#vector algebra basics

[hxrpooner]

people on this webbed site, and honestly non math people in general, seem to completely devalue math as a field and it makes me so sad. guys i promise not all math is high school trigonometry. please don't run away. i promise i'm not going to make you calculate the height of a ferris wheel. let's draw a truth table together. i can show you the world. shining shimmering splendid

[anonymousdormhacks]

It is absolutely amazing how much upper div computer science/engineering relies on vectors and probabilities, considering those two classes were basically footnotes in my school's computer science curriculum

[algebraic-dumbass]

this might be a dumb question but like. how do you learn math without a class/curriculum to follow. i have a pretty solid calculus understanding and I want to pursue more advanced math but like im not sure where to start. what even is like category theory it sounds so cool but so scary???. do you have any recommendations on specific fields to begin to look into/whether its best to learn via courses or textbooks or lectures/etc.? any advice would be super appreciated!! dope blog by the way

thanks for the compliment!

first of all it's not a dumb question. trust me i'm the algebraic-dumbass I know what I'm talking about. okay so uh. how does one learn math without a class? it's already hard to learn math WITH a class, so uhhh expect to need motivation. i would recommend making friends with people who know more math than you so you have like, a bit more motivation, and also because math gets much easier if you have people you can ask questions to. Also, learning math can be kind of isolating - most people have no clue what we do.

That said, how does one learn more advanced math?

Well i'm gonna give my opinion, but if anyone has more advice to give, feel free to reblog and share. I suppose the best way to learn math on your own would be through books. You can complement them with video lectures if you want, a lot of them are freely available on the internet. In all cases, it is very important you do exercises when learning: it helps, but it's also the fun part (math is not a spectator sport!). I will say that if you're like me, working on your own can be quite hard. But I will say this: it is a skill, and learning it as early as possible will help you tremendously (I'm still learning it and i'm struggling. if anyone has advice reblog and share it for me actually i need it please)

Unfortunately, for ""basic"" (I'm not saying this to say it's easy but because factually I'm going to talk about the first topics you learn in math after highschool) math topics, I can't really give that much informed book recommendations as I learned through classes. So if anyone has book recommandations, do reblog with them. Anyways. In my opinion the most important skill you need to go further right now is your ability to do proofs!

That's right, proofs! Reasoning and stuff. All the math after highschool is more-or-less based on explaining why something is true, and it's really awesome. For instance, you might know that you can't write the square root of 2 as a fraction of two integers (it's irrational). But do you know why? Would you be able to explain why? Yes you would, or at least, you will! For proof-writing, I have heard good things about The Book of Proof. I've also heard good things about "The Art of Problem Solving", though I think this one is maybe a bit more competition-math oriented. Once you have a grasp on proofs, you will be ready to tackle the first two big topics one learns in math: real analysis, and linear algebra.

Real analysis is about sequences of real numbers, functions on the real numbers and what you can do with them. You will learn about limits, continuity, derivatives, integrals, series, all sorts of stuff you have already seen in calculus, except this time it will be much more proof-oriented (if you want an example of an actual problem, here's one: let (p_n) and (q_n) be two sequences of nonzero integers such that p_n/q_n converges to an irrational number x. Show that |p_n| and |q_n| both diverge to infinity). For this I have heard good things about Terence Tao's Analysis I (pdf link).

Linear algebra is a part of abstract algebra. Abstract algebra is about looking at structures. For instance, you might notice similarities between different situations: if you have two real numbers, you can add them together and get a third real number. Same for functions. Same for vectors. Same for polynomials... and so on. Linear algebra is specifically the study of structures called vector spaces, and maps that preserve that structure (linear maps). Don't worry if you don't get what I mean right away - you'll get it once you learn all the words. Linear algebra shows up everywhere, it is very fundamental. Also, if you know how to multiply matrices, but you've never been told why the way we do it is a bit weird, the answer is in linear algebra. I have heard good things about Sheldon Axler's Linear Algebra Done RIght.

After these two, you can learn various topics. Group theory, point-set topology, measure theory, ring theory, more and more stuff opens up to you. As for category theory, it is (from my pov) a useful tool to unify a lot of things in math, and a convenient language to use in various contexts. That said, I think you need to know the "lots of things" and "various contexts" to appreciate it (in math at least - I can't speak for computer scientists, I just know they also do category theory, for other purposes). So I don't know if jumping into it straight away would be very fun. But once you know a bit more math, sure, go ahead. I have heard a lot of good things about Paolo Aluffi's Algebra: Chapter 0 (pdf link). It's an abstract algebra book (it does a lot: group theory, ring theory, field theory, and even homological algebra!), and it also introduces category theory extremely early, to ease the reader into using it. In fact the book has very little prerequisites - if I'm not mistaken, you could start reading it once you know how to do proofs. it even does linear algebra! But it does so with an extremely algebraic perspective, which might be a bit non-standard. Still, if you feel like it, you could read it.

To conclude I'd say I don't really belive there's a "correct" way to learn math. Sure, if you pursue pure math, at some point, you're going to need to be able to read books, and that point has come for me, but like I'm doing a master's, you can get through your bachelor's without really touching a book. I believe everyone works differently - some people love seminars, some don't. Some people love working with other people, some prefer to focus on math by themselves. Some like algebra, some like analysis. The only true opinion I have on doing math is that I fully believe the only reason you should do it is for fun.

Hope I was at least of some help <3

[la-cosmonauta-extraterrestre]

How to learn physics as an adult

I'm creating this post in response to some posts @un-ionizetheradlab made the other day, but I'm creating this as a guide to anyone this is relevant for. It's going to be a long post, but pick and choose what to do from this list based on what works for you and what your goals are, whether it's just to gain basic scientific literacy or become a physicist (or something in between). Also remember that it's a journey not a sprint, so it's ok if you don't understand physics at first (and if it makes you feel better, physics was one of my worst subjects in school and now I have a master's degree in physics). Without further ado:

First Thing's First

There are some mathematical methods you need to learn to understand physics; there's no way around this:

Vectors: This is the most important thing to learn for physics, how to use vectors. It seemed every mathematics or physics class I took in my first year of my physics degree started with an introduction to vectors, and for good reason. You can learn about how vectors work on Khan Academy for free.

Matrices and Tensors: Once you've mastered vectors, learn about matrices and linear algebra, and perhaps go on to learning about tensors once you're at it. You can at least get the basics about matrices from Khan Academy, but you might want to invest in a linear algebra textbook.

Calculus: I said vectors are the most important thing to learn for physics, but it actually might be calculus. If you have absolutely no previous knowledge of calculus, you can watch the video "Calculus at a Fifth Grade Level" on YouTube; it's a little more advanced than fifth grade level but can give you a good feel for what calculus is about. Once you've done that, there are multiple calculus courses available on Khan Academy. There's also a calculus course available on Brilliant, but it might only be available through the paid version.

Ordinary Differential Equations and Boundary Value Problems: You don't need to learn these right away, but if you want to do physics at the upper undergraduate level, you'll need to learn these at some point.

2. Learning to Think Like a Scientist

Some suggestions of apps and things to watch if you don't know much about science so you can start thinking like a scientist:

SciShow: If you don't know much science at all, SciShow on YouTube is a good place to start, I used to watch it and as I recall it's more focused on life sciences but there's some physics videos there, too.

Ciencias De La Ciencia: This is sort of a Spanish version of SciShow but it's more physics-focused. At least some of the videos have subtitles if you don't know Spanish.

Cosmos: If you haven't seen Cosmos (either the old version with Carl Sagan or the new version with Neil DeGrasse Tyson), it's very good and at least some of the episodes are available for free online. It's more pop-science and history of science than actual science content, but at least they make a point of using anecdotes from the history of science to illustrate how the scientific method works.

Sabine Hossenfelder: Highly recommend her YouTube channel; she's one of the most intellectually honest scientific communicators in the world nowadays. Her videos are a good illustration of how to think like a scientist. She also has a blog and has written a few books.

Brilliant: This is an app with mathematics and science courses that places an emphasis on problem-solving. Most of the courses are only available on the paid version of the app (but you should be able to get a discount on it if you're subscribed to any mainstream science YouTubers), but even the free version gives you access to a few courses, plus a forum where people post problems. I had this app back in the day and liked solving problems on the forum (no idea if it's changed since then).

3. Books to Study

If you're committed to learning physics you should study from some textbooks:

Physics LibreTexts: This is a whole collection of university-level physics textbooks for free online. It's an invaluable resource for learning physics. Use it to learn classical mechanics, electricity and magnetism, modern physics (but don't jump into quantum mechanics straight away if you're just starting out in physics).

Landau and Lifshitz Course of Theoretical Physics: This was the physics school curriculum in the Soviet Union; it's a little dated now but if you're just learning the basics it can't be beat given the excellent pedagogy. It's easy enough to find copies of it online, especially on Russian sites. Most if not all of the textbooks in the series have been translated into English, but if you know any Russian, the original is easy to follow.

Introduction to Electrodynamics by David J. Griffiths: Once you've learned mechanics, modern physics and some electricity and magnetism get yourself a copy of this textbook; you can get used editions on Amazon for a reasonable enough price. American physics majors are obsessed with this textbook, refer to it as the Bible, and for good reason.

Every Life is on Fire by Jeremy England: This isn't a textbook, but reading it took me back to my statistical mechanics class and it's way more readable than any actual statmech textbook so if you are interested in learning statmech, this book is a good start. It's actually a general reading book about England's ideas about the origin of life, interspersed with some parallels to the Hebrew Bible because England is also a rabbi. He actually has some interesting ideas about the philosophy of science, though they can be difficult to get behind, so if you're interested go listen to a podcast where they interview him (obligatory I don't condone the Kahaneist politics he sometimes promotes).

4. Learn About Research and Experiments

Physics is an experimental science, so expose yourself to some experiments:

Look for PDFs of high school physics labs online. You can find some for free and it should be cheap enough to do the experiments at home.

Read scientific papers on topics that interest you to try and understand what's happening today. If you find them difficult to understand, try reading older papers and go from there, for example, in undergrad I did a research internship relating to neutron stars, but I found some of the recent scientific papers difficult to understand, but reading the 1938 paper "On Massive Neutron Cores" by Oppenheimer and Volkoff helped me to understand neutron stars better. (When I returned to some of those same papers during my master's degree, I was proud to have understood them well.)

5. Take University-Level Physics Courses

You can take university-level physics courses without committing to a degree:

Search online for MOOCs (Massive Open Online Courses). You can find MOOCs on multiple sites about many physics topics, and they're often free (sometimes you have to pay for them).

If you live in the United States, you can take physics classes at your local community college.

You can enroll in online physics courses through Open University, based in London but you can take the courses from anywhere. It's expensive, but you pay by the credit so you don't have to pay for a whole year of tuition if you're just taking one course.

If you happen to have free time in the summer and the money for it, many American universities (and elite British universities) offer summer courses that one can enroll in even if they don't attend the university. These are usually in-person classes.

6. Get a Physics Degree

Getting a physics degree is ultimately the only course of action if you've decided to become a physicist, the recommended course of action if you're ultimate goal is a PhD in the history of science or philosophy of science, and a good idea if learning physics has made you want a career in science communication of science education. There's no shame in being a non-traditional age student; in both my bachelor's and master's degree in physics I knew students who were non-traditional age. The downside of this is that it's a bad financial decision to get a degree, especially if it's a second bachelor's degree, but there are ways to lessen the financial burden of a degree:

If you attend an American university with American tuition, you can usually get an on-campus job, though that's pocket change compared to the costs of tuition.

On the bright side, if you already have a bachelor's degree you can probably get credit for general requirements at American universities, so a second bachelor's degree in physics might not take long.

You can also do a part-time degree while you work at many universities.

Just some general advice, if you go the American university route go to a university with a Society of Physics Students and get a student membership in American Physical Society; you get all kinds of benefits like access to Physics Today magazine, scholarships, internships, conferences, an honour society induction.

All that said, it's difficult to attend an American university without losing money. For that matter it's difficult to attend any university in the world without losing money, but you can lessen that burden by going to a country where university is cheaper. There the limiting factor is going to be language; although English is the international language of physics and the medium of most postgraduate physics degrees around the world, physics bachelor's degrees are usually in the local language. Some possible exceptions I found to this, for those who are not fluent in a language other than English:

Apparently there are world-class English-medium physics degree programmes in France? I figure there must be some kind of catch given the way the French are about their language, but given the high research output France has in physics, this is worth getting into.

There are English-medium physics bachelor's degrees in the Czech Republic, and tuition there is pretty affordable (for the English-language degrees; it's free for Czech-language degrees if you happen to be fluent in Czech). I don't know Czechia to have a lot of physics research output today, but back in the day Prague was a major centre of scientific research (Einstein briefly lived and worked as a physics professor in Prague), so it you're goal is to do a PhD in the history of science...I'm just gonna say that there's an English-medium physics bachelor's degree programme at Charles University and you'd have time when you're not studying to explore the city and it's history (but you should learn some Czech if you're going to live there).

University degrees in South Africa are usually English-medium, and tuition there is pretty affordable. There's also a fair amount of research output from South African universities. (Though I understand not wanting to live in South Africa.)

[susidestroyerofworlds]

what’s your favorite number

this is a really hard question. Its probably 2, since it makes for an amazing base and is just generally very useful, like, when you have a+a, which is common, that turns into 2a. It's also the successor to 1, making it the first number thats not just defined directly by axioms like 0 and 1 and rather a consequence.

2 is also aesthetically fun, the way it swoops at the top and then sits on a sturdy base, with both a round and a sharp corner. So, that's my answer,

2

But! If we go further we can find lots of other numbers I also love. 6 is a really nice number because its small but not "weird" in any immediate way. It's not an identity, it's not prime, it's not a square, etc. If you have a formula or something and you want to check it with a number 6 usually works without running into common edge cases.

Leaving the Naturals, -1 is an amazing integer. Really useful for anything that alternates, I love the roots of unity and it's the first nontrivial of those. All the roots of unity are pretty amazing tho, highly recommend.

Rising further we get to the Rationals, which are honestly kinda boring imo. Shoutout to ½ for being the reciprocal of 2 but that's it.

Now, shit is gonna get Real. The Reals are an amazing set, lots of useful properties, a really cool field (the Rationals are also a field but they suck so the Reals get to be the cool field). The Reals contain just about every number you normally interact with, including all-time highs such as pi or e. All algebraic and transcendental numbers are in here, including my favourite, sqrt(2). The square root of 2 is real cool. It's fun geometrically, being a factor of the length of the hypotenuse of a right triangle where the legs are the same length. It's cool algebraically as being the easiest to introduce number thats real but not rational, I have fond childhood memories of learning that the Reals are Dedekind complete (of course that word was only introduced in high school) with this beautiful number as the example. It's reciprocal is also really cool and useful. sqrt(2) even has its own Wikipedia article.

Leaving behind the Reals, the Complex Numbers are just fun as a category. No longer ordered, basically acting as if R² was a field, just really fun. I don't really have a favourite complex number but I am a sucker for units for i gets nominated as my favourite here. All complex numbers are really cool. Shoutout to |0> and |1> (ket-zero and ket-one) for being cool in quantum computing and being vectors in a space using the field of the complex numbers.

There is lots more sets but I am not familiar enough with numbers to even come close to doing them justice. I hope your questions is answered, have a nice day <3

[questioning-my-identity-element]

Do you have any recommendations for learning about algebra beyond high school level? I'm in calculus right now and we just hit definite integrals, if that helps.

sure, ill give a handful of answers depending on your goals.

i would categorize "beyond high school algebra" into linear algebra (which is lower-div college level) and abstract algebra (which is upper-div college level). one could argue that linear algebra is a subfield of abstract algebra, but i am not going to.

linear algebra: lines, planes, hyperplanes, etc can all be fit into essentially the same framework of linear (or affine) equations. think of y=mx+b, except y, x, and b are vectors and m is a matrix. linear algebra is essentially just the study of problems like these and structures that are relevant to their study (e.g., vector spaces). this is inarguably the most important field of mathematics.

abstract algebra: this is a very broad subject which broadly studies structures and the way they interact with other structures. it is hard to get a good feel for abstract algebra without actually doing it, so here is a blog post from my website. you may not be familiar with the notation, but you can find hopefully everything important on the Wikipedia page for sets and functions.

my answer to the question is under the cut. i would encourage other mathblrs to add their opinions though

(A) you want to learn some (but not all) cool algebra without the painful detail

while abstract algebra can be touched with your background, there are certain topics which depend deeply on linear algebra. broadly, this is because linear algebra underlies almost all math. this splits my answer into two parts depending on if (A) you want to learn some (but not a lot of) cool algebra without the painful detail or (B) you want to deeply understand algebra.

try some general audience videos, like from numberphile. generally, videos by good presenters are amazing at teaching you the cool stuff. if you find this to be too little, pick up a lecture series on group theory for undergrads and try your best to follow along. if you can't keep up but still want to pursue it, go to (B).

(B) you want to deeply understand algebra

this is my recommended plan of action:

(1a) pick up some textbook on "discrete math and intro to proofs" (example) and work through a few problems in each section. abstract algebra in general has many prerequisites, and discrete math fills in the vast majority. the topics you should look out for are: proof techniques such as induction, functions and relations, and some basic combonatorics/counting.

(1b) pick up a textbook on "linear algebra and applications" (example) and do the same. i encourage taking a textbook directed at sciences for a few reasons: it's easier, applications can sometimes spark other interests, and most importantly applications give a deeper intuition for the meaning of the math. where possible, try to use your new proof skills to prove the things discussed!

(2) pick up a more serious book on abstract algebra and/or linear algebra and do as many exercises as possible (as in attempt every problem in every section). the standard reference for abstract algebra is dummit and foote but i prefer jacobson. for linear, i would just suggest linear algebra done right

[azdoine]

You probably already know this, but for stoichiometry, instead of trying to solve Rv_1 = Pv_2, where R is the matrix of reactants and P the matrix of products, you can simply solve [R; -P] v = 0, which is cleaner, because you can just mechanically apply Gaussian elimination, instead of trying to guess and check. The visual intuition is simply that of finding some null-subspace within molecule-space (because within the null-space total atom/element count is constant)

:3 mhm, mhm; once someone pointed me at the linear algebra methods, I actually tried solving the problem sets I was given with Gaussian elimination, but I ended up going around in circles and giving myself headaches because I was shite at the recognized notation and I was using techniques I didn't really understand the implications of.

I ended up basically rebuilding similar ideas up out of geometric intuition wrt those vectors in moleculespace-

-using the matrix grids of numbers solely as writing shorthands and visual aids to extract linear equations, and cranking out elementary algebra on the linear equations.

my vague impression from independent research (i.e. stackexchange) is that all of this is totally babytier & direly insufficient for real srsbzns chemistry, b/c the ordering relationship between reactants and products in each equation is non-fundamental - that atomic species are conserved at this level (not accounting for nuclear reactions), but the recombinations of those species are determined by stochastic hops over gradients in the energy landscape, and can therefore somehow generate more pathologically complex balanced reactions than I'm prepared to understand with these techniques.

well, in the absence of a rationale to convince my chemistry department that I'm ready to sit in on higher-level lectures and get credits for it, that's why I'm self-studying more pure math in the mean time (i.e. watching 3blue1brown and very slowly working through the exercises in Linear Algebra Done Right, lol)

[tomorrowillbeyou]

what algebra plagues you good sir

Basically the question is like there's a vector space V and endomorphism φ and you are supposed to prove that ker(φ○φ) = ker(φ) iff V = ker(φ)⊕im(φ). i knoewww there's like probably a really obvious solution but i can't think of anything at all ive just been writing random observations about it and staring at them -_-

[get-thee-to-a-shrubbery]

a post for trannies with at least passing familiarity with the basics of linear algebra:

male and female aren't a binary, or even two ends of a line segment. they're basis vectors. every gender we've come up with so far is defined largely by its position relative to male and female

[spectrallysequenced]

Hi, I'm an undergrad CS and maths student. Usually I enjoy algebra more than analysis, and I greatly enjoyed learning about computability theory and a little earlier, group theory. Unfortunately, right now I'm taking a course (aka module) in real analysis and another in complex analysis. My algebra course is primarily about linear algebra, we learn about fields, rings, and all the linear algebra stuff like vector spaces and the like. I am surprisingly enjoying complex analysis, but I'm not the biggest fan of linear algebra, do you have any advice on how to stick through it until I get to some of the more fun generalised things, or is it gonna be like linear algebra from here on out?

It's hard to say without knowing more about why you dislike linear algebra at the moment. If your class is computation-based, I get it, it's boring. Hopefully you're learning proof-based linear algebra, but if you've already learned about groups, this might be a bit boring for you, since the start of linear algebra is pretty simple, and there's some "dirty work" to be done. If you've seen group theory, a lot of things might look familiar or even identical to you. I would try to see how many parallels you can draw for yourself between the theories. A lot of the underlying properties/theorems here can actually be stated much more generally (e.g. the isomorphism theorems). Edit: I accidentally posted this too early, here's the rest: Linear algebra does underlie a lot of mathematics (I'd say most of it, honestly), including a lot of analysis (Maybe not introductory complex analysis in particular, but multivariable analysis uses LOT of linear algebra and basic theory of inner product spaces). Additionally, a lot of arguments in algebra (and analysis!) reduce to (or use) linear algebra. A lot of arguments don't, but it seems to permeate the rest of your mathematical future. I can't say a lot more about how to motivate you in particular without knowing the course structure, and what exactly you aren't enjoying. I will say that with time and practice linear algebra will become pretty much second nature, and a very useful tool, and more advanced linear/multilinear algebra can actually be pretty fun if you've enjoyed algebra so far, but I will conclude by saying that more advanced algebra really really does not feel like linear algebra, even if some areas use it/generalisations of it quite a lot. Feel free to follow up with more info/dm me!

[enetarch-quantum-physics]

Topics to study for Quantum Physics

Calculus

Taylor Series

Sequences of Functions

Transcendental Equations

Differential Equations

Linear Algebra

Separation of Variables

Scalars

Vectors

Matrixes

Operators

Basis

Vector Operators

Inner Products

Identity Matrix

Unitary Matrix

Unitary Operators

Evolution Operator

Transformation

Rotational Matrix

Eigen Values

Coefficients

Linear Combinations

Matrix Elements

Delta Sequences

Vectors

Basics

Derivatives

Cartesian

Polar Coordinates

Cylindrical

Spherical

LaPlacian

Generalized Coordinate Systems

Waves

Components of Equations

Versions of the equation

Amplitudes

Time Dependent

Time Independent

Position Dependent

Complex Waves

Standing Waves

Nodes

AntiNodes

Traveling Waves

Plane Waves

Incident

Transmission

Reflection

Boundary Conditions

Probability

Probability

Probability Densities

Statistical Interpretation

Discrete Variables

Continuous Variables

Normalization

Probability Distribution

Conservation of Probability

Continuum Limit

Classical Mechanics

Position

Momentum

Center of Mass

Reduce Mass

Action Principle

Elastic and Inelastic Collisions

Physical State

Waves vs Particles

Probability Waves

Quantum Physics

Schroedinger Equation

Uncertainty Principle

Complex Conjugates

Continuity Equation

Quantization Rules

Heisenburg's Uncertianty Principle

Schroedinger Equation

TISE

Seperation from Time

Stationary States

Infinite Square Well

Harmonic Oscillator

Free Particle

Kronecker Delta Functions

Delta Function Potentials

Bound States

Finite Square Well

Scattering States

Incident Particles

Reflected Particles

Transmitted Particles

Motion

Quantum States

Group Velocity

Phase Velocity

Probabilities from Inner Products

Born Interpretation

Hilbert Space

Observables

Operators

Hermitian Operators

Determinate States

Degenerate States

Non-Degenerate States

n-Fold Degenerate States

Symetric States

State Function

State of the System

Eigen States

Eigen States of Position

Eigen States of Momentum

Eigen States of Zero Uncertainty

Eigen Energies

Eigen Energy Values

Eigen Energy States

Eigen Functions

Required properties

Eigen Energy States

Quantification

Negative Energy

Eigen Value Equations

Energy Gaps

Band Gaps

Atomic Spectra

Discrete Spectra

Continuous Spectra

Generalized Statistical Interpretation

Atomic Energy States

Sommerfels Model

The correspondence Principle

Wave Packet

Minimum Uncertainty

Energy Time Uncertainty

Bases of Hilbert Space

Fermi Dirac Notation

Changing Bases

Coordinate Systems

Cartesian

Cylindrical

Spherical - radii, azmithal, angle

Angular Equation

Radial Equation

Hydrogen Atom

Radial Wave Equation

Spectrum of Hydrogen

Angular Momentum

Total Angular Momentum

Orbital Angular Momentum

Angular Momentum Cones

Spin

Spin 1/2

Spin Orbital Interaction Energy

Electron in a Magnetic Field

ElectroMagnetic Interactions

Minimal Coupling

Orbital magnetic dipole moments

Two particle systems

Bosons

Fermions

Exchange Forces

Symmetry

Atoms

Helium

Periodic Table

Solids

Free Electron Gas

Band Structure

Transformations

Transformation in Space

Translation Operator

Translational Symmetry

Conservation Laws

Conservation of Probability

Parity

Parity In 1D

Parity In 2D

Parity In 3D

Even Parity

Odd Parity

Parity selection rules

Rotational Symmetry

Rotations about the z-axis

Rotations in 3D

Degeneracy

Selection rules for Scalars

Translations in time

Time Dependent Equations

Time Translation Invariance

Reflection Symmetry

Periodicity

Stern Gerlach experiment

Dynamic Variables

Kets, Bras and Operators

Multiplication

Measurements

Simultaneous measurements

Compatible Observable

Incompatible Observable

Transformation Matrix

Unitary Equivalent Observable

Position and Momentum Measurements

Wave Functions in Position and Momentum Space

Position space wave functions

momentum operator in position basis

Momentum Space wave functions

Wave Packets

Localized Wave Packets

Gaussian Wave Packets

Motion of Wave Packets

Potentials

Zero Potential

Potential Wells

Potentials in 1D

Potentials in 2D

Potentials in 3D

Linear Potential

Rectangular Potentials

Step Potentials

Central Potential

Bound States

UnBound States

Scattering States

Tunneling

Double Well

Square Barrier

Infinite Square Well Potential

Simple Harmonic Oscillator Potential

Binding Potentials

Non Binding Potentials

Forbidden domains

Forbidden regions

Quantum corral

Classically Allowed Regions

Classically Forbidden Regions

Regions

Landau Levels

Quantum Hall Effect

Molecular Binding

Quantum Numbers

Magnetic

Withal

Principle

Transformations

Gauge Transformations

Commutators

Commuting Operators

Non-Commuting Operators

Commutator Relations of Angular Momentum

Pauli Exclusion Principle

Orbitals

Multiplets

Excited States

Ground State

Spherical Bessel equations

Spherical Bessel Functions

Orthonormal

Orthogonal

Orthogonality

Polarized and UnPolarized Beams

Ladder Operators

Raising and Lowering Operators

Spherical harmonics

Isotropic Harmonic Oscillator

Coulomb Potential

Identical particles

Distinguishable particles

Expectation Values

Ehrenfests Theorem

Simple Harmonic Oscillator

Euler Lagrange Equations

Principle of Least Time

Principle of Least Action

Hamilton's Equation

Hamiltonian Equation

Classical Mechanics

Transition States

Selection Rules

Coherent State

Hydrogen Atom

Electron orbital velocity

principal quantum number

Spectroscopic Notation

=====

Common Equations

Energy (E) .. KE + V

Kinetic Energy (KE) .. KE = 1/2 m v^2

Potential Energy (V)

Momentum (p) is mass times velocity

Force equals mass times acceleration (f = m a)

Newtons' Law of Motion

Wave Length (λ) .. λ = h / p

Wave number (k) ..

k = 2 PI / λ

= p / h-bar

Frequency (f) .. f = 1 / period

Period (T) .. T = 1 / frequency

Density (λ) .. mass / volume

Reduced Mass (m) .. m = (m1 m2) / (m1 + m2)

Angular momentum (L)

Waves (w) ..

w = A sin (kx - wt + o)

w = A exp (i (kx - wt) ) + B exp (-i (kx - wt) )

Angular Frequency (w) ..

w = 2 PI f

= E / h-bar

Schroedinger's Equation

-p^2 [d/dx]^2 w (x, t) + V (x) w (x, t) = i h-bar [d/dt] w(x, t)

-p^2 [d/dx]^2 w (x) T (t) + V (x) w (x) T (t) = i h-bar [d/dt] w(x) T (t)

Time Dependent Schroedinger Equation

[ -p^2 [d/dx]^2 w (x) + V (x) w (x) ] / w (x) = i h-bar [d/dt] T (t) / T (t)

E w (x) = -p^2 [d/dx]^2 w (x) + V (x) w (x)

E i h-bar T (t) = [d/dt] T (t)

TISE - Time Independent

H w = E w

H w = -p^2 [d/dx]^2 w (x) + V (x) w (x)

H = -p^2 [d/dx]^2 + V (x)

-p^2 [d/dx]^2 w (x) + V (x) w (x) = E w (x)

Conversions

Energy / wave length ..

E = h f

E [n] = n h f

= (h-bar k[n])^2 / 2m

= (h-bar n PI)^2 / 2m

= sqr (p^2 c^2 + m^2 c^4)

Kinetic Energy (KE)

KE = 1/2 m v^2

= p^2 / 2m

Momentum (p)

p = h / λ

= sqr (2 m K)

= E / c

= h f / c

Angular momentum ..

p = n h / r, n = [1 .. oo] integers

Wave Length ..

λ = h / p

= h r / n (h / 2 PI)

= 2 PI r / n

= h / sqr (2 m K)

Constants

Planks constant (h)

Rydberg's constant (R)

Avogadro's number (Na)

Planks reduced constant (h-bar) .. h-bar = h / 2 PI

Speed of light (c)

electron mass (me)

proton mass (mp)

Boltzmann's constant (K)

Coulomb's constant

Bohr radius

Electron Volts to Jules

Meter Scale

Gravitational Constant is 6.7e-11 m^3 / kg s^2

History of Experiments

Light

Interference

Diffraction

Diffraction Gratings

Black body radiation

Planks formula

Compton Effect

Photo Electric Effect

Heisenberg's Microscope

Rutherford Planetary Model

Bohr Atom

de Broglie Waves

Double slit experiment

Light

Electrons

Casmir Effect

Pair Production

Superposition

Schroedinger's Cat

EPR Paradox

Examples

Tossing a ball into the air

Stability of the Atom

2 Beads on a wire

Plane Pendulum

Wave Like Behavior of Electrons

Constrained movement between two concentric impermeable spheres

Rigid Rod

Rigid Rotator

Spring Oscillator

Balls rolling down Hill

Balls Tossed in Air

Multiple Pullys and Weights

Particle in a Box

Particle in a Circle

Experiments

Particle in a Tube

Particle in a 2D Box

Particle in a 3D Box

Simple Harmonic Oscillator

Scattering Experiments

Diffraction Experiments

Stern Gerlach Experiment

Rayleigh Scattering

Ramsauer Effect

Davisson–Germer experiment

Theorems

Cauchy Schwarz inequality

Fourier Transformation

Inverse Fourier Transformation

Integration by Parts

Terminology

Levi Civita symbol

Laplace Runge Lenz vector

[lipshits-continuous]

Prime numbers of the ask game let's go!

This is gonna be a long old post haha /pos

2. What math classes did you do best in?:

It's joint between Analysis in Many Variables (literally just Multivariable calculus, I don't know why they gave it a fancy name) and Complex Analysis. Both of which I got 90% in :))

3. What math classes did you like the most?

Out of the ones I've completely finished: complex analysis

Including the ones I'm taking at the moment:

Topology

5. Are there areas of math that you enjoy? What are they?

Yes! They are Topology and Analysis. Analysis was my favourite for a while but topology is even better! (I still like analysis just as much though, topology is just more). I also really like group theory and linear algebra

7. What do you like about math?

The abstractness is really nice. Like I adore how abstract things can be (which is why I really like topology, especially now we're moving onto the algebraic topology stuff). What's better is when the abstract stuff behaves in a satisfying way. Like the definition of homotopy just behaves so nicely with everything (so far) for example.

11. Tell me a funny math story.

A short one but I am not the best at arithmetic at times. During secondary school we had to do these tests every so often that tested out arithmetic and other common maths skills and during one I confidently wrote 8·3=18. I guess it's not all that funny but ¯⁠\⁠_⁠(⁠ツ⁠)⁠_⁠/⁠¯

13. Do you have any stories of Mathematical failure you’d like to share?

I guess the competition I recently took part in counts as a failure? It's supposed to be a similar difficulty to the Putnam and I'm not great at competition maths anyway. I got 1/60 so pretty bad. But it was still interesting to do and I think I'll try it again next year so not wholly a failure I think

17. Are there any great female Mathematicians (living or dead) you would give a shout-out to?

Emmy Noether is an obvious one but I don't you could understate how cool she is. I won't name my lecturers cause I don't want to be doxxed but I have a few who are really cool! One of them gave a cool talk about spectral geometry the other week!

19. How did you solve it?

A bit vague? Usually I try messing around with things that might work until one of them does work

23. Will P=NP? Why or why not?

Honestly I'm not really that well versed in this problem but from what I understand I sure hope not.

29. You’re at the club and Grigori Perlman brushes his gorgeous locks of hair to the side and then proves your girl’s conjecture. WYD?

✨polyamory✨

31. Can you share a math pickup line?

Are you a subset of a vector space of the form x+V? Because you're affine plane

37. Have you ever used math in a novel or entertaining way?

Hmm not that I can think of /lh

41. Which is better named? The Chicken McNugget theorem? Or the Hairy Ball theorem?

Hairy Ball Theorem

43. Did you ever fail a math class?

Not so far

47. Just how big is a big number?

At least 3 I'd say

53. Do you collect anything that is math-related?

Textbooks! I probably have between 20 and 30 at the moment! 5 of which are about topology :3

59. Can you reccomend any online resources for math?

The bright side of mathematics is a great YouTube channel! There is a lot of variety in material and the videos aren't too long so are a great way to get exposed to new topics

61. Does 6 really *deserve* to be called a perfect number? What the h*ck did it ever do?

I think it needs to apologise to 7 for mistakingly accusing it of eating 9

67. Do you have any math tatoos?

I don't have any tattoos at all /lh

71. 👀

A monad is a monoid in the category of endofunctors

73. Can you program? What languages do you know?

I used to be decent at using Java but I've not done for years so I'm very rusty. I also know very basic python

Thanks for the ask!!

[kiiamn]

Minecraft in Desmos

Links:

Empty world except for corner blocks to orient yourself. You can place blocks on the invisible floors/walls/ceiling.

Full world with stuff I made while debugging. Very laggy.

Instructions:

Hit 'ctrl+alt+p' then tab to access the controls. Inputs are:

Turning: plain arrow keys,

Forwards movement: 'shift+up'

Jump: 'shift+down'

Change held block: 'shift+right'

Place: 'shift+left' with block in hand

Break: 'shift+left' with nothing in hand

I've been working on this on and off for a few years, and things turned out pretty good I think. I'm pretty sure there's no desmos community on here, but why not, right? Rambling below the cut.

memory

The memory is waaay over engineered. I thought the world was going to be bigger than it ended up being. The bottleneck turned out to be how fast desmos could run rather than the memory.

I still managed to get a 520,000 bit object in desmos, so that's cool. (It could be bigger, I think you can get 540,000 but eh.)

rendering:

The rendering is fairly standard, I generate sides for all the blocks (stored as a point (block, direction)), then kill the ones that face another block. After placing/breaking, I update the sides for the block and all surrounding blocks.

I thought of doing backspace culling, but it didn't seem worth it.

The actual rendering is just a bunch of functions (rotation, translation, projection) then displaying them based on distance from player.

movement/collision detection:

Movement and collision detection were an absolute pain. I'm sooo glad that all the vectors are orthogonal to each other. I can just treat each vector separately and eliminate them when they result in a collision. There's some edge cases that won't work, but that's fine.

Gravity was fine, a bunch of if statements for when you should fall, move and stop.

place/break:

My favorite part is the place/break detection! I'm pretty sure if I was doing this on not-Desmos, we could get the data for which sides are colliding with the centre while rendering, but I can't do that.

2d version here:

You start with a parametric line going from the player to a point one block in front of them. You have t=0 at the player, and as t increases to 1, you linearly interpolate to the point. (If you put a negative number you go backwards, and t>1 goes beyond the point)

You can do basic algebra to find the t value when the line intersects with the gridlines surrounding the player, then sort the intersection-type (horizontal or vertical in 2d) by t value. (ie, how close they are to the player.) (I also remove the points behind the player and too far from them.)

The list can be used to see how the block the line is 'in' changes over t. (If the line crosses a horizontal gridline, the next block would either be above or below the previous one.)

And since a) it's in order, and b) the line always starts from the player, you can find the nth block that the player is looking at by adding the first n translations together.

With that, I can easily find the first block on the list that isn't an air block.

[void-magician]

Computer Science bachelor here, so I know some basics of related math like vector math and logic proofs, but never deep dove into math. Can you give me a basic explanation of a mathematical concept that you think is underrated or more interesting than people give it credit for?

oh my god i'm so glad you asked me this holy shit. without question, the algebraic property of cancellation.

basically the idea is if you have some operation, we'll call it *, then you can say that if a * b = a * c, then b = c.

it's a very intuitive and straightforward idea, but proving it abstractly is much more difficult than it might seem, and it's such an incredibly necessary part of so many more complex, foundational theorems and concepts without which most of modern mathematics wouldn't exist at all.

[fangirlsovertoomanythings]

I have come to the conclusion that while I enjoy Math, you need to tie it to computers to make me care for it. I've been watching online Linear Algebra classes for like an hour and I couldn't care less for what's being said

Last year, I've taught myself basic Linear Algebra for fun 'cause I wanted to mess up with Python. I ate that shit up. This revision process sucks so bad 'cause there's no Python anymore, it's just vectors and matrices and coordinates and

[nhaneh]

I think another one of the big things that separates previous "thought-replacing" tools such as calculators or, y'know, the written word, from these newfangled "AI" chatbots is that the former still generally tend to require you to maintain a level of understanding to be used?

Thing is, a calculator does not actually solve problems for you - it lets you outsource the computational work of crunching the actual numbers, sure, but you still need to understand the actual problem at hand in order to make use of that ability. A graphing calculator is not going to help you calculate the angular direction and magnitude of a vector from a pair of X and Y values unless you already have a basic understanding of trigonometry, nor is it going to help you find the intersections between f(x) = x and f(x) = 2 sin(x) unless you know about linear algebra.

Likewise, losing out on rote memorisation of maths by always having a calculator around does not prevent you from being able to calculate arbitrary numbers like, say, 1234 * 14 - it just makes it take much longer because you have to do it manually by hand rather than just having a bunch of partial solutions pre-memorised. It's basically replacing a time-saving method in your brain with a time-saving gadget in your pocket - either approach still requires you to know what kind of problem you're attempting to solve before they'll actually help you solve the problem at hand.

"AI" chatbots, on the other hand, lets you outsource the entire process - thinking, understanding, all of it. They'll give you an answer, no matter of whether or not you've even correctly described the problem. If you don't understand the problem you're asking them to solve, then not only can you not verify whether the answer they give you is in any way corrector not, you also can't verify whether or not you're even asking the right questions in the first place.

In computing, there's something that's known as the GIGO principle - Garbage In, Garbage Out. What it means is that the quality of the data output can never be higher than that of the input - if you ask the wrong questions you'll get the wrong answers, if you input the wrong numbers then you'll get wrong results. If you tell a calculator to divide by zero, it will give you an error telling you it can't be done. "AI" chatbots don't work like that - there's no reliable screening to test whether what you asked even makes sense in the first place, no sanity checking to make sure all the inputs are correct and in the right place, just a bunch of statistical probabilities.

Despite what much of the fawning tech media might tell you, the things being passed around as "AI" right now are not things that are capable of independent thought - they're statistical models that crunch absurd amounts of data. As much as they might be capable of providing a mostly coherent-looking conversation, they don't actually know or understand anything - if they did, then it'd only take one or a handful of examples to teach them something, instead of the thousands upon thousands that it does. If you ask an "AI" chatbot to solve a problem for you, it is still you who are responsible for knowing what you're asking and for understanding the answer that it gives, because it will not and can not do those things for you.

generative AI literally makes me feel like a boomer. people start talking about how it can be good to help you brainstorm ideas and i’m like oh you’re letting a computer do the hard work and thinking for you???