# Physics and maths tutor

## Maths Revision

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## UK Legacy A-levels

So physics is a huge subject that covers many different topics going from galaxies in the depths of space right down to subatomic particles. And if you don’t already know physics its difficult sometimes to see how all these different subjects are related to each other.

So this is my attempt to show that in a map, so this is the map of physics. I hope you enjoy it. Physics can be broadly broken down into three main parts: Classical Physics, Quantum Physics, and Relativity. We’ll start with classical physics and a good person to start with is Issac Newton.

His laws of motion describe how everything made of matter moves about, and his law of universal gravitation tied together with the motion of planets in the sky with the falling of objects on Earth into one elegant and general description. He also invented calculus, a supremely powerful mathematical tool that has been used over the centuries to derive new physics.

Calculus is really part of mathematics but physics and mathematics are inseparable. Math is the language of physics, you can imagine it like the bedrock that the world of physics is built from. Newton also made strides in the field of optics which is the physics of light and how it travels through different materials. It explains, refraction seen in prisms and lenses which are used to focus light in telescopes, microscopes, and cameras.

Telescopes enabled us to peer into the depths of space and observe the wild array of objects there and develop astrophysics and cosmology. Optics is closely related to the theory of waves, which is basically how energy can travel through disturbances of a medium, like ripples on the surface of a pond or sound through the air. Light doesn’t need a medium to travel through, it can travel through the vacuum of space, but it still follows the same principles as all waves namely reflection, refraction, and diffraction.

This leads us to electromagnetism: the description of magnets, electricity, or more generally, electric and magnetic fields. It was a Physicist called James Clerk Maxwell who discovered that these are two aspects of the same thing and derived the wonderfully elegant rules of electromagnetism and theorized that light was an electromagnetic wave. Electromagnetism also explains all of electricity.

Jumping back a little bit, classical mechanics is related to Newton’s laws and covers the properties and motion of solid objects, how they move when forces hit them, what happens when they are joined together, like in gears or buildings, or bridges. Fluid mechanics is the description of the flow of liquids and gasses. Using fluid mechanics you can work out how much lift is generated from an airplane’s wing, or how aerodynamic a car is. Fluid mechanics is notoriously difficult, mostly because the motions of tiny things like molecules get really complicated really fast.

This leads us to the Chaos theory. Chaos theory is the description of large complex systems and how small differences in initial conditions can lead to very different final outcomes. Thermodynamics is the description of energy and how it passes from one form to another. It also includes entropy which is a measure of order and disorder and basically tells you how useful different kinds of energy are.

Energy is a fundamental property to physics and although I have written energy here, I should have written it everywhere on this map because everything has energy. So that is all of classical physics, the picture of the Universe we had around the year 1900. It told us we lived in the Universe where everything ran a sort of like clockwork, if you could measure everything accurately enough the future was kind of predetermined. However, not everything was solved, there were just a few holes in experiments that hinted at something more.

The orbit of Mercury was slightly too fast and some strange things happened on the smallest scales with electrons and light which were all unexplained. Physicists at the time thought that they would solve and explain these problems soon enough but poking at them they unraveled the new domains of relativity and quantum physics and turned our understanding of the Universe completely on its head. Albert Einstein was the genius who developed the theories of special and general relativity.

Special relativity predicts that the speed of light is constant for all observers which means that when you travel really fast weird stuff starts happening like time slowing down. It also states that energy and matter are different aspects of the same thing through the famous formula E=mc2. General relativity says that space and time are part of the same fabric called spacetime and that the force of gravity comes from objects bending spacetime, making other objects fall in towards them.

While relativity describes the very big, other physicists were busy at work on the very small in the world of Quantum Physics. Atomic theory probed the nature of the atom, and more and more detailed descriptions of the atom were developed. From a tiny sphere to electron orbits, to energy levels, and then to the electrons being wave-like charge distributions.

Condensed matter physics describes the quantum physics of many atoms together in solids and liquids, and is where many great technologies have come from like computers, lasers, and quantum information science. Nuclear physics describes how the nucleus of atoms behaves and explains radiation, nuclear fission, the splitting of the atom used in our nuclear power plants, and nuclear fusion which takes place in the Sun and will hopefully soon be harnessed here on Earth.

Particle physics probes even deeper to find the fundamental subatomic particles that everything is made of and are described in the standard model of particle physics. Quantum field theory captures all of quantum physics and combines it with the special theory of relativity and is the best description of the Universe we have.

Unfortunately, Quantum field theory doesn’t include gravity and so physicists don’t know how to join together quantum physics and the general theory of relativity leading to the giant chasm of ignorance. One day in the future we hope to close this chasm and come up with a theory of all of physics we call it quantum gravity, and there are many attempts to do this some examples are string theory or loop quantum gravity and there is many more. But quantum gravity isn’t the only thing we observe but don’t understand, there are also the major puzzles of dark energy and dark matter which seem to make up 95% of the Universe.

So all of our physics only really describes 5% of what we know about and everything else, at the moment, is a mystery. There are many other mysteries out there like the Big Bang, and no doubt there are things beyond that that we don’t even know that we don’t know.

Which gets to the lofty cloud which floats over all of physics: philosophy. Although many physicists make fun of philosophy, it is the big philosophical questions that motivate a lot of physics, like, “What is the fundamental nature of reality?” “How come the Universe even exists?” “Do we have free will if we are just made of physics?” or “How do we know that the way that we do physics and science actually gets to the fundamental truth of the Universe?” And, just, why is all of the physics the way it is? Well, those are the big questions, ones which we may or may never answer, but that is no reason to give up trying, after all, physicists are not quitters. And that was the map of physics.

The mathematics we learn in school doesn’t quite do the field of mathematics justice. We only get a glimpse at one corner of it, but mathematics as a whole is a huge and wonderfully diverse subject.

The origin of mathematics lies in counting. In fact, counting is not just a human trait, other animals are able to count as well and evidence for human counting goes back to prehistoric times with checkmarks made in bones. There were several innovations over the years with the Egyptians having the first equation, the ancient Greeks made strides in many areas like geometry and numerology, and negative numbers were invented in China. And zero as a number was first used in India.

Then in the Golden Age of Islam Persian mathematicians made further strides and the first book on algebra was written. Then mathematics boomed in the renaissance along with the sciences. Now there is a lot more to the history of mathematics than what I have just said, but I’m gonna jump to the modern age and mathematics as we know it now. Modern mathematics can be broadly be broken down into two areas, pure maths: the study of mathematics for its own sake, and applied maths: when you develop mathematics to help solve some real-world problem. But there is a lot of crossovers.

In fact, many times in history someone’s gone off into the mathematical wilderness motivated purely by curiosity and kind of guided by a sense of aesthetics. And then they have created a whole bunch of new mathematics which was nice and interesting but doesn’t really do anything useful.

But then, say a hundred years later, someone will be working on some problem at the cutting edge of physics or computer science and they’ll discover that this old theory in pure maths is exactly what they need to solve their real-world problems! Which is amazing, I think! And this kind of thing has happened so many times over the last few centuries.

It is interesting how often something so abstract ends up being really useful. But I should also mention, pure mathematics on its own is still a very valuable thing to do because it can be fascinating and on its own can have a real beauty and elegance that almost becomes like art. Okay enough of this highfalutin, lets get into it. Pure maths is made of several sections.

The study of numbers starts with the natural numbers and what you can do with them with arithmetic operations. And then it looks at other kinds of numbers like integers, which contain negative numbers, rational numbers like fractions, real numbers which include numbers like pi which go off to infinite decimal points, and then complex numbers and a whole bunch of others.

Some numbers have interesting properties like Prime Numbers, or pi, or exponential. There are also properties of these number systems, for example, even though there is an infinite amount of both integers and real numbers, there are more real numbers than integers.

So some infinities are bigger than others. The study of structures is where you start taking numbers and putting them into equations in the form of variables. Algebra contains the rules of how you then manipulate these equations. Here you will also find vectors and matrices which are multi-dimensional numbers, and the rules of how they relate to each other are captured in linear algebra.

Number theory studies the features of everything in the last section on numbers like the properties of prime numbers. Combinatorics looks at the properties of certain structures like trees, graphs, and other things that are made of discreet chunks that you can count. Group theory looks at objects that are related to each other in, well, groups.

A familiar example is a Rubik’s cube which is an example of a permutation group. And order theory investigates how to arrange objects following certain rules like, how something is a larger quantity than something else. The natural numbers are an example of an ordered set of objects, but anything with any two-way relationship can be ordered. Another part of pure mathematics looks at shapes and how they behave in spaces. The origin is in geometry which includes Pythagoras and is close to trigonometry, which we are all familiar with from school.

Also, there are fun things like fractal geometry which are mathematical patterns that are scale-invariant, which means you can zoom into them forever, and they always look kind of the same. Topology looks at different properties of spaces where you are allowed to continuously deform them but not tear or glue them. For example, a Möbius strip has only one surface and one edge whatever you do to it. And coffee cups and donuts are the same things – topologically speaking.

Measure theory is a way to assign values to spaces or sets tying together numbers and spaces. And finally, differential geometry looks at the properties of shapes on curved surfaces, for example, triangles have got different angles on a curved surface, and brings us to the next section, which is changes. The study of changes contains calculus which involves integrals and differentials which looks at areas spanned out by functions or the behavior of gradients of functions.

And vector calculus looks at the same things for vectors. Here we also find a bunch of other areas like dynamical systems which look at systems that evolve in time from one state to another, like fluid flows or things with feedback loops like ecosystems.

And chaos theory studies dynamical systems that are very sensitive to initial conditions. Finally, the complex analysis looks at the properties of functions with complex numbers. This brings us to applied mathematics. At this point, it is worth mentioning that everything here is a lot more interrelated than I have drawn. In reality, this map should look like more of a web tying together all the different subjects but you can only do so much on a two-dimensional plane so I have laid them out as best I can.

Okay, we’ll start with physics, which uses just about everything on the left-hand side to some degree. Mathematical and theoretical physics has a very close relationship with pure maths. Mathematics is also used in the other natural sciences with mathematical chemistry and biomathematics which look at loads of stuff from modeling molecules to evolutionary biology.

Mathematics is also used extensively in engineering, building things has taken a lot of maths since Egyptian and Babylonian times. Very complex electrical systems like airplanes or the power grid use methods in dynamical systems called control theory. Numerical analysis is a mathematical tool commonly used in places where mathematics becomes too complex to solve completely. So instead you use lots of simple approximations and combine them all together to get good approximate answers.

For example, if you put a circle inside a square, throw darts at it, and then compare the number of darts in the circle and square portions, you can approximate the value of pi. But in the real world numerical analysis is done on huge computers.

The game theory looks at what the best choices are given a set of rules and rational players and it’s used in economics when the players can be intelligent, but not always, and other areas like psychology, and biology. Probability is the study of random events like coin tosses or dice or humans, and statistics is the study of large collections of random processes or the organization and analysis of data.

This is obviously related to mathematical finance, where you want to model financial systems and get an edge to win all those fat stacks. Related to this is optimization, where you are trying to calculate the best choice amongst a set of many different options or constraints, which you can normally visualize as trying to find the highest or lowest point of a function.

Optimization problems are second nature to us humans, we do them all the time: trying to get the best value for money, or trying to maximize our happiness in some way. Another area that is very deeply related to pure mathematics in computer science and the rules of computer science were actually derived in pure maths and is another example of something that was worked out way before programmable computers were built.

Machine learning: the creation of intelligent computer systems uses many areas in mathematics like linear algebra, optimization, dynamical systems, and probability. And finally, the theory of cryptography is very important to computation and uses a lot of pure maths like combinatorics and number theory.

So that covers the main sections of pure and applied mathematics, but I can’t end without looking at the foundations of mathematics. This area tries to work out the properties of mathematics itself and asks what the basis of all the rules of mathematics is. Is there a complete set of fundamental rules, called axioms, from which all of the mathematics comes from? And can we prove that it is all consistent with itself? Mathematical logic, set theory, and category theory try to answer this and a famous result in mathematical logic are Gödel’s incompleteness theorems which, for most people, means that Mathematics does not have a complete and consistent set of axioms, which mean that it is all kinda made up by us humans.

This is weird seeing as mathematics explains so much stuff in the Universe so well. Why would a thing made up by humans be able to do that? That is a deep mystery right there. Also, we have the theory of computation which looks at different models of computing and how efficiently they can solve problems and contains complexity theory which looks at what is and isn’t computable and how much memory and time you would need, which, for most interesting problems, is an insane amount.

Ending So that is the map of mathematics. Now the thing I have loved most about learning maths is that feeling you get where something that seemed so confusing finally clicks in your brain and everything makes sense: like an epiphany moment, kind of like seeing through the matrix.

In fact, some of my most satisfying intellectual moments have been understanding some part of mathematics and then feeling like I had a glimpse at the fundamental nature of the Universe in all of its symmetrical wonder.

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