Self-teaching Mathematics -- advice please

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  • #1
NovaeSci
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Hi all,

I'm partaking in a PT Astronomy degree and I'm curious about self-teaching myself maths for Physics/Astrophysics. I'm currently working through the Foundation section in Stroud's Engineering Mathematics. I am of course familiar with the usual recommendations of Riley's and Boas's Mathematical Methods textbooks, but what would be the most ideal way to study the topics in terms of order?

The reason I ask is that I want to understand the topics as thoroughly as I can, rather than just have a big book containing everything. Preferably books that as I'm working through the Riley and/or Boas textbooks I can have books that just deal with those topics in a much deeper way. I'm also looking for the resources to practice questions on each topics as much as I can to cement what I learn - Mainly Applied , but also interested in Pure. Just a well rounded knowledge as a whole.

Even though my level will be Pre-Calculus, so I'll be looking for textbooks starting at basic Calculus, it would also be quite cool to own a few books on the topics pre-calculus to learn about the basic topics in more depth and to help me become totally fluent.

Thanks in advance :)
 

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  • #2
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FOr some of your math needs:

1) Openstax.org has various math books available

2) KhanAcademy and Mathispower4u.com has a large collection of short math videos on every topic upto and including 1st year college namely Calc 1,2,3 LinAlg, DiffEqns, Stat
 
  • #4
PeroK
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Even though my level will be Pre-Calculus, so I'll be looking for textbooks starting at basic Calculus, it would also be quite cool to own a few books on the topics pre-calculus to learn about the basic topics in more depth and to help me become totally fluent.
What topics are you talking about specifically? When you say pre-Calculus, what level have you reached?
 
  • #5
NovaeSci
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Thanks for your responses. I'm currently working through the Foundation section (topics below) and should be done in a month or 2. So basically looking at what to study after these topics. But at the same time, if there are any topics listed below people would advise to learn on a deeper level, the happy to take advice :)

PART I: FOUNDATION TOPICS
Arithmetic
Introduction to Algebra
Expressions and Equations
Graphs
Linear Equations
Polynomial Equations
Binomials
Partial Fractions
Trigonometry
Functions
Trigonometric and Exponential Functions
Differentiation
Integration


The main part of the text includes the below topics. But after the foundation topics, I really want to start studying the topics in a self-contained way with plenty of questions to practice and to go deeply into the topic for full understanding. Sure I could learn from this book and then buy Advanced Engineering Mathematics, but I don't want to be shown how to do it, be give a dozen questions, then straight on to the next topic. You see books dedicated to certain topics that have hundreds of pages, rather than say a dozen or so in the "all-in-one" books.

PART II
Complex Numbers 1
Complex Numbers 2
Hyperbolic Functions
Determinants
Matrices
Vectors
Differentiation
Differentiation Applications
Tangents, Normals and Curvature
Sequences
Series 1
Series 2
Curves and Curve Fitting
Partial Differentiation 1
Partial Differentiation 2
Integration 1
Integration 2
Reduction Formulas
Integration Applications 1
Integration Applications 2
Integration Applications 3
Approximate Integration
Polar Coordinate Systems
Multiple Integrals
First-Order Differential Equations
Second-Order Differential Equations
Introduction to Laplace Transforms
Statistics
Probability


Thanks again :)
 
  • #7
NovaeSci
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So just to be sure, instead of studying the foundation section in Engineering Mathematics, I would first study the below books, before Calculus, in the respective order?

  1. Algebra by Gelfand, Shen
  2. Geometry I and II by Kiselev
  3. Trigonometry by Gelfand and Saul
  4. Geometry by Lang, Murrow
  5. Then Basic Mathematics by Lang before embarking on Calculus

Then of course along with the textbooks I can use the below exercise books by Shaum.

  1. Elementary algebra
  2. Geometry
  3. Trigonometry
  4. Precalculus

Would this pretty much cover everything I need in a much better way than an all in one book does? Just if anyone has any opinions on the choice, or other recommendations?
 
  • #8
PeroK
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So just to be sure, instead of studying the foundation section in Engineering Mathematics, I would first study the below books, before Calculus, in the respective order?

  1. Algebra by Gelfand, Shen
  2. Geometry I and II by Kiselev
  3. Trigonometry by Gelfand and Saul
  4. Geometry by Lang, Murrow
  5. Then Basic Mathematics by Lang before embarking on Calculus

Then of course along with the textbooks I can use the below exercise books by Shaum.

  1. Elementary algebra
  2. Geometry
  3. Trigonometry
  4. Precalculus

Would this pretty much cover everything I need in a much better way than an all in one book does? Just if anyone has any opinions on the choice, or other recommendations?
That looks a lot of books before you get to calculus!

You only need a high-school maths syllabus, surely? You've some physics to learn as well. And, ideally, some chemistry.
 
  • #9
NovaeSci
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This was what was recommended on the articles posted by @fresh_42 . It's essentially only 4 books. From the looks of them, they go further into the topic, compared to how All-in-one books just show you how to solve it and move on. The 5th book is purely just an extra. The last 4 are actually just books that provide exercises. I've manage to source them dead cheap anyway. Just curious if these are still recommended, or if there are new textbooks which are now better than these. As mentioned, I'm just starting out, so wanting to grasp topics as deeply as I can :)
 
  • #10
fresh_42
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This was what was recommended on the articles posted by @fresh_42 . It's essentially only 4 books. From the looks of them, they go further into the topic, compared to how All-in-one books just show you how to solve it and move on. The 5th book is purely just an extra. The last 4 are actually just books that provide exercises. I've manage to source them dead cheap anyway. Just curious if these are still recommended, or if there are new textbooks which are now better than these. As mentioned, I'm just starting out, so wanting to grasp topics as deeply as I can :)
Yes, this was the ideal. You must not forget, that each article was written by an author who concentrated on a specific field. Gathering all of them leads to an unrealistic workload. You will certainly have to know some geometric basics and trigonometry if your goal is engineering. But you will not have to learn it excessively. There are about a dozen useful formulas and a dozen useful theorems. That's it. We live in a world where you can look up additional information easily whenever needed. Of course you should know some basic as Pythagoras, Thales, some symmetries on triangles, the law of sine and cosine and a few more. If not, and you will meet them: look it up. I think the OpenStaxx books are a good start. If you can read them easily, then you can go ahead.
 
  • #11
NovaeSci
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It's actually Astrophysics/Physics I'm getting in to, not engineering. just the Engineering Mathematics book, alongside Principles of Physics (Fundamentals in US) are recommended for the Physics/Maths module. So would this route not be advisable for someone following a theoretical/applied science pathway? Since I want to preferably research Astrophysics/Cosmology PG, I'm just wanting to make sure my maths is as complete as it can be for an undergraduate level. The tougher stuff will of course come PG, but I'm guessing by then I'll be well versed in what maths I then need to further learn from experience. I just want to get the best undergraduate foundation that a mathematical/theoretical physicist would learn.
 
  • #12
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It is quite simple for physicist: there is no way to avoid calculus, hence you will need to know everything which is necessary to learn the calculus syllabus: real, complex, vector. The OpenStaxx books are - which I think - meant to bring American students on a comparable level when they enter college. At least I take them for that.

It is hard to recommend you something specific, since we cannot really know where you are at the moment. My advice would be thus to start with a single book of your choice. If you already know what's in there, choose another one, until you start having difficulties. They are necessary to learn. Then come to PF and ask specific questions. This way we could give you far better advises than on the general level in this thread.
 
  • #13
vela
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I'm partaking in a PT Astronomy degree and I'm curious about self-teaching myself maths for Physics/Astrophysics. I'm currently working through the Foundation section in Stroud's Engineering Mathematics. I am of course familiar with the usual recommendations of Riley's and Boas's Mathematical Methods textbooks, but what would be the most ideal way to study the topics in terms of order?
If you were taking classes, a book like Boas would be two years in your future. You need to learn basic calculus and differential equations first. You never answered @PeroK's question about your level of preparation, but if you're ready for a calculus course, I suggest you just dive into calculus. You can review more basic topics as needed.

Keep in mind learning math and physics isn't a linear process. You don't typically learn everything there is to know about, say, algebra before moving on to trig. You learn about a wide variety of topics at an introductory level, and then revisit topics later to learn about them in greater depth and detail. And then, quite possibly, you do that again. As you go through this process, you see how the individual pieces fit together and become acquainted with more complex and abstract concepts. Most important, you develop critical thinking skills that may help you see the holes in your own knowledge and give you the ability to fill them in on your own.

This is my long-winded way of saying that while your goal seems admirable, your time may be better spent moving on to new subjects.
 
  • #14
NovaeSci
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Thanks again for the advice. I actually got an A* in GCSE Maths, but that was 16 years ago. I'm a bit rusty, but I'm picking up the foundation topics mentioned in a previous post pretty fast. It was more when I passed all those years ago, it was just reading how to solve a problem and jotting down the answer. There was no real philosophy, or why certain things are as they are, when I was studying. Was more about memorising steps, etc. I'm just bringing myself up to scratch, so everything is fresh again. But wanting to try and learn on a deeper level. The degree I'm on is part-time, so I'm just focussing on Pre-Calculus between now and September, then I will move on to Calculus when I start my next modules.

You have all been very helpful and provided me with some food for thought :)
 

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