Studying Share self-studying mathematics tips

[megatyler30]

Do you think writing notes in LaTeX would be a good method of learning (the subject and to better be able to use LaTeX)?
In my case, it'll be for (classical) Nonequilibrium Thermodynamics (classical as in it focuses on continuum methods and using few results from quantum) using the book Nonequilibrium Thermodynamics by Donald Fitts (Note: there are no exercises). Fitts focuses on Fluids, it would be awesome if I could find a similar book that focuses on solids as I want it to include at least some of both.

My idea was to
(i) Read through the section for understanding.
(ii) Type up notes from second read-through in LaTeX using my own words whenever possible but more or less same organization/structure as author.
(iii) Once done, use other sources (no luck finding, help please?) to add to it and make the structure my own and add in the solid side of things.

Would this be a good method? Any suggestions on similar books and/or classical Nonequilibrium Thermodynamics on solids.

 

[micromass]

Do you think writing notes in LaTeX would be a good method of learning (the subject and to better be able to use LaTeX)?
In my case, it'll be for (classical) Nonequilibrium Thermodynamics (classical as in it focuses on continuum methods and using few results from quantum) using the book Nonequilibrium Thermodynamics by Donald Fitts (Note: there are no exercises). Fitts focuses on Fluids, it would be awesome if I could find a similar book that focuses on solids as I want it to include at least some of both.

My idea was to
(i) Read through the section for understanding.
(ii) Type up notes from second read-through in LaTeX using my own words whenever possible but more or less same organization/structure as author.
(iii) Once done, use other sources (no luck finding, help please?) to add to it and make the structure my own and add in the solid side of things.

Would this be a good method? Any suggestions on similar books and/or classical Nonequilibrium Thermodynamics on solids.

Seems like a solid study method. Sadly I cannot recommend any books. But be sure to post in the textbook forum.

 

[Perpetuella]

Hiya o/

2nd year college student here. I've been through calculus A - C(the typical required courses). I didn't learn the material that well the first time through. In addition(and more importantly imo) I feel as though I have no mathematical intuition. To try and remedy this I was considering self studying either apostol or spivak's or courants calculus books(or all of them vOv) this summer. I've looked at them a bit and they honestly seem somewhat daunting. I guess my question is two-fold then:

1. Do you recommend any of these(or none at all)?
2. Do you have any tips on where to start to foster "mathematical intuition"?

Thanks a bunch!

 

[micromass]

Hiya o/

2nd year college student here. I've been through calculus A - C(the typical required courses). I didn't learn the material that well the first time through. In addition(and more importantly imo) I feel as though I have no mathematical intuition. To try and remedy this I was considering self studying either apostol or spivak's or courants calculus books(or all of them vOv) this summer. I've looked at them a bit and they honestly seem somewhat daunting. I guess my question is two-fold then:

1. Do you recommend any of these(or none at all)?
2. Do you have any tips on where to start to foster "mathematical intuition"?

Thanks a bunch!

You already know a bit of calculus, so you could in principle go through the books. However, they are quite difficult books, so don't be discouraged if you indeed find them daunting. In your situation, I recommend Apostol. Be aware though that the problems in Apostol are very different from the problems in your average calculus class. Namely, you will be asked to give proofs of assertions, not just computations. This requires a mindset that is very different, and which you - I hope- find more enjoyable than the usual calculus. Certainly don't worry if you get stuck a lot and if you go slow, that is normal. It would be nice if you had somebody who you could ask for help now and then.

How to get mathematical intuition? I'm afraid the answer is "by practice and experience". King Ptolemy once asked Proclus if there was no easy way to learn math. Proclus replied that "there is not royal road to geometry". This is -sadly enough- true. The only way you can understand math is by blood, sweat and tears. But boy is it worth it!

 

[megatyler30]

Another suggestion would be to tackle an intro to proof book first. I'll leave it to micromass to pick suggestions; I was forced to pick it up in number theory.

 

[Intrastellar]

How to get mathematical intuition? I'm afraid the answer is "by practice and experience". King Ptolemy once asked Proclus if there was no easy way to learn math. Proclus replied that "there is not royal road to geometry". This is -sadly enough- true. The only way you can understand math is by blood, sweat and tears. But boy is it worth it!

What do you mean by intuition here ?

 

[ohwilleke]

On the issue of answer sets, the reason is simple. If you are self-instructing yourself, it is easy to think that you have the answer right when you don't. When I self-studied, I always completed the entire set of problems with answers available before checking any of them, no matter how long it took. I would never use the answer key to develop my own initial answer to the problem.

Usually, I'd get about 90%-95% of the problems I did myself right, but I learned a great deal from the 5%-10% of cases where my answer did not match the one in the answer key and I had to spend time puzzling what caused me to get the wrong answer so that I could correct my error. About half of the problems that I got wrong were dumb mistakes with arithmetic or lack of attention to detail in some other respect. But, about half of the problems that I got wrong signaled a misunderstood concept. Without a real human being to serve an an advisor or grader, I don't know how you can prevent yourself from getting the wrong answer to a problem and not realizing it and getting off on the wrong foot as you build on that foundation to the next section or concept that relies upon that knowledge.

I also acknowledge that this is harder to do with advanced topics. Advanced textbooks tend to spell out concepts less completely, tend to be less rigorously policed for errors in the text that are easily corrected by the instructor in a classroom setting, and tend not to have answer sets. One curative in that situation is to read published academic journal articles that use the body of knowledge that you are studying at the same time that you work through a textbook to provide a third party reality check and to actively engage in online forums like this one. This tends to lead to a less linear means of learning the material, but is often a necessary curative for textbooks that are thin on exposition and read like warmed over lecture notes.

 

[Loststudent22]

What are your thoughts on sitting in on a class even if you have taken the class already? I have time available to me during the summer where I can sit in on two math classes I have taken before and did well but would like a review them and the instructor is known to be very tough so I was curious to see his teaching style. I could just review it myself with a book but the summer class goes at a much faster pace and its only a few week commitment and I feel it would force me to review.

 

[ohwilleke]

Sitting in on a class I'd already taken would drive me absolutely batty, worse than solitary confinement, but I understand that most people are not like that.

 

[Cygnus_A]

How long does it take you folks to make it through a whole book solving selected problems?

 

[micromass]

What are your thoughts on sitting in on a class even if you have taken the class already? I have time available to me during the summer where I can sit in on two math classes I have taken before and did well but would like a review them and the instructor is known to be very tough so I was curious to see his teaching style. I could just review it myself with a book but the summer class goes at a much faster pace and its only a few week commitment and I feel it would force me to review.

If you can take the absolute boringness of seeing the material again, go for it! It could be a useful exercise. But I kind of think that self-studying (so you can choose what to go into more) is more worth it.

 

[micromass]

How long does it take you folks to make it through a whole book solving selected problems?

Depends on the book, really. But it can take me several months.

 

[ohwilleke]

How long does it take you folks to make it through a whole book solving selected problems?

I would typically finish a 16 week college course amount of material in 10-14 weeks. In part this is because taking a lengthy vacation from it is much more perilous than doing the same in a regular course. Get off track for too long and you may never get back. This said, the pacing of self-study for me is much more erratic in terms of material covered than a classroom course. I might spend three weeks on something that only takes a week in a classroom, because I'm stumped, and then breeze through two or three weeks of classroom material in a week because it all made sense to me.

 

[Hellmut1956]

I have gone through the description of the courses and first think I met was that you have to decide upfront if you want to learn pure mathematics or related to physics. I believe this is something you should seriously think about and make an as far as possible educated decision about it. The next think the document correctly states is, that the studying of the courses you have to take is very heavy!

This leads me to my suggestion. This suggestion reflects that due to health problems I am not able to sustain concentrated study efforts as required to follow the courses offered in a bachelor study of mathematics. but I have studied the course book, equivalent to the document yo link to, from the technical university of Munich, the institute of mathematics. Analysis and Linear Algebra are fundamental mathematical tools to grasp what ever is presented in courses later. So I found the video lectures from a german professor about Analysis to be the best fit to my way of learning. He bases his lectures of the 2 books from Terence Tao, UCLA, available for free legal download from the home page of professor Terence Tao. He starts rigidly to have its students learn to think as a mathematician by learning to apply the techniques of proofing starting with the natural numbers.Terence Taos course of analysis is course with honours, but thanks to his rigid methodology you do not phase the obstacle it represents at least for me, that mathematicians tend to see often issues as trivially obvious!

As to Linear Algebra I can only confirm, what has been written earlier in this thread, the courses from Gilbert Strang offered as part of MIT's OpenCourseware course offering. Different from what the one writing about his experience with this course and the "B" rating he received applying the knowledge from Gilbert Strang's teaching to his examin, I would only take this OpenCourseware offerings as a mean for preparing yourself for the courses at the university you will be assisting in person. Most of the other courses listed in the document you offered the link to are also in the course offering from MIT! It is my solid believe, that never ever you should learn for passing exams at school or university, but to learn and study to understand the topics for yourself. So preparing for the visit at the university studying with great effort and dedication the courses offered for example from MIT will take the pressure from you to follow all the courses you have to assist you at the university, as you would also already have a solid understanding, but it enables you to ask the questions left not fully understood as part of the self study and to grasp the concepts teached more extensively. A side effect will be that to follow the courses will not be as stressful as it would otherwise be and that your results in the exams I am sure will be excellent!

I found out myself, that in the 4 decades since I learned the mathematics a lot of my knowledge has eroded. So I did a step back and took the Calculus Single Variable and the Calculus Multivariable courses offered by MIT, if I remember properly those are the courses 18.01SC and 18.02SC. The courses follow the books from Gilbert Strang available for free and legal as PDF archives in the Internet. I even took this opportunity to learn to solve the numerous example problems in Gilbert Strang's book on Calculus with the software tool Mathematica in parallel to solving those assignments in writing manually to learn the contents and to learn to apply Mathematica to them!

 

[Loststudent22]

Is it reasonable to work through Calculus of Several Variables by Lang even though I was taught out of a easier book(anton) for single variable? I could pick up a copy of the first book from the library and probably go through it pretty quick since I know the topic.

 

[Hellmut1956]

Here the link where you can down load the complete book as a pdf archive for free and legal!

 

[Cruz Martinez]

Something I've been having some problems with, is how to avoid forgetting the proofs of particularly tricky theorems. Do you have any advise?

 

[tommyxu3]

When I tried to learn something before, what made me worried most was that there didn't exist many ones around me interesting in what I did. Thanks to the forums like this PF and others such as Art of Problem Solving, I have more opportunities to discuss my opinions with others! That's such a blessing for me!

 

[JorisL]

Something I've been having some problems with, is how to avoid forgetting the proofs of particularly tricky theorems. Do you have any advise?

The cliché; Try to understand what the proof does.

For me this often means making sketches for example visualizing the domains of functions when I have to look at compositions.
If it is in any way related to some geometric problem make a simple picture which shows potential problems.
This is not always possible or easy (a simple example is showing that the biggest fraction of the volume of a sphere is concentrated around the equator in high dimensions)

Next up is that you have to revisit theorems and such you are using. Here the pictures can speed things up again.
As they say a picture says more than a thousand words.

Finally think about the limits, if you need for example that a function is $C^\infty$, what goes wrong if this condition is not satisfied.
This is equally important as the picture for me. (and usually reasonably fun to do)

Note that this is my perception and method. For you another method could be better.

 

[Randy Johnson]

I am 17, and have finished all the mathematics courses my high school provides. I am staying back a year to play sports and work, but I am concerned I will forget or become "rusty" on the calculus, vectors, and functions, I have done so far. Is there any good textbooks you can recommend? I am very good at math and physics, and pick up on things quickly.

 

[micromass]

I am 17, and have finished all the mathematics courses my high school provides. I am staying back a year to play sports and work, but I am concerned I will forget or become "rusty" on the calculus, vectors, and functions, I have done so far. Is there any good textbooks you can recommend? I am very good at math and physics, and pick up on things quickly.

Is there anything specific you want to learn? What is your current knowledge? What do you intend to do with the math later in life?

 

[Randy Johnson]

Is there anything specific you want to learn? What is your current knowledge? What do you intend to do with the math later in life?

Very general, more to get a head start on university than anything else. My current knowledge for vectors is lines and planes intersections and relations. For calculus is basic derivatives (exponential, trigonometric, and polynomial up to 3 prime) , limits, graph sketching. I'm ok with proofs.
In physics we've done basic projectile, forces, electricity, waves and magnetism.
I initially planned to take some form of engineering (such as mechanical, physics, or electrical), in university, but recently I have been considering taking a general physics or math.

 

[micromass]

OK cool. Then I can recommend either studying a bit of calculus, or linear algebra (or both!).

You know calculus of course, but you can go deeper in it, for example, you can do integrals (which are among the most beautiful mathematical objects ever created!). I recommend the free book by Keisler: https://www.math.wisc.edu/~keisler/calc.html It will cover a lot of what you know, but I do recommend going over that stuff since Keisler has a truly original approach to calculus. Namely, he works with infinitesimals, which were the historic method of calculus, and which still remain very very useful in physics. If you truly understand this book, then you will have a very good intuition for calculus.

Linear algebra is a kind of generalization of geometry, but by using algebra. You know lines and planes in usual 3D-space, linear algebra generalizes it to higher dimensions. This is useful not only because it is cool, but also because many real-life phenomena have higher dimensions (you should see a dimension as a parameter or a degree of freedom). For this I recommend Lang's introduction to linear algebra (do not get his "linear algebra" which is more advanced). This will teach you vectors (which you know), matrices, vector spaces, etc. If you're comfortable with vectors and matrices AND proofs, and you don't mind a challenge, then you can't beat Treil's linear algebra done wrong (freely available again): http://www.math.brown.edu/~treil/papers/LADW/LADW.html

 

[Randy Johnson]

OK cool. Then I can recommend either studying a bit of calculus, or linear algebra (or both!).

You know calculus of course, but you can go deeper in it, for example, you can do integrals (which are among the most beautiful mathematical objects ever created!). I recommend the free book by Keisler: https://www.math.wisc.edu/~keisler/calc.html It will cover a lot of what you know, but I do recommend going over that stuff since Keisler has a truly original approach to calculus. Namely, he works with infinitesimals, which were the historic method of calculus, and which still remain very very useful in physics. If you truly understand this book, then you will have a very good intuition for calculus.

Linear algebra is a kind of generalization of geometry, but by using algebra. You know lines and planes in usual 3D-space, linear algebra generalizes it to higher dimensions. This is useful not only because it is cool, but also because many real-life phenomena have higher dimensions (you should see a dimension as a parameter or a degree of freedom). For this I recommend Lang's introduction to linear algebra (do not get his "linear algebra" which is more advanced). This will teach you vectors (which you know), matrices, vector spaces, etc. If you're comfortable with vectors and matrices AND proofs, and you don't mind a challenge, then you can't beat Treil's linear algebra done wrong (freely available again): http://www.math.brown.edu/~treil/papers/LADW/LADW.html

Thank you so much, I appreciate your time and advice. This truly means the world to me. Also, do you have any advice, suggestions, or anything else you'd like to share about future studies in university or future career paths?

 

[atyy]

You know calculus of course, but you can go deeper in it, for example, you can do integrals (which are among the most beautiful mathematical objects ever created!). I recommend the free book by Keisler: https://www.math.wisc.edu/~keisler/calc.html It will cover a lot of what you know, but I do recommend going over that stuff since Keisler has a truly original approach to calculus. Namely, he works with infinitesimals, which were the historic method of calculus, and which still remain very very useful in physics. If you truly understand this book, then you will have a very good intuition for calculus.

It's absolutely correct, but shouldn't an introduction start off with the more standard approach? Do Robinson's infinitesimals really correspond to physicists' infinitesimals?

 

[micromass]

I have much advise, but I don't really know what you're looking for. But here's some things I would have liked to hear:

1) Get in touch with the profs. Many profs are more approachable than you think (while some are absolutely not!). Get to know them, go to office hours, ask questions, etc. And I don't (only) mean to talk about the class, but talk about other things in physics/math too.

2) Don't be discouraged by your class mates. While in undergrad, and while teaching undergrad I have seen many classes with bright students, but with an atmosphere that is very bad. Many would care about the grades only, and others openly disliked the courses. This reflected on the entire class. Don't let yourself be discouraged by them.

3) Don't care about your grades (only). I have mentioned this before in point (1), but there is more than grades. Grades =/= understanding (although there is a correlation). Focus on understanding the topic, not only on getting good grades.

4) Think before you ask a question. Don't just go to a prof and start asking a lot of questions without first thinking about it for a long time. Of course, if you REALLY don't know, then ask the prof and don't be afraid to do so. But it is worthwhile to think things through first.

5) Be sure to have fun too. Life isn't only about learning.

 

[micromass]

It's absolutely correct, but shouldn't an introduction start off with the more standard approach?

That is a matter of taste, I guess. If you do it right then there is very little difference between the two approaches. Both approaches have limits, derivatives, integrals, etc. The only essential difference is how limits are defined. There are other differences too, but it shouldn't be too difficult to translate between the two approaches.

A first course in calculus should not focus much on epsilon-delta definitions. It should be brought up, but it is too difficult for the students at that stage. Infinitesimals on the other hand are much more intuitive and do provide a solid basis for calculus.

I do think it would be a mistake to focus only on one type of approach. Both approaches have their benefits. The standard approach is beneficial because it doesn't need mysterious infinitesimal numbers, and because everybody works with this. The nonstandard approach is more intuitive, more historical, and offers useful ways of thinking. Keisler has both approaches.

Do Robinson's infinitesimals really correspond to physicists' infinitesimals?

Not exactly, but they come closer to physics' calculus than the standard approach. Face it, who uses epsilon-delta definitions in physics? A lot of the techniques in nonstandard calculus come up in physics, for example integrals as sums, infinitesimals, geometry with infinitesimals. So I feel that students might feel more comfortable if they already saw infinitesimals in calculus, even though they're not exactly the same thing. If you want a better correspondence with physics, then you'll have to look at constructive mathematics, and especially at synthetic differential geometry. But that would be completely unsuitable to teach newcomer to calculus. http://math.andrej.com/2008/08/13/intuitionistic-mathematics-for-physics/comment-page-1/

 

[one]

Good evening everyone,
Over the summer I have been trying to study some extra mathematics to prepare for my next school year, but laziness has impeded my progress :)
I would like to know if there is some place on PhysicsForums for "study groups"; say, if several people wanted to study a specific subject, they could come up with a schedule of what to read and then they could have a thread for discussion, questions and such. Is there anything like this set up already?

 

[ohwilleke]

Something I've been having some problems with, is how to avoid forgetting the proofs of particularly tricky theorems. Do you have any advise?

Unless you are going to be tested on them, there is no good reason to memorize proofs of particularly tricky theorems. The whole point is a theorem is to avoid having to reinvent it from scratch each time you encounter a situation where it applies. Doing it once gives you confidence in the method that produced the theorem and a sense of how that kind of proof is done, but isn't something that you need to use on a regular basis. If there is some reason to avoid forgetting the proofs, however, recopying them into a journal from your local book store (I prefer the ones with pink unicorns myself), is an easy way to refresh your memory when you need to.

 

[arturodonjuan]

One thing I'd like to share from my experiences is that far too many of my peers get discouraged from self-studying mathematics because they "get lost". A mathematics textbook is (typically) not a story book, and so it shouldn't be read like one. I've learned that I personally need to go very slowly through every single statement/paragraph, theorem, lemma, and corollary very carefully, making sure I understand what is being presented before I move on. I know this may seem completely obvious, but I think it's worth saying given the ridiculous number of people that get discouraged by self-studying mathematics for this reason.