Learning mathematics in an interdisciplinary program

[ElectronicTeaCup]

Hello everyone! I finished a masters in integrative neuroscience about a year back, which was supposed to have a very strong mathematics tilt. Despite this, and the two semesters of mathematics, I feel that it did not help me out much. I ended up doing my masters thesis in a lab of physicists and have been doing me doctoral work (which is quite interdisciplinary; I do experiments, data analysis and work with a mathematical models) there since Jan 2020.

In the last two years I've been trying to brush up my mathematics from the basics. My bachelors was in medicine and surgery, so I had a huge gap (almost 10 years) of no mathematics. A lot of the work I do involves simulations, and I superficially understand the mathematics behind it. But I really, really want to understand the nitty gritty details—to think mathematically about physical problems and how they can be modeled. More than a desire, it's become a need for me. I dream of the day I can do advanced mathematics such as dynamical systems to understand neural systems in a more formal manner, rather than the clunky paragraphs I read in med school.

I originally started with MIT's online calculus resource, however, it proved very difficult and then I started on Morris Kline's Calculus: an intuitive and physical approach. It was going well, however, I felt that I kept stumbling on basics. It was then that I discovered this side, and saw Micromass' post on learning calculus, and decided to do the relevant chapters from Lang's basic mathematics.

After doing that I went back to Kline (April this year) and restarted the book. The earlier chapters were a little easier than the first try, however, now I find it difficult to gage when I am moving too slowly or am getting stuck on problems that I need more background material on. At this point, I am unsure if I should just push through, or try to get some kind of tutor to mentor me through this? One of the problems with my current book is that I do not know which parts I should be really remembering, since there are no tests/exams.

For example, am I supposed to know by heart how I derive an eclipse and hyperbola? Other issues include constantly getting stuck with algebraic manipulation and just finding it difficult in general to solve questions. Alternatives also exist: I could do Calculus made easy to get an overview, and do something like Strang's Calculus book since the online resources include tests and exams.

I believe I have the work ethic and have committed a minimum of 15 hours a week to this. I would love to hear any advice regarding this.

 

[fresh_42]

I would have expected more probability theory (stochastics) and statistics, rather than calculus. Of course, calculus is needed everywhere, since it is the basic language of mathematics outside of mathematics. So 'Calculus made easy' is certainly a good idea. But as neuroscientist you should have some decent knowledge in statistics, e.g. in hypothesis testing ...

... and differential equations in case you deal with virology or models in general.

 

[ElectronicTeaCup]

Thanks for the reply. The current model we use consists of a system of differential equations, thus the direct motivation towards calculus. Plus, like you said, it is the prerequisite/foundation for other advanced mathematics. But my question is more about my particular method, is it best to self study by myself at this point or try to get a tutor to complement my studies. Alternatively I am open to other paths such as other books.

 

[fresh_42]

I learned analysis in the first semesters and the prof basically went along some paperbacks (Otto Foster: Analysis I,II,III) which could as well be studied alone. But they are not in english. The corresponding english book seems to be Michael Spivak, Calculus considering the frequency of recommendations here.

If you have specific problems you can always come on over and ask questions in the homework section of PF (just don't forget to add some of your own attempts or thoughts at the attempt to solve the problem). If it doesn't work this way for you, you could still look for a tutor. I am not sure whether these books cover vector calculus and complex calculus, but these two areas are certainly necessary to understand differential equation systems and possible solutions to some physical problems.

 

[ElectronicTeaCup]

I am not sure whether these books cover vector calculus and complex calculus, but these two areas are certainly necessary to understand differential equation systems and possible solutions to some physical problems.

Yes, Kline's book only covers single variate calculus. It is supposed to serve as my first step into learning more math.

 

[fresh_42]

Yes, Kline's book only covers single variate calculus. It is supposed to serve as my first step into learning more math.

This is of course where you want to start, and develop a feeling how the arguments work. Vector calculus is basically the same, only that one has to manage more directions. This has some technical consequences: norm instead of absolute value, partial derivatives instead of just one differentiation, and so on.

Complex calculus, however, is different. Changing from the reals to complex numbers has some properties which aren't just another direction: $i^2 = -1$ connects the real and imaginary direction, so they cannot be considered separated.

 

[StoneTemplePython]

I do experiments, data analysis and work with a mathematical models) there since Jan 2020...

I originally started with MIT's online calculus resource, however, it proved very difficult ...

After doing that I went back to Kline (April this year) and restarted the book. The earlier chapters were a little easier than the first try, however, now I find it difficult to gage when I am moving too slowly or am getting stuck on problems that I need more background material on. At this point, I am unsure if I should just push through, or try to get some kind of tutor to mentor me through this? One of the problems with my current book is that I do not know which parts I should be really remembering, since there are no tests/exams.

where you are at on this is the first two years of undergrad at most US universities. If you want to kill 2 birds with one stone -- (i) avoid the planning fallacy, (ii) have tests -- don't start from scratch. This means finding a course (OCW or otherwise) that you can clone and do the assignments and tests -- and then check your answers against the 'official ones'. There are a huge number of them for intro calculus and linear algebra (which I assume you'll need in large doses).

re: the planning fallacy -- not something most people think about but it gets at the heart of whether you are 'moving too slowly', what parts of the book it is ok to skip, how many problems to do, etc. Many people will have many different opinions on this but what you want is the view of someone who has successfully taught that course multiple times (and refined their approach over time). There's bonus points if that person is a very gifted mathematician as they'll have some extra insights coloring selection of subsections in a book.

If you find the course itself to be very difficult, that indicates either (i) finding an easier course or (ii) getting a tutor to help plug gaps. It's a tough judgment call.

 

[ElectronicTeaCup]

where you are at on this is the first two years of undergrad at most US universities. If you want to kill 2 birds with one stone -- (i) avoid the planning fallacy, (ii) have tests -- don't start from scratch. This means finding a course (OCW or otherwise) that you can clone and do the assignments and tests -- and then check your answers against the 'official ones'.

If you find the course itself to be very difficult, that indicates either (i) finding an easier course or (ii) getting a tutor to help plug gaps. It's a tough judgment call.

It is a tough call, since now I will have to again restart if I am to pick a new course. Other than MIT OCW, I haven't been able to find courses freely available in such an organized manner (notes/text + tests + exams). The MIT ocw course was very tough, and I was very demotivated by how much I was struggling. Also, I discovered that I just like textbooks more than video lectures. I am finding it a bit difficult to find courses around textbooks, but this might be the way to go. Something like this where there are tests and exams based on a popular textbook.

I definitely like the idea of tests, it would really let me know how I'm doing. Currently I am making my own 'tests' from exercises, but this doesn't really emphasize exercises that are most important.

 

[gleem]

Have you talked to your faculty, fellow grad students, or postdocs about what you should know and to what depth?

 

[atyy]

This book has problems with solutions. It also has exposition of the basic concepts. And it's not too expensive: Schaum's Outline of Calculus

You can use the Schaum's book as a companion to a standard text like Strang's calculus book: Calculus

Then try a differential equations text. There are several standards, one of which is
Differential Equations: Computing and Modeling

Then try Strogatz's book on dynamical systems: Nonlinear Dynamics And Chaos

When you get stuck, ask specific questions to people you know or in the fora here.

 

[MidgetDwarf]

I would suggest Moise: Calculus. It is a balance between Spivak/Courant/Apostol and computational calculus taught in most universities. Moise is closer to Courant, then say Stewart. Moreover, he motivates ideas very clearly and gives proofs. It is very readable. The proofs are well motivated and clear. If you found Kline to hard, then you will definitely find Spivak difficult, and maybe unreadable. I did not like Kline. He is too verbose for my taste.

If you need to review basics Ie., pre-calculus material. Maybe give Axler: Pre- Calculus in a nutshell a try. It comes with solutions to some problems, so good for someone starting self study at that level. Don't just look at the solution when you get. Maybe play with the problem for a day, then re-read the section prevalent to the problem, and try it again. If you are stuck, then maybe take a gander at the solution.

 

[MidgetDwarf]

It is a tough call, since now I will have to again restart if I am to pick a new course. Other than MIT OCW, I haven't been able to find courses freely available in such an organized manner (notes/text + tests + exams). The MIT ocw course was very tough, and I was very demotivated by how much I was struggling. Also, I discovered that I just like textbooks more than video lectures. I am finding it a bit difficult to find courses around textbooks, but this might be the way to go. Something like this where there are tests and exams based on a popular textbook.

I definitely like the idea of tests, it would really let me know how I'm doing. Currently I am making my own 'tests' from exercises, but this doesn't really emphasize exercises that are most important.

With a person with a PhD. I don't think you need the organization that MIT OCW offers. You should now be able to understand that learning is difficult, expectations should be based on reality (ie., not trying to learn Quantum Mechanics ,at the Griffith level, in one year, when starting from an intro mechanics course., and rolling your sleeves up and keep pencil and paper moving.

 

[atyy]

Neuromatch Academy
An online school for Computational Neuroscience
Started in response to the COVID-19 pandemic, Neuromatch Academy is a non-profit course in computational neuroscience offered online from July 13-31, 2020.
https://www.neuromatchacademy.org/

The summer school is over, but you can still see their recommendations for what to study as preparation: https://github.com/NeuromatchAcademy/precourse

And the pre-recorded lectures are at: https://www.youtube.com/channel/UC4LoD4yNBuLKQwDOV6t-KPw/about

 

[madness]

Have you tried reading Izhikevich's book Dynamical Systems in Neuroscience? It's available for free https://www.izhikevich.org/publications/dsn.pdf.

I wouldn't recommend using it as your primary source for learning the maths, but it might help you build some intuition since you're coming from a neuroscience background.