++++ math/integrals.org | 20 +++++++++ math/levi_cevita.org | 26 +++++++++++ math/math.css | 85 ++++++++++++++++++++++++++++++++++++ math/matrices.org | 32 ++++++++++++++ math/orbit.org | 93 ++++++++++++++++++++++++++++++++++++++++ math/polar_coordinates.org | 36 ++++++++++++++++ math/spherical_coordinates.org | 39 +++++++++++++++++ math/trigonometry.org | 51 ++++++++++++++++++++++ math/vectors.org | 58 +++++++++++++++++++++++++ 12 files changed, 521 insertions(+) create mode 100644 math/cylindrical_coordinates.org create mode 100644 math/derivatives.org create mode 100644 math/index.org create mode 100644 math/integrals.org create mode 100644 math/levi_cevita.org create mode 100644 math/math.css create mode 100644 math/matrices.org create mode 100644 math/orbit.org create mode 100644 math/polar_coordinates.org create mode 100644 math/spherical_coordinates.org create mode 100644 math/trigonometry.org create mode 100644 math/vectors.org (limited to 'math') diff --git a/math/cylindrical_coordinates.org b/math/cylindrical_coordinates.org new file mode 100644 index 0000000..fab185b --- /dev/null +++ b/math/cylindrical_coordinates.org @@ -0,0 +1,37 @@ +#+TITLE:Cylindrical Coordinates +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} +#+LATEX_HEADER: \newcommand{\deriv}[2]{\frac{\text{d}#1}{\text{d}#2}} +#+LATEX_HEADER: \newcommand{\unitv}[1]{\bm{\hat{e}}_#1} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) +* Coordinate transformations +\begin{align*} +x &= r \cos\varphi\\ +y &= r \sin\varphi\\ +z &= z +\end{align*} + +* Local unit vectors +\begin{align*} +\unitv{r} &= \cos\theta \unitv{x} + \sin\theta \unitv{y}\\ +\unitv{\theta} &= -\sin\theta \unitv{x} + \cos\theta \unitv{y}\\ +\unitv{z} &= \unitv{z} +\end{align*} +* Kinematic in cylindrical coordinates +** Time derivatives of the local unit vectors +\begin{align*} +\deriv{\unitv{r}}{t} &= \dot{\theta}\unitv{\theta}\\ +\deriv{\unitv{\theta}}{t} &= -\dot{\theta}\unitv{r}\\ +\deriv{\unitv{z}}{t} &= 0 +\end{align*} +** Position vector and its time derivatives +\begin{align*} +\bm{r} &= r\unitv{r} + z\unitv{z}\\ +\bm{v} &= \dot{r}\unitv{r} + r\dot\theta\unitv{\theta} + \dot{z}\unitv{z}\\ +\bm{a} &= \left(\ddot{r}-r\dot{\theta}^2\right)\unitv{r} + \left(2\dot{r}\dot{\theta}+r\ddot{\theta}\right)\unitv{\theta} + \ddot{z}\unitv{z} +\end{align*} diff --git a/math/derivatives.org b/math/derivatives.org new file mode 100644 index 0000000..4d5d4be --- /dev/null +++ b/math/derivatives.org @@ -0,0 +1,31 @@ +#+TITLE:Table of Derivatives +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} +#+LATEX_HEADER: \newcommand{\dcoff}[1]{\frac{\text{d}}{\text{d}x} #1} +#+LATEX_HEADER: \newcommand{\sdcoff}[1]{\frac{\text{d}#1}{\text{d}x}} +#+LATEX_HEADER: \newcommand{\deriv}[2]{\frac{\text{d}}{\text{d}x} #1 &= #2} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) + +* General Properties of the Derivative +Let $f$ and $g$ be real valued functions and $c$ some real constant: +\begin{align*} +\dcoff{(cf)} &= c\sdcoff{f}\\ +\dcoff{(f \pm g)} &= \sdcoff{f} \pm \sdcoff{g}\\ +\dcoff{(fg)} &= \sdcoff{f}g + f\sdcoff{g}\\ +\dcoff{\left(\frac{f}{g}\right)} &= \frac{\sdcoff{f}g - f\sdcoff{g}}{g^2} +\end{align*} + +* Trigonometric Funtions +\begin{align*} +\deriv{\sin(x)}{\cos(x)}\\ +\deriv{\cos(x)}{-\sin(x)}\\ +\deriv{\tan(x)}{\sec^2(x)}\\ +\deriv{\sec(x)}{\sec(x)\tan(x)}\\ +\deriv{\csc(x)}{\csc(x)\cot(x)}\\ +\deriv{\csc(x)}{-\csc^2(x)} +\end{align*} diff --git a/math/index.org b/math/index.org new file mode 100644 index 0000000..069c4b6 --- /dev/null +++ b/math/index.org @@ -0,0 +1,13 @@ +#+TITLE: Math and Physics articles +#+SETUPFILE: ../math_options.org + +- [[file:vectors.org][Cross Product]] +- [[file:cylindrical_coordinates.org][Cylindrical Coordinates]] +- [[file:matrices.org][Matrix Properties]] +- [[file:orbit.org][Orbit]] +- [[file:polar_coordinates.org][Polar Coordinates]] +- [[file:spherical_coordinates.org][Spherical Coordinates]] +- [[file:derivatives.org][Table of Derivatives]] +- [[file:integrals.org][Table of Integrals]] +- [[file:levi_cevita.org][The Levi Cevita Symbol]] +- [[file:trigonometry.org][Trigonometric identities]] \ No newline at end of file diff --git a/math/integrals.org b/math/integrals.org new file mode 100644 index 0000000..6a32903 --- /dev/null +++ b/math/integrals.org @@ -0,0 +1,20 @@ +#+TITLE:Table of Integrals +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} +#+LATEX_HEADER: \newcommand{\intg}[2]{\int #1 \text{d}x &= #2 + C} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) + +* Trigonometric Funtions +\begin{align*} +\intg{\sin(x)}{-\cos(x)}\\ +\intg{\cos(x)}{\sin(x)}\\ +\intg{\tan(x)}{-\ln(\cos(x))}\\ +\intg{\sec(x)}{\ln(\sec(x) + \tan(x))}\\ +\intg{\csc(x)}{-\ln(\csc(x) + \cot(x))}\\ +\intg{\cot(x)}{\ln(\sin(x))} +\end{align*} diff --git a/math/levi_cevita.org b/math/levi_cevita.org new file mode 100644 index 0000000..aaa2074 --- /dev/null +++ b/math/levi_cevita.org @@ -0,0 +1,26 @@ +#+TITLE: The Levi Cevita Symbol +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) +* Levi Cevita symbol +\begin{equation*} +\varepsilon_{ijk} = \left\{\begin{array}{rl} +1,&(i,j,k) \in \{(1,2,3), (2,3,1), (3,1,2)\}\\ +-1,&(i,j,k) \in \{(3,2,1), (2,1,3), (1,3,2)\}\\ +0,&\text{otherwise} +\right{} +\end{array} +\end{equation*} + +\begin{equation*} +\varepsilon_{ijk} = \varepsilon_{jki} = \varepsilon_{kij} = - \varepsilon_{ikj} = - \varepsilon_{jik} = - \varepsilon_{kji} = 1 +\end{equation*} + +\begin{align*} +\sum_{k=1}^3 \varepsilon_{ijk}\varepsilon_{kmn} = \delta_{im} \delta_{jn} - \delta_{in} \delta_{jm} +\end{align*} diff --git a/math/math.css b/math/math.css new file mode 100644 index 0000000..d4e1054 --- /dev/null +++ b/math/math.css @@ -0,0 +1,85 @@ +/* +p.author +p.date +p.creator +.title +.subtitle +.todo +.done +.WAITING +.timestamp +.timestamp-kwd +.timestamp-wrapper +.tag +._HOME +.target +.linenr +.code-highlighted +div.outline-N +div.outline-text-N +.section-number-N +.figure-number +.table-number +.listing-number +div.figure +pre.src +pre.example +p.verse +div.footnotes +p.footnote +.footref +.footnum +.org-svg +*/ + +html { + height: 100%; + color: #cccccc; + background-color: hsl(240, 30%, 10%); +} + +body { + margin: auto; + max-width: 64em; + min-height: 100%; + display: flex; + flex-direction: column; + background-color: hsl(240, 30%, 15%); +} + +main { + padding: 2em 2em 2em 2em; + flex-grow: 1; +} + +h1 { + text-align: center; +} + +#content { + max-width: 50em; + margin: auto; +} + +div.outline-2:first-of-type { + padding: 0.25em 1em; + background-color: hsl(60,30%,30%); +} + +.equation-container { + display: block; + padding: 1em 2em; + background-color: hsl(240, 30%, 20%); + overflow: auto; + border: 1px solid hsl(240, 30%, 50%) +} + +img { + max-width: unset; +} + +.equation-container img { + object-fit: contain; + display: block; + margin: auto; +} diff --git a/math/matrices.org b/math/matrices.org new file mode 100644 index 0000000..a7ba58b --- /dev/null +++ b/math/matrices.org @@ -0,0 +1,32 @@ +#+TITLE:Matrix Properties +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) +* Basic properties +\begin{align*} +A + B = B + A +\end{align*} +* Dot product +\begin{align*} +(A^\text{T})^\text{T} &= A\\ +(A + B)^\text{T} &= A^\text{T} + B^\text{T}\\ +(AB)^\text{T} &= B^\text{T}A^\text{T} +\end{align*} +* Transpose +\begin{align*} +\bm{a} \cdot \bm{b} &= \overline{\bm{b}} \cdot \bm{a}\\ +\bm{a} \cdot \bm{b} &= \bm{a}^\text{T} \bm{b} +\end{align*} + +* Hermitian transpose +\begin{align*} +A^\ast &= \left[\overline{a_{ij}}\right]\\ +(\lambda A)^\ast &= \left[\overline{\lambda a_{ij}}\right] = \overline{\lambda} \left[a_{ij}\right]^\ast = \overline{\lambda}\,\overline{A}\\ +A^\dag &= (A^\ast)^\text{T}\\ +A^{\dag\dag} &= A +\end{align*} diff --git a/math/orbit.org b/math/orbit.org new file mode 100644 index 0000000..88079a6 --- /dev/null +++ b/math/orbit.org @@ -0,0 +1,93 @@ +#+TITLE:Orbit +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} +#+LATEX_HEADER: \usepackage{amssymb} +#+LATEX_HEADER: \newcommand{\deriv}[2]{\frac{\text{d}#1}{\text{d}#2}} +#+LATEX_HEADER: \newcommand{\unitv}[1]{\bm{\hat{e}}_#1} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) +* Deriving Kepler's first law from Newton's law of universal gravitation + +The movement of an object with mass $m$ orbiting another body with mass $M$ is +given by Newton's law of gravitation. If $m \ll M$ it is possible to consider +the position of the larger object constant and use it as the origin of our +coordinate system. Then the following equation applies for movement of the +smaller object: + +\begin{equation*} +m\ddot{\bm{r}} = - \frac{GMm}{r^3}\bm{r} \Leftrightarrow \ddot{\bm{r}} = - \frac{GM}{r^3}\bm{r} +\end{equation*} + +In order to solve this differential equation we first consider the angular +momentum of or object around its orbit. + +\begin{equation*} +\bm{L} = \bm{r} \times m \dot{\bm{r}} +\end{equation*} + +In the abscense of external toques, because the only force acting on the object +is parallel to its position, the angular momentum is conserved. + +\begin{equation*} +\deriv{\bm{L}}{t} = \dot{\bm{r}} \times m \dot{\bm{r}} + \bm{r} \times m \ddot{\bm{r}} = 0 +\end{equation*} + +We now multiply both sides of our equation from the right by the angular +momentum and develop the right side of the equation using vector identities. + +\begin{align*} +\ddot{\bm{r}} \times \bm{L} &= -GM\frac{\bm{r} \times \left(\bm{r} \times m \dot{\bm{r}}\right)}{r^3}\\ +&= - \frac{GMm}{r^3} \left(\left(\bm{r} \cdot \dot{\bm{r}}\right)\bm{r} - \left(\bm{r} \cdot \bm{r}\right)\dot{\bm{r}}\right)\\ +&= GMm\left(\frac{\left(\bm{r} \cdot \bm{r}\right)\dot{\bm{r}}}{r^3} - \frac{\left(\bm{r} \cdot \dot{\bm{r}}\right)\bm{r}}{r^3}\right)\\ +&= GMm\left(\frac{\dot{\bm{r}}}{r} - \frac{\left(\bm{r} \cdot \dot{\bm{r}}\right)\bm{r}}{r^3}\right)\\ +&= GMm\left(\frac{1}{r} \deriv{\bm{r}}{t} + \deriv{}{t}\left(\frac{1}{r}\right)\bm{r}\right)\\ +&= GMm\deriv{}{t}\left(\frac{\bm{r}}{r}\right) +\end{align*} + +We observe that each side of our equation is a derivative of a quantity. We know +integrate both sides and take the integrations constant into account. + +\begin{align*} +& \deriv{}{t} \left(\dot{\bm{r}} \times \bm{L}\right) = GM \deriv{}{t}\left(\frac{\bm{r}}{r}\right)\\ +\Leftrightarrow \quad & \dot{\bm{r}} \times \bm{L} = GM \frac{\bm{r}}{r} + \bm{a} +\end{align*} + +Our objective is now to solve the equation for $r$, so we multiply both sides by $\bm{r}$: + +\begin{align*} +\dot{\bm{r}} \times \left(\bm{r} \times \dot{\bm{r}}\right) &= GM \frac{\bm{r}}{r} + \bm{a}\\ +\bm{r} \cdot \left(\dot{\bm{r}} \times \left(\bm{r} \times \dot{\bm{r}}\right)\right) &= GMr + \bm{r} \cdot \bm{a} +\end{align*} + +By applying a cyclic permutation of the resulting triple product and using the +known property of the scalar product we now express the equation only in terms +of the magnitudes of the vectors. + +\begin{align*} +\left(\bm{r} \times \dot{\bm{r}}\right) \cdot \left(\bm{r} \times \dot{\bm{r}}\right) &= GMr + \bm{r} \cdot \bm{a}\\ +\left(\frac{L}{m}\right)^2 &= GMr + \bm{r} \cdot \bm{a}\\ +\left(\frac{L}{m}\right)^2 &= GMr + r a \cos\theta +\end{align*} + +The last steps are to solve for $r$ + +\begin{align*} +r &= \left(\frac{L}{m}\right)^2 \frac{1}{GM + a \cos\theta}\\ +&= \left(\frac{L}{m}\right)^2 \frac{1}{GM} \frac{1}{1 + \frac{a}{GM} \cos\theta}\\ +&= \left(\frac{L}{m}\right)^2 \frac{1}{GM} \frac{1}{1 + e\cos\theta} +\end{align*} + +Finally we reach our result. Objects orbiting according to Newton's Law of +Gravitation follow paths that correspond to the conic sections. Here is Kepler's +first Law a special case, where our object has a stable orbit around the larger +body. + +\begin{equation*} +r &= \frac{L^2}{GM m^2} \frac{1}{1 + e\cos\theta} +\end{equation*} + +* Deriving a physical interpretation of the excentricity of the orbit diff --git a/math/polar_coordinates.org b/math/polar_coordinates.org new file mode 100644 index 0000000..ff07be3 --- /dev/null +++ b/math/polar_coordinates.org @@ -0,0 +1,36 @@ +#+TITLE:Polar Coordinates +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} +#+LATEX_HEADER: \newcommand{\deriv}[2]{\frac{\text{d}#1}{\text{d}#2}} +#+LATEX_HEADER: \newcommand{\unitv}[1]{\bm{\hat{e}}_#1} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) +* Coordinate transformations +\begin{align*} +x &= r \cos\theta\\ +y &= r \sin\theta +\end{align*} + +* Local unit vectors +\begin{align*} +\unitv{r} &= \cos\theta \unitv{x} + \sin\theta \unitv{y}\\ +\unitv{\theta} &= -\sin\theta \unitv{x} + \cos\theta \unitv{y} +\end{align*} + +* Kinematic in polar coordinates +** Time derivatives of the local unit vectors +\begin{align*} +\deriv{\unitv{r}}{t} &= \dot{\theta}\unitv{\theta}\\ +\deriv{\unitv{\theta}}{t} &= -\dot{\theta}\unitv{r} +\end{align*} +** Position vector and its time derivatives +\begin{align*} +\bm{r} &= r\unitv{r}\\ +\bm{v} &= \dot{r}\unitv{r} + r\dot\theta\unitv{\theta}\\ +\bm{a} &= \left(\ddot{r} - r\dot\theta^2\right)\unitv{r} ++ \left(2\dot{r}\dot\theta + r\ddot\theta\right)\unitv{\theta} +\end{align*} diff --git a/math/spherical_coordinates.org b/math/spherical_coordinates.org new file mode 100644 index 0000000..1c7fd9c --- /dev/null +++ b/math/spherical_coordinates.org @@ -0,0 +1,39 @@ +#+TITLE:Spherical Coordinates +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} +#+LATEX_HEADER: \newcommand{\deriv}[2]{\frac{\text{d}#1}{\text{d}#2}} +#+LATEX_HEADER: \newcommand{\unitv}[1]{\bm{\hat{e}}_#1} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) +* Coordinate transformations +\begin{align*} +x &= r \sin\theta \cos\varphi\\ +y &= r \sin\theta \sin\varphi\\ +z &= r \cos\theta +\end{align*} + +* Local unit vectors +\begin{align*} +\bm{\hat{e}}_r &= \sin\theta \cos\varphi \bm{\hat{e}}_x + \sin\theta \sin\varphi \bm{\hat{e}}_y + \cos\theta \bm{\hat{e}}_z\\ +\bm{\hat{e}}_\theta &= \cos\theta \cos\varphi \bm{\hat{e}}_x + \cos\theta \sin\varphi \bm{\hat{e}}_y - \sin\theta \bm{\hat{e}}_z\\ +\bm{\hat{e}}_\varphi &= - \sin\theta \sin\varphi \bm{\hat{e}}_x + \sin\theta \cos\varphi \bm{\hat{e}}_y +\end{align*} +* Kinematic in spherical coordinates +** Time derivatives of the local unit vectors +\begin{align*} +\deriv{\unitv{r}}{t} &= \dot{\theta}\unitv{\theta} + \dot{\varphi}\sin\theta \unitv{\varphi}\\ +\deriv{\unitv{\theta}}{t} &= -\dot{\theta}\unitv{r} + \dot{\varphi}\cos\theta \unitv{\varphi}\\ +\deriv{\unitv{\varphi}}{t} &= -\dot{\varphi} \left(\sin\theta\unitv{r} + \cos\theta\unitv{\theta}\right) +\end{align*} +** Position vector and its time derivatives +\begin{align*} +\bm{r} &= r\unitv{r}\\ +\bm{v} &= \dot{r}\unitv{r} + r\dot\theta\unitv{\theta} + r\dot\varphi\sin\theta\unitv{\varphi}\\ +\bm{a} &= \left(\ddot{r} - r\dot\theta^2 - r\dot\varphi^2\sin^2\theta\right)\unitv{r} ++ \left(2\dot{r}\dot\theta + r\ddot\theta - r\dot\varphi^2\sin\theta\cos\theta\right)\unitv{\theta} ++ \left(2\dot{r}\dot\varphi\sin\theta + 2r\dot\theta\dot\varphi\cos\theta + r\ddot\varphi\sin\theta\right)\unitv{\varphi} +\end{align*} diff --git a/math/trigonometry.org b/math/trigonometry.org new file mode 100644 index 0000000..3e81eeb --- /dev/null +++ b/math/trigonometry.org @@ -0,0 +1,51 @@ +#+TITLE:Trigonometric identities +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) + +* Pythagorean identities +\begin{align*} +&\cos^2\left(x\right) + \sin^2\left(x\right) = 1\\ +&\tan^2\left(x\right) + 1 = \sec^2\left(x\right)\\ +&1 + \cot^2\left(x\right) = \csc^2\left(x\right) +\end{align*} +* Sum of angles +\begin{align*} +&\sin\left(a \pm b\right) = \sin\left(a\right)\cos\left(b\right) \pm \cos\left(a\right)\sin\left(b\right)\\ +&\cos\left(a \pm b\right) = \cos\left(a\right)\cos\left(b\right) \mp \sin\left(a\right)\sin\left(b\right)\\ +&\tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a) \tan(b)} +\end{align*} +* Multiple angles +\begin{align*} +&\sin(2\theta) = 2\sin(\theta)\cos(\theta)\\ +&\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta +\end{align*} +* Half-angle formulae +\begin{align*} +&\sin \frac{\theta}{2} = \pm \sqrt{\frac{1-\cos\theta}{2}}\\ +&\cos \frac{\theta}{2} = \pm \sqrt{\frac{1+\cos\theta}{2}} +\end{align*} +* Power-reduction formulae +\begin{align*} +&\sin^2\theta = \frac{1 - \cos(2\theta)}{2}\\ +&\cos^2\theta = \frac{1 + \cos(2\theta)}{2} +\end{align*} +* Product-to-sum formulae +\begin{align*} +&2 \cos\theta \cos\varphi = \cos(\theta-\varphi) + \cos(\theta+\varphi)\\ +&2 \sin\theta \sin\varphi = \cos(\theta-\varphi) - \cos(\theta+\varphi)\\ +&2 \sin\theta \cos\varphi = \sin(\theta+\varphi) + \sin(\theta-\varphi)\\ +&2\cos\theta\sin\varphi = \sin(\theta+\varphi) - \sin(\theta-\varphi) +\end{align*} +* Sum-to-product formulae +\begin{align*} +&\sin\theta \pm \sin\varphi = 2 \sin\left(\frac{\theta\pm\varphi}{2}\right) \cos\left(\frac{\theta\mp\varphi}{2}\right)\\ +&\cos\theta + \cos\varphi = 2 \cos\left(\frac{\theta+\varphi}{2}\right) \cos\left(\frac{\theta-\varphi}{2}\right)\\ +&\tan\theta \pm \tan\varphi = \frac{\sin(\theta\pm\varphi)}{\cos\theta \cos\varphi}\\ +&\cos\theta - \cos\varphi = - 2 \sin\left(\frac{\theta+\varphi}{2}\right) \sin\left(\frac{\theta-\varphi}{2}\right) +\end{align*} diff --git a/math/vectors.org b/math/vectors.org new file mode 100644 index 0000000..49d8132 --- /dev/null +++ b/math/vectors.org @@ -0,0 +1,58 @@ +#+TITLE:Cross Product +#+SETUPFILE: ../math_options.org +#+LATEX_HEADER: \usepackage{bm} +#+LATEX_HEADER: \usepackage{mathtools} + +* Disclaimer +This site as of now just a technology demonstration and its claims +should not be taken as true (even though I myself am pretty confident +they are) + +* Definition +We shall define the cross product in terms of two properties we are +interested in: + +- Distributive: $\bm{a} \times \left(\bm{b} + \bm{c}\right) = \bm{a} \times \bm{b} + \bm{a} \times \bm{c}$ +- Orthogonal: $\bm{a} \times \bm{b} = \bm{c} \implies \bm{c} \cdot \bm{a} = \bm{0} \land \bm{c} \cdot \bm{b} = \bm{0}$ + +It is worth mentioning that given a pair of vectors in $\mathbb{R}^3$ +there exist an infinite amount of vectors that satisfy these +properties, so it is also necessary to introduce the following +relations between the basis vectors to properly define the cross +product. + +\begin{align*} +\bm{e}_1 &= \bm{e}_2 \times \bm{e}_3\\ +\bm{e}_2 &= \bm{e}_3 \times \bm{e}_1\\ +\bm{e}_3 &= \bm{e}_1 \times \bm{e}_2\\ +\bm{e}_i \times \bm{e}_i &= \bm{0},\qquad\text{For}\quad i = 1,2,3 +\end{align*} + +We introduce the Levi-Civita symbol to condense our calculations. + +\begin{equation*} +\epsilon_{ijk} \coloneqq \bm{e}_i \cdot \left(\bm{e}_j \times \bm{e}_k\right) +\end{equation*} + +Based on this we may now derive a way to compute the cross product of +two vectors + +\begin{align*} +\left[\bm{a} \times \bm{b}\right]_i &= \left[\left(\sum_j a_j \bm{e}_j\right) \times \left(\sum_k b_k \bm{e_k}\right)\right]_i\\ +&= \left[\sum_{jk} a_j b_k \left(\bm{e}_j \times \bm{e}_k\right)\right]_i\\ +&= \sum_{jk} a_j b_k \bm{e}_i \cdot \left(\bm{e}_j \times \bm{e}_k\right)\\ +&= \sum_{jk} \epsilon_{ijk} a_j b_k\\ +\bm{a} \times \bm{b} &= \sum_{ijk} \epsilon_{ijk} a_i b_j \bm{e}_k +\end{align*} + +* Properties +\begin{align*} +&\bm{a} \times \bm{b} = - \bm{b} \times \bm{a}\\ +&\bm{a} \cdot \left(\bm{b} \times \bm{c}\right) += \bm{b} \cdot \left(\bm{c} \times \bm{a}\right) += \bm{c} \cdot \left(\bm{a} \times \bm{b}\right)\\ +&\bm{a} \times \left(\bm{b} \times \bm{c}\right) += \left(\bm{a} \cdot \bm{c}\right)\bm{b} - \left(\bm{a} \cdot \bm{b}\right)\bm{c}\\ +&\left(\bm{a} \times \bm{b}\right) \times \left(\bm{c} \times \bm{d}\right) += \left(\bm{a}\cdot\bm{c}\right) \left(\bm{b}\cdot\bm{d}\right) - \left(\bm{a}\cdot\bm{d}\right) \left(\bm{b}\cdot\bm{c}\right) +\end{align*} -- cgit v1.2.3