From c335c05f511a373681d8644500d7750a519f58fa Mon Sep 17 00:00:00 2001 From: Preston Pan Date: Thu, 13 Jul 2023 22:58:59 +0800 Subject: add a lot of things --- mindmap/duality.org | 9 ++--- mindmap/electrostatics.org | 97 ++++++++++++++++++++++++++++++++++++++++++++++ mindmap/emergence.org | 25 +++++++++++- mindmap/mathematics.org | 14 ++++++- mindmap/physics.org | 36 +++++++++++++++++ 5 files changed, 173 insertions(+), 8 deletions(-) create mode 100644 mindmap/electrostatics.org create mode 100644 mindmap/physics.org (limited to 'mindmap') diff --git a/mindmap/duality.org b/mindmap/duality.org index a47e951..cef6faa 100644 --- a/mindmap/duality.org +++ b/mindmap/duality.org @@ -130,9 +130,6 @@ 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. -*** P-ality, where P is Prime -I've yet to experiment with other P-alities, but I'm sure they work too. For now, I will say that they probably work, but won't be as elegant as -a duality, for other reasons: -** Two is small, and Close to e -The last thing that makes two unique is simply that it is the smallest prime number, which means thinking about the concept is relatively easy. It is -also an approximation of the constant e, or euler's constant, which has implications in computer science and storing data as well. +** 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 +it more elegant compared to some other modalities. diff --git a/mindmap/electrostatics.org b/mindmap/electrostatics.org new file mode 100644 index 0000000..e71a05e --- /dev/null +++ b/mindmap/electrostatics.org @@ -0,0 +1,97 @@ +:PROPERTIES: +:ID: 32f0b8b1-17bc-4c91-a824-2f2a3bbbdbd1 +:END: +#+title: electrostatics +#+author: Preston Pan +#+html_head: +#+html_head: +#+html_head: +#+options: broken-links:t + +* What is Electricity? +Because this is an introduction and not a lesson in quantum mechanics, I will say that electricity is broadly +defined by the charge that an object has that corresponds the force that it both feels and also gives +to other objects. Charge is measured in coulombs and can be negative or positive, which leads us to the man himself: + +* Coulomb's Law +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. +Then let \( \vec{r} = \vec{r_{1}} - \vec{r_{2}} \) be the distance between the charges. For simplicity, we assume +these charges have no mass or volume; we call these /point charges/. The equation for the force +on \( P_{1} \) is as follows: + +\begin{align*} +\vec{F(\vec{r})} = \frac{1}{4\pi\epsilon_{0}}\frac{qQ}{r^{2}} \hat{r}. +\end{align*} + +Where \( \hat{r} \) is the unit vector pointing in the direction of \( P_{2} \). Note that there are a +couple of interesting things about this force. First, it is an inverse square law, and the formula looks a lot like the one for gravitation, +only charge can be negative and mass cannot. Second, it is symmetrical, +in the sense that the force felt by \( P_{2} \) is going to be the same, only \( \hat{r} \) +is pointing in the other direction. Also, note that due to linearity, this force calculation follows the /superposition principle/. +That is, if we have different electrostatic forces acting on one particle: + +\begin{align*} +\vec{F_{tot}} = \vec{F_{1}} + \vec{F_{2}} + … = \sum_{i=1}^{n} \vec{F_{i}}. +\end{align*} + +Wait, where does the \( \frac{1}{4\pi\epsilon_{0}} \) term come from? Well, the surface area of a sphere +is \( 4\pi r^{2}\) , which explains both the inverse square law and this \( 4\pi \) term in the denominator, +but what about \( \epsilon_{0} \), what does it even mean? + +Well, it is simply a conversion of units from /speed of light/ terms to /SI unit terms/. If you +think of it like that, you will never need to know what the units actually are, although I'm +sure you can find that online. Just know that it is called the permeability of free space, and +it is defined in terms of the speed of light and a constant relating to magnetism: + +\begin{align*} +\epsilon_{0}\mu_{0} = \frac{1}{c^{2}} +\end{align*} + +but since this is electrostatics and not electrodynamics, you will not have to worry about +magnetic constants. Again, it is just a shift from speed of light units to our mortal units. + +** Electric Field +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: + +\begin{align*} +\vec{F} = Q\vec{E} +\end{align*} + +Where: + +\begin{align*} +\vec{E} = \frac{\vec{F}}{Q} +\end{align*} + +Therefore, the value of \( \vec{E} \) for a point charge must be: + +\begin{align*} +\vec{E} := \frac{1}{4\pi\epsilon_{0}}\frac{q}{r^{2}}\hat{r}. +\end{align*} + +The result is we find a way to express force in a /test charge independent way/. This is useful +because we often want to find the force if an arbitrary object with an arbitrary charge is next +to the particle in question, instead of focusing specifically on two charges. + +Note that it is trivial to prove that \( \vec{E} \) also follows the superposition principle. + + +** Continuous Charge Distributions +Now that we have a working definition of \( \vec{E} \), we can now find the electric field of an object +that has a continuous charge distribution. Note that there aren't actually infinite charges in real +world objects which is what we are assuming by taking an integral over some space of charge, but +it's close enough because there are so many individual charges in real world objects. Assuming we are in +three dimensions: + +\begin{align*} +\vec{E(\vec{r})} = \frac{1}{4\pi\epsilon_{0}} \int_{space} \frac{\sigma(\vec{r_{2}})}{r^{2}}\hat{r}d\tau +\end{align*} + +Where \( \tau \) is the patch of volume we are integrating over, and \( \sigma \) is the charge density function, +which takes a position vector and returns the charge at that vector. Of course, surface and line integrals +have their own analogues -- simply replace \( d\tau \) with \( da \) or \( dl \), and make sure your charge +distribution is in the correct amount of dimensions. diff --git a/mindmap/emergence.org b/mindmap/emergence.org index 9a146cf..9c975a5 100644 --- a/mindmap/emergence.org +++ b/mindmap/emergence.org @@ -6,4 +6,27 @@ #+html_head: * Emergence systems are… -Many things. +Existent in many forms and at many levels. They are the fundamental building +block of everything that exists, I argue, and nothing in this theory of +everything will contradict this statement! + +** What is Emergence? +Emergence can be broadly defined as a system whose components organize themselves +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: + +*** 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 +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. diff --git a/mindmap/mathematics.org b/mindmap/mathematics.org index 55c3dc5..d6d7065 100644 --- a/mindmap/mathematics.org +++ b/mindmap/mathematics.org @@ -12,4 +12,16 @@ With a couple of set theory axioms. One might describe it as an extrapolation framework without grounding (i.e. a set of implications; if p then q, but never specifying if p is a property -of a real system or not). +of a real system or not). Therefore, mathematics +is suitable for modeling other things if we believe +those other things have some rules and are logically +consistent. + +There are many subfields in math, ranging from group theory +to complex analysis. However, much of the time we are able to find +morphisms between these different fields in mathematics, which we +model using category theory. + +** Mathematical Models +*** [[id:ece8bf94-4e3c-4939-a77a-9949c1ec0dc6][Physics]] +This is a stub. diff --git a/mindmap/physics.org b/mindmap/physics.org new file mode 100644 index 0000000..b70f829 --- /dev/null +++ b/mindmap/physics.org @@ -0,0 +1,36 @@ +:PROPERTIES: +:ID: ece8bf94-4e3c-4939-a77a-9949c1ec0dc6 +:END: +#+title: physics +#+author: Preston Pan +#+html_head: +#+html_head: +#+html_head: +#+options: broken-links:t + +* Laws of Nature +By engaging in physics modeling, we are postulating that there are rules that +"nature" follows, and that these rules are logically consistent. At least, +mathematical models postulate this, and non-mathematical models of the universe +have been relegated to the study of metaphysics. + +* Fields in Physics +** Classical Mechanics +Classical mechanics deals with fields of physics that do not deal with time dilation or quantum +weirdness. It is called classical mechanics because most of these theories were invented before +their quantum or relativistic counterparts, and usually serve as a baseline understanding +for the later topics. +*** Newtonian Mechanics +It all started when Newton created three (but really two) fundamental laws of the universe that +governed all things. +*** Lagrangian Mechanics +This is a different strain of classical mechanics. Instead of looking at direction and vectors, +it looks at a single fundamental principle, even more fundamental than inertia and conservation of momentum: +optimization. We believe that nature always optimizes for some parameters in all physical phenomena. +*** Classical Electrodynamics +Classical electrodynamics attempts to explain the electromagnetic force from a classical perspective. What is +light? How does electricity actually work? All these questions you will find (half) answered in the study +of electrodynamics. +**** [[id:32f0b8b1-17bc-4c91-a824-2f2a3bbbdbd1][Electrostatics]] +This is a study of the force of particles with charge on another in a motionless context. It is the baseline +for studying more complicated electromagnetic theory. -- cgit