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Diffstat (limited to 'mindmap')
28 files changed, 647 insertions, 49 deletions
diff --git a/mindmap/Fourier Transform.org b/mindmap/Fourier Transform.org new file mode 100644 index 0000000..9996abd --- /dev/null +++ b/mindmap/Fourier Transform.org @@ -0,0 +1,12 @@ +:PROPERTIES: +:ID: 262ca511-432f-404f-8320-09a2afe1dfb7 +:END: +#+title: Fourier Transform +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +The Fourier Transform is a generalization of the Fourier Series. diff --git a/mindmap/Lagrangian mechanics.org b/mindmap/Lagrangian mechanics.org new file mode 100644 index 0000000..d306be7 --- /dev/null +++ b/mindmap/Lagrangian mechanics.org @@ -0,0 +1,106 @@ +:PROPERTIES: +:ID: 83da205c-7966-417e-9b77-a0a354099f30 +:END: +#+title: Lagrangian mechanics +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t +* Introduction +The Lagrangian, $L: (\mathbb{R}, \mathbb{R} \rightarrow \mathbb{R}, \mathbb{R} \rightarrow \mathbb{R}) \rightarrow \mathbb{R}$ is simply a functional: +\begin{align*} +L = L(t, f(t), f'(t)) +\end{align*} +Where the Lagrangian represents some metric by which we calculate how optimized $f(x)$ is. The action: +\begin{align*} +J[f] = \int_{a}^{b}L(t, f(t), f'(t))dt \\ +\end{align*} +Defines the actual relationship between $f(t)$ and its level of optimization, where $a$ and $b$ represent the start +and end points for a certain curve. For example, if you wanted to minimize the surface area of something, $a$ and $b$ +would be the starting and end points of the surface. +* Euler-Lagrange Equation +We first define some function: +\begin{align*} +g(t) := f(t) + \epsilon \nu(t) +\end{align*} +Where $f(t)$ is our optimized function and $\nu(t)$ represents some function we add to $f(t)$ such that we perturb it +by some small amount. Now $\epsilon$ is a small number such that the perturbation is small. Note that when $\epsilon = 0$, $g(t) = f(t)$, +our optimized function. +\begin{align*} +J[g] = \int_{a}^{b}L(t, g(t), g'(t))dt +\end{align*} +Now $J[g]$ is optimized when $g(t)$ is a maximum or minimum with respect to the Lagrangian. $\frac{dJ}{d\epsilon}$ represents the extent to which +the action changes when the perturbation changes. When $\epsilon = 0$, $g(t) = f(t)$, which means $\frac{dJ}{d\epsilon}$ evaluated at $\epsilon = 0$ +should be zero, by definition of maxima and minima. +\begin{align*} +\frac{dJ[g]}{d\epsilon} = \int_{a}^{b}\frac{dL}{d\epsilon}dt +\end{align*} +By the multivariable chain rule: +\begin{align*} +\frac{dL}{d\epsilon} = \frac{\partial L}{\partial t}\frac{dt}{d\epsilon} + \frac{\partial L}{\partial g}\frac{dg}{d\epsilon} + \frac{\partial L}{\partial g'}\frac{dg'}{d\epsilon} +\end{align*} +because $t$ does not depend on $\epsilon$, $g = f + \epsilon\nu$, and $g' = f' + \epsilon\nu'$: +\begin{align*} +\frac{dL}{d\epsilon} = \frac{\partial L}{\partial g}\nu(t) + \frac{\partial L}{\partial g'}\nu'(t) +\end{align*} +now substituting back into the integral: +\begin{align*} +\frac{dJ}{d\epsilon} = \int_{a}^{b}(\frac{\partial L}{\partial g}\nu(t) + \frac{\partial L}{\partial g'}\nu'(t))dt +\end{align*} +applying integration by parts to the right side: +\begin{align*} +\frac{dJ}{d\epsilon} = \int_{a}^{b}\frac{\partial L}{\partial g}\nu(t)dt + \nu(t)\frac{\partial L}{\partial g'}\bigg|_{a}^{b} - \int_{a}^{b}\nu(t)\frac{d}{dt}\frac{\partial L}{\partial g'}dt +\end{align*} +now $\nu(t)$ can be any perturbation of $f(t)$ but the boundary conditions must stay the same (every function that we are considering for optimization must have the same start and end points); +therefore, $\nu(a) = \nu(b) = 0$. We can evaluate the bar to be 0 as a result. Doing this, combining the integral, then factoring out $\nu(t)$: +\begin{align*} +\frac{dJ}{d\epsilon} = \int_{a}^{b}\nu(t)(\frac{\partial L}{\partial g} - \frac{d}{dt}\frac{\partial L}{\partial g'})dt +\end{align*} +Now we finally set $\epsilon = 0$. This means $g(t) = f(t)$, $g'(t) = f'(t)$, and $\frac{dJ}{d\epsilon} = 0$: +\begin{align*} +0 = \int_{a}^{b}\nu(t)(\frac{\partial L}{\partial f} - \frac{d}{dt}\frac{\partial L}{\partial f'})dt +\end{align*} +And now because $\nu(t)$ can be an arbitrarily large or small valued function as long as the boundary conditions remain the same and the left hand side +must be zero, we get the Euler-Lagrange equation: +\begin{align*} +\frac{\partial L}{\partial f} - \frac{d}{dt}\frac{\partial L}{\partial f'} = 0 +\end{align*} +This is because the integral implies that for all selections for this function $\nu(t)$, $\nu(t)(\frac{dL}{df} - \frac{d}{dt}\frac{dL}{dg'}) = 0$. Because $\nu(t)$ can be any +function assuming it satisfies the boundary conditions, this can only be the case if $\frac{dL}{df} - \frac{d}{dt}\frac{dL}{dg'} = 0$. +In physics, we re-cast $f$ as $q$ and $f'$ as $\dot{q}$, where $q$ and $\dot{q}$ are the /generalized coordinates/ and /generalized velocities/ respectively. +* The Hamiltonian +The Hamiltonian represents the total energy in the system; it is the [[id:23df3ba6-ffb2-4805-b678-c5f167b681de][Legendre Transformation]] of the Lagrangian. Applying the Legendre Transformation to the +Lagrangian for coordinate $\dot{q}$: +\begin{align*} +L = \frac{1}{2}m\dot{q}^{2} - V(q) \\ +H = \frac{\partial L}{\partial \dot{q}}\dot{q} - L +\end{align*} +the Hamiltonian is defined as: +\begin{align*} +H(q, p) = \sum _{i}p_{i}\dot{q_{i}} - L(q, \dot{q}) +\end{align*} +Or: +\begin{align*} +H(q, p) = \frac{p^{2}}{2m} + V(q) +\end{align*} +where $p$ is the generalized momentum, and $q$ is a generalized coordinate. This results in two differential equations, the first of which is: +\begin{align*} +\frac{\partial H}{\partial p_{i}} = \dot{q_{i}} +\end{align*} +which follows directly from the Hamiltonian definition. Then, from the Euler-Lagrange equation: +\begin{align*} +L = \sum_{i}p_{i}\dot{q_{i}} - H \\ +\frac{\partial(\sum_{i}p_{i}\dot{q_{i}} - H)}{\partial q_{i}} - \frac{d}{dt}\frac{\partial(\sum_{i}p_{i}\dot{q_{i}} - H)}{\partial \dot{q_{i}}} = 0 \\ +- \frac{\partial H}{\partial q_{i}} = \frac{dp_{i}}{dt} \\ +\frac{\partial H}{\partial q_{i}} = - \frac{dp_{i}}{dt} +\end{align*} +Although the generalized coordinate system in question does not have to be linear, we can encode all the differential +equations for all the coordinates at once with the [[id:4bfd6585-1305-4cf2-afc0-c0ba7de71896][del operator]]: +\begin{align*} +\vec{\nabla}_{p}H = \frac{d\vec{q}}{dt} \\ +\vec{\nabla}_{q}H = -\frac{d\vec{p}}{dt} +\end{align*} +this notation isn't standard and I kind of made it up, but I think it works, as long as you don't take the divergence +or the curl of this system to really mean anything. Note that in both the Hamiltonian formulation and Lagrangian formulation, +the differential equations reduce to [[id:6e2a9d7b-7010-41da-bd41-f5b2dba576d3][Newtonian mechanics]] if we are working in a linear coordinate system with energy conservation. diff --git a/mindmap/Legendre Transformation.org b/mindmap/Legendre Transformation.org new file mode 100644 index 0000000..f9bc51f --- /dev/null +++ b/mindmap/Legendre Transformation.org @@ -0,0 +1,31 @@ +:PROPERTIES: +:ID: 23df3ba6-ffb2-4805-b678-c5f167b681de +:END: +#+title: Legendre Transformation +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Definition +The Legendre Transformation represents a function in terms of the y-intercept of the tangent line at every point on the function. +we start with the equation for a tangent line: +\begin{align*} +y = mx + b +\end{align*} +However, the Legendre transform actually solves for $b$. For a general function $f(x)$ we define +the tangent line to a point on that function to be: +\begin{align*} +y = y'(x)x - b +\end{align*} +where subtracting $b$ is the convention, for some reason. Then solving for b: +\begin{align*} +b = y'(x)x - y +\end{align*} +The actual Legendre Transform requires $b$ to be a function of $y'$, therefore: +\begin{align*} +x(f') = (f'(x))^{-1} \\ +L\{f(x)\} = b(f') = f'x(f') - f((x(f')) +\end{align*} +In [[id:83da205c-7966-417e-9b77-a0a354099f30][Lagrangian mechanics]], the Hamiltonian can be defined as the Legendre transform of the Lagrangian. diff --git a/mindmap/Lorentz Force.org b/mindmap/Lorentz Force.org new file mode 100644 index 0000000..23aa782 --- /dev/null +++ b/mindmap/Lorentz Force.org @@ -0,0 +1,21 @@ +:PROPERTIES: +:ID: 658f3916-6b7f-4897-85c6-9acc82b13214 +:END: +#+title: Lorentz Force +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Definition +The Lorentz Force is an experimental law given by the equation: +\begin{align*} +\vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) +\end{align*} +Where $q$ is the electric charge, $\vec{E}$ is the [[id:63656810-537f-42fc-a38a-1468d763b39a][Electric Field]], and $\vec{B}$ is the Magnetic Field. +For a continuous charge distribution: +\begin{align*} +\vec{F} = \rho\vec{E} + \vec{J} \times \vec{B} +\end{align*} +Where $\vec{J}$ is the electric current. diff --git a/mindmap/Maxwell's Equations.org b/mindmap/Maxwell's Equations.org new file mode 100644 index 0000000..187f7a5 --- /dev/null +++ b/mindmap/Maxwell's Equations.org @@ -0,0 +1,78 @@ +:PROPERTIES: +:ID: fde2f257-fa2e-469a-bc20-4d11714a515e +:END: +#+title: Maxwell's Equations +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Introduction +Maxwell's Equations are a set of four differential equations in multiple dimensions that produce a complete classical +theory of electromagnetic phenomena. + +* Derivations +These are the derivations of all four laws in their differential forms based on the [[id:2a543b79-33a0-4bc8-bd1c-e4d693666aba][inverse square]] law for [[id:32f0b8b1-17bc-4c91-a824-2f2a3bbbdbd1][electrostatics]] ([[id:5388f4e8-7bb8-452e-b997-fe9892aefcf3][Coulomb's Law]]), the [[id:658f3916-6b7f-4897-85c6-9acc82b13214][Lorentz Force]], +as well as the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] and electromagnetic induction (which are just special cases of [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]]) as initial assumptions: +** Gauss' Law +This is given by the divergence of an [[id:2a543b79-33a0-4bc8-bd1c-e4d693666aba][inverse square]] field, specifically for an electric field which is the same as in [[id:32f0b8b1-17bc-4c91-a824-2f2a3bbbdbd1][electrostatics]]: +\begin{align*} +\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_{0}} +\end{align*} +** Divergence of Magnetic Field +The divergence of the magnetic field is the same as in [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatics]]: +\begin{align*} +\vec{\nabla} \cdot \vec{B} = 0 +\end{align*} +** Ampere's Law with Modifications +The [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatic]] magnetic field is given by the Bio-Savart Law, which can be derived from [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]] and [[id:5388f4e8-7bb8-452e-b997-fe9892aefcf3][Coulomb's Law]]: +\begin{align*} +\vec{B} = \frac{\mu_{0}}{4\pi}\int_{V}\frac{\vec{J} \times \hat{r}}{r^{2}}d\tau +\end{align*} +Now the curl of this field is given by [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatics]]: +\begin{align*} +\vec{\nabla} \times \vec{B} = \mu_{0}\vec{J} +\end{align*} +However, if you take the divergence of this equation, the left hand side reduces to zero by the definition of the [[id:4bfd6585-1305-4cf2-afc0-c0ba7de71896][del operator]], but the +right hand side does not always: +\begin{align*} +\vec{\nabla} \cdot \mu_{0}\vec{J} = \mu_{0} (\vec{\nabla} \cdot \vec{J}) \neq 0 +\end{align*} + +Given this problem, a correction is given via the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]]: +\begin{align*} +\vec{\nabla} \cdot \vec{J} = -\frac{\partial \rho}{\partial t} \\ +\epsilon_{0}(\vec{\nabla} \cdot \vec{E}) = \rho \\ +\vec{\nabla} \cdot \vec{J} = -\epsilon_{0}\vec{\nabla} \cdot \frac{\partial\vec{E}}{\partial t} +\end{align*} +So therefore when we account for the fact that $\vec{\nabla} \cdot \vec{\nabla} \times \vec{B} = 0$ +\begin{align*} +\vec{\nabla} \times \vec{B} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t} +\end{align*} +** Faraday's Law of Induction +By definition of electromagnetic induction (and to make Ampere's law consistent with relativity): +\begin{align*} +\vec{\nabla} \times \vec{E} = - \frac{\partial\vec{B}}{\partial t} +\end{align*} +Instead of assuming induction as an axiom, it is possible to fix Ampere's equation with the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] first, and then +assume Lorentz symmetry. This explanation is a work in progress. +* Implications +Maxwell's Equations can be used to calculate all electromagnetic phenomena on the macro scale all the way down to the atom. +In practice, solving Maxwell's Equations can be analytically impossible, so several simplifying assumptions are often made. +To recap, these are the four equations: +\begin{align*} +\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\epsilon_{0}} \\ +\vec{\nabla} \times \vec{E} = -\frac{\partial\vec{B}}{\partial t} \\ +\vec{\nabla} \cdot \vec{B} = 0 \\ +\vec{\nabla} \times \vec{B} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial\vec{E}}{\partial t} +\end{align*} +* Speed of Light +Maxwell's Equations can be shown to reproduce the speed of light in a vacuum, where: +\begin{align*} +\mu_{0}\epsilon_{0} = \frac{1}{c^{2}} +\end{align*} +* Relativity +It is known that Maxwell's Equations are consistent with [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]] and can be expressed +in terms of curved spacetime. In fact, if relativity is taken as an axiom, it can be proven that the electric +and magnetic fields are descriptions of the same phenomena; this can be taken as a specific example of a [[id:1b1a8cff-1d20-4689-8466-ea88411007d7][duality]]. diff --git a/mindmap/Newtonian mechanics.org b/mindmap/Newtonian mechanics.org index c78b1e3..7d4b414 100644 --- a/mindmap/Newtonian mechanics.org +++ b/mindmap/Newtonian mechanics.org @@ -54,12 +54,13 @@ In general, the total momentum is defined to be: \end{align*} And in real life, we observe that things can transfer momentum. That is: \begin{align*} -\vec{p}_{1} = -\vec{p}_{2} +\vec{p}_{1} = -\vec{p}_{2} + \vec{p}_{i} \end{align*} -When these two objects have the same position vector \( \vec{r}_{1} = \vec{r}_{2} \) (if they are point masses; don't have volume but have mass which is idealistic but works as an approximation). -Because this operation of momentum transfer is symmetrical: +Where $\vec{p}_{i}$ is the initial momentum of the objects. +this happens when these two objects have the same position vector \( \vec{r}_{1} = \vec{r}_{2} \) (if they are point masses; don't have volume but have mass which is idealistic but works as an approximation). +This operation of momentum transfer is symmetrical: \begin{align*} -\vec{p}_{2} = -\vec{p}_{1} +\vec{p}_{2} = -\vec{p}_{1} + \vec{p}_{i} \end{align*} Note that the fact that this operation is symmetrical must be the case to preserve the property: \begin{align*} diff --git a/mindmap/conservative force.org b/mindmap/conservative force.org index 83d1c36..9b01117 100644 --- a/mindmap/conservative force.org +++ b/mindmap/conservative force.org @@ -15,7 +15,7 @@ A conservative force has this property: \end{align*} In other words, work done by \(\vec{f}\) is path independent, because in any closed loop integral, you go from point \(\vec{a}\) to point \(\vec{b}\) and then back. If these forwards and backwards -paths end up canceling no matter what path you take, then it is clear that \(\vec{f}\) will do the +paths end up canceling no matter what path you take, then it is clear that \(\vec{f}\) will be the same amount of force no matter what path you take. Using Stokes' theorem: \begin{align*} \int_{S}(\vec{\nabla} \times \vec{f}) \cdot d\vec{a} = \oint\vec{f} \cdot d\vec{l} diff --git a/mindmap/continuity equation.org b/mindmap/continuity equation.org new file mode 100644 index 0000000..579e09a --- /dev/null +++ b/mindmap/continuity equation.org @@ -0,0 +1,24 @@ +:PROPERTIES: +:ID: a871e62c-b4a0-4674-9dea-d377de2f780b +:END: +#+title: continuity equation +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Derivation and Motivation +In continuum mechanics, the continuity equation is a statement about the inability for particles to teleport between +two different points in space. In other words, each particle must take a path between two points. In particular, if +$Q_{enc} = \int_{V} \rho(r')d\tau$ or $Q_{enc}$ is the total amount of particles inside some surface where $\rho$ is the density: +\begin{align*} +\oint_{S} \vec{J} \cdot d\vec{a} = -\frac{\partial Q_{enc}}{\partial t}, +\end{align*} +or in other words, the amount that the current goes through some closed surface must be proportional to the loss of particles +inside of the enclosure. This is of course because of conservation of mass (which is in and of itself conservation of energy). +Using the [[id:44e65b69-e5d5-464a-b1f3-8a914e1b7e9e][divergence theorem]]: +\begin{align*} +\int_{V}\vec{\nabla} \cdot \vec{J}d\tau = - \int_{V}\frac{\partial\rho}{\partial t}d\tau \\ +\vec{\nabla} \cdot \vec{J} = -\frac{\partial\rho}{\partial t} +\end{align*} diff --git a/mindmap/del operator.org b/mindmap/del operator.org index 7410e30..3d9e2f5 100644 --- a/mindmap/del operator.org +++ b/mindmap/del operator.org @@ -86,6 +86,15 @@ vector field "perpendicular" to the divergence of the field. In fact, if you hav you can represent this field as an addition of a curl-less field and a divergence-less field. Another way to think of it is that you are measuring the strength of rotational component of the vector field about a certain axis. +** directional derivative +:PROPERTIES: +:ID: e255eb0a-246b-4a4b-8db8-ac0d15d9cc3c +:END: +The directional derivative is defined as follows: +\begin{align*} +\vec{f} \cdot \vec{\nabla} = \sum_{i=0}^{n}f_{i}\frac{\partial}{\partial x_{i}} +\end{align*} +Which represents a superposition of states which corresponds to the direction you want to take the derivative in. ** Laplacian :PROPERTIES: :ID: 65004429-a6b7-41f2-8489-07605841da3d @@ -98,6 +107,6 @@ It returns a scalar field and is the multivariable analogue to the second deriva and gradient have been described, I feel it is trivial to understand the Laplacian. ** Product Rules -The product rules pertaining to the del operator are consistent with that of linear algebra. +The product rules pertaining to the del operator are consistent with that of linear algebra and single variable derivative rules. For example, \( \vec{\nabla} \times \vec{\nabla}f = 0\). You can show this yourself quite easily, so I find no need to go over it here. -When in doubt, just assume the del works the same way as any old vector, and you will usually be correct. +When in doubt, just assume the del works the same way as any old vector except you apply the [[id:d1e245f4-0b04-450e-8465-a9c85fe57f7e][product rule]], and you will usually be correct. diff --git a/mindmap/derivative.org b/mindmap/derivative.org index be84116..d046459 100644 --- a/mindmap/derivative.org +++ b/mindmap/derivative.org @@ -7,7 +7,6 @@ #+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> #+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> #+options: broken-links:t -#+OPTIONS: tex:dvipng * Derivation Let's say we want to know the rate of change of the [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] \(f(x) = x^{2}\). Because this [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] is not @@ -76,14 +75,17 @@ We derive many of them here. = \frac{d}{dx}f(x) + \frac{d}{dx}g(x) \end{align*} of course, subtraction works in the same way. -** Multiplication Rule +** product rule +:PROPERTIES: +:ID: d1e245f4-0b04-450e-8465-a9c85fe57f7e +:END: \begin{align*} \frac{d}{dx}(f(x)g(x)) = \lim_{h\to0}\frac{f(x + h)g(x + h) - f(x)g(x)}{h} = \lim_{h\to0}\frac{f(x + h)g(x + h) - f(x)g(x + h) + f(x)g(x + h) - f(x)g(x)}{h} \\ = \lim_{h\to0}\frac{g(x + h)(f(x + h) - f(x)) + f(x)(g(x + h) - g(x))}{h} \\ = g(x)\lim_{h\to0}\frac{f(x + h) - f(x)}{h} + f(x)\frac{g(x + h) - g(x)}{h} = g(x)f'(x) + g'(x)f(x) \end{align*} And using the this rule as well as the chain rule and power rule which we will show later, the division rule is easily acquired. -** Chain Rule +** chain rule :PROPERTIES: :ID: ffd1bc3d-ab64-4916-9c09-0c89d2731b6d :END: diff --git a/mindmap/divergence theorem.org b/mindmap/divergence theorem.org new file mode 100644 index 0000000..b2c6660 --- /dev/null +++ b/mindmap/divergence theorem.org @@ -0,0 +1,17 @@ +:PROPERTIES: +:ID: 44e65b69-e5d5-464a-b1f3-8a914e1b7e9e +:END: +#+title: divergence theorem +#+author: Preston Pan +#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t + +* Definition +The [[id:12a2d5b3-f98c-45e5-9107-5560288b5aa8][divergence]] theorem is a generalization of the one-dimensional fundamental theorem of calculus: +\begin{align*} +\int_{V}\vec{\nabla} \cdot \vec{f}(\vec{r})d\tau = \oint_{S}\vec{f}(\vec{r}) \cdot d\vec{a} +\end{align*} +it is a statement about conservation in physical systems: the outflow of force from the inside of a closed boundary has +to be equal to the amount of force at the boundaries. diff --git a/mindmap/duality.org b/mindmap/duality.org index cef6faa..c18db08 100644 --- a/mindmap/duality.org +++ b/mindmap/duality.org @@ -71,7 +71,9 @@ p \neq \neg p. This statement filters for binary, or as I would call it, dual mode frameworks, and gets around the principle of explosion. We have an intuitive understanding of truth and falsehood, and we can use those general terms whenever there is a mutually exclusive divide. In short, you can view the logical framework as an abstraction of all other dual frameworks. I propose that you can do analysis on all dual frameworks in much the same -way group theory does analysis on groups. +way [[id:ba7b95b0-0ce6-4b33-9a79-5e5fddaea710][group]] theory does analysis on groups. [[id:a6bc601a-7910-44bb-afd5-dffa5bc869b1][Mathematics]] in general has the same recursive binary structure to it, because it is based on a couple +of axioms and utilizes logic as an extrapolation mechanism. By [[id:4ed61028-811e-4425-b956-feca6ee92ba1][inheritance]], everything that utilizes [[id:a6bc601a-7910-44bb-afd5-dffa5bc869b1][mathematics]] is also an inherently dualistic +structure. * Programming Explains Duality Of course, there is logic in programming, but that is kind of boring. What I am going to explain here is a recursive, binary structure known @@ -120,7 +122,7 @@ into small tasks, which is needed for recursion to be finite. * Why duality, and not Any other Modality? This is a good question, and one that I've still yet to answer completely. However, I would still like to try my hand at this, because there are things that make the number two specially suited for the task of subdividing. -** Two is a Natural Number +** Two is a [[id:2d6fb5ac-a273-4b33-949c-37380d03c076][Natural Number]] From a biological perspective, we're probably more used to dealing with whole numbers. We did not even come up with the concept of any others until much later, and negative numbers, and even zero, were a construct invented much later as well. Yes, there are an infinite number of natural numbers, but at least it's a filter we can use. @@ -131,5 +133,5 @@ prime can be represented by a smaller factor of that number. For example, 4-alit What's interesting is that one is a factor of everything. This represents the "null filter", or "anti filter", which doesn't filter any data and simply represents it all as one thing. Very interesting. ** Two is small and not One -The number two is also the smallest natural number that is not one. This means it is the simplest way to subdivide any particular object. This makes +The number two is also the smallest [[id:2d6fb5ac-a273-4b33-949c-37380d03c076][natural number]] that is not one. This means it is the simplest way to subdivide any particular object. This makes it more elegant compared to some other modalities. diff --git a/mindmap/electrostatics.org b/mindmap/electrostatics.org index 55310de..6625d4c 100644 --- a/mindmap/electrostatics.org +++ b/mindmap/electrostatics.org @@ -14,6 +14,9 @@ defined by the charge that an object has that corresponds the force that it both to other objects. Charge is measured in coulombs and can be negative or positive, which leads us to the man himself: * Coulomb's Law +:PROPERTIES: +:ID: 5388f4e8-7bb8-452e-b997-fe9892aefcf3 +:END: In order to define the phenomena of electric force in the real world, we use this experimentally verified law known as Coulomb's Law. Let \( \vec{r_{1}} \) be the displacement of a charge \( Q \), and let \( \vec{r_{2}} \) be the displacement of a charge \( q \), where these two charges are named \( P_{1}\) and \( P_{2} \) respectively. @@ -53,6 +56,9 @@ but since this is electrostatics and not electrodynamics, you will not have to w magnetic constants. Again, it is just a shift from speed of light units to our mortal units. ** Electric Field +:PROPERTIES: +:ID: 63656810-537f-42fc-a38a-1468d763b39a +:END: Okay, now we can continue to defining the /electric field/ of a particle. Let's call \( P_{1} \) our /test charge/, and \( P_{2} \) our /source charge/. If we now want to measure the force on \( P_{1} \), our equation is going to be the same. However, we can define a field \( \vec{E(\vec{r})} \) such that: diff --git a/mindmap/emergence.org b/mindmap/emergence.org index 9c975a5..a2ea32e 100644 --- a/mindmap/emergence.org +++ b/mindmap/emergence.org @@ -4,6 +4,9 @@ #+title: emergence #+author: Preston Pan #+html_head: <link rel="stylesheet" type="text/css" href="style.css" /> +#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script> +#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> +#+options: broken-links:t * Emergence systems are… Existent in many forms and at many levels. They are the fundamental building @@ -17,16 +20,42 @@ into a larger structure via simpler rules that each component follows. Entire systems such as [[id:a6bc601a-7910-44bb-afd5-dffa5bc869b1][mathematics]] can be explained in terms of a couple axioms, where all the theorems arising from those axioms are emergent from those axioms. At least, that is a relatively simple explanation, and does not capture the full -beauty of emergent systems. Therefore, I call apoun an example from daily life: +beauty of emergent systems. Therefore, I call upon an example from daily life: *** Societies At every scale, societies exhibit properties of emergence. For example, families and small communities comprise small areas of town, which comprise cities, which then make up provinces, and finally countries. In this particular example, we self organize into self-similar [[id:8f265f93-e5fd-4150-a845-a60ab7063164][recursive]] hierarchies. This is for a good reason; -in order for societies to scale, there need to be abstractions. Each level in the +in order for societies to scale, there needs to be abstractions. Each level in the hierarchy conveys more but less exact information, until we get to the national level which deals the most with aggregates. *** Markets Emergent -To use a particular hierarchy example, markets are emergent from barter in goods. +In order to define the emergence of markets, we must first define the emergence of +property rights. Property rights emerge when a society scales and is able to extract +value from more and more natural resources. Property rights come from two different +things: a need for certain people to use certain resources more or less often +(specialization, division of labor causes this, as well as belongings), and a need +for some incentive for people to maintain resources. When property is traded, a market +emerges, as well as a currency; a currency is simply the most fungible and liquid +asset that has a market. When goods are traded, both parties gain value from the trade. +In this way, the distribution of resources becomes more optimal the more consensual trade happens. +Capital, or resources used to produce other resources, are sold to people that believe +they have the skills to utilize the capital, from people that believe that the capital is suboptimally +allocated to them. Companies are simply contractually enforced percentage ownerships of capital. + +Prices are adjusted in markets according to supply and demand; they are a signal of the relative scarcity +of the good in question. High prices communicate to consumers that they should look for alternatives +due to the scarcity of the good, and low prices communicate to consumers that the resource is currently +widely available. + +This whole study of markets has lead to the study of human behavior known as economics. Although this +naive view of markets gives a good intuition about human behavior, it is an incomplete story. Economists +have made more particular statements about the state of economic affairs with the tools of [[id:a6bc601a-7910-44bb-afd5-dffa5bc869b1][mathematics]], +under neokeynesian theory. + +*** Languages Emergent + +* Key Takeaway +Emergence is the bootstrapping problem that attempts to solve the problem of induction. diff --git a/mindmap/factorial.org b/mindmap/factorial.org index 345fe39..2f738b1 100644 --- a/mindmap/factorial.org +++ b/mindmap/factorial.org @@ -5,4 +5,11 @@ #+author: Preston Pan #+options: num:nil #+html_head: <link rel="stylesheet" type="text/css" href="../style.css" /> -#+options: tex:dvipng + +* Introduction +The factorial [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] $n!: \mathbb{N} \rightarrow \mathbb{N}$ describes the amount of ways one can arrange $n$ differentiable objects. In practice: +\begin{align*} +0! = 1 \\ +n! = (n - 1)! \times n +\end{align*} +is a [[id:8f265f93-e5fd-4150-a845-a60ab7063164][recursive]] definition of the factorial. diff --git a/mindmap/function.org b/mindmap/function.org index 07b86cb..716a1ec 100644 --- a/mindmap/function.org +++ b/mindmap/function.org @@ -8,8 +8,6 @@ #+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> #+options: broken-links:t - - * Definition A function \( f(x) \) is a set \( S \) of ordered pairs that map the first value of the ordered pair to the second value in the ordered pair, where the first value may not have duplicates in the set \(S\). The map from the first value to the diff --git a/mindmap/group.org b/mindmap/group.org index 5fb0498..fb24bf8 100644 --- a/mindmap/group.org +++ b/mindmap/group.org @@ -28,3 +28,6 @@ An inverse is defined as follows: \begin{align*} \forall a \exists a^{-1} : a * a^{-1} = e \end{ |