My background is physicist/engineer, and at university we studied only particular topics in mathematics. Some of the courses were never used and faded away with time, and I need to refresh the knowledge when they are needed again. In certain areas I have to build it up almost from very beginning. Currently, I often skip some parts or do not really understand where certain formula comes from. It is very annoying. My first question is: what would be an efficient strategy to study in this case?
There are similar questions How to start with mathematics? or I want to start mathematics from scratch. What should I begin with? they are somewhat useful, but still pretty general. I need only particular topic, and I want a structured knowledge on one particular topic. Maybe is is worthy to explain by example what I mean by "structured knowledge". When I started to work with integral equations, there was a need to refresh knowledge on linear algebra (system of equations, matrix properties, singular values) and numerical techniques (iterative solvers, preconditioning techniques), but it was difficult to place this knowledge into some kind of a system. To do this properly, I would have to read a book or take a course on linear algebra. Unfortunately, there is no time for that, and the complete book/course will give much more information which may fade away without any use again. Instead, I would like to build kind of Knowledge Map for specific area and highlight the spots I recently did. Also such a map would be very useful to build prerequisites and collect learning materials when I start to learn a new topic.
One of the starting points to build such a map and/or strategy would be Mathematics Subject Classification. Unfortunately, relations/prerequisites between the subjects is not always clear in this classification. Also it is not very useful to collect the material: neither Amazon, not OpenCourseWare use it. And the second question is: can anyone suggest some resources to elaborate on this strategy?
Best Answer
You might want to search out taxonomies on mathematics, for example (I do not believe any will be complete): http://en.wikipedia.org/wiki/Lists_of_mathematics_topics
You can also see: http://en.wikipedia.org/wiki/Mathematics_Subject_Classification
In addition, I would recommend reading some books:
What Is Mathematics? An Elementary Approach to Ideas and Methods Richard Courant (Author), Herbert Robbins (Author), Ian Stewart (Editor)
How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) G. Polya (Author)
How to Think Like a Mathematician: A Companion to Undergraduate Mathematics Kevin Houston (Author)
I think you should certainly practice, practice, practice, especially proof strategies. See my post here for useful references: how to be good at proving?
You should look to universities with Open Course Ware as they have posted free courses online. See the following for a list of some of those schools: http://en.wikipedia.org/wiki/OpenCourseWare
The rest depends on which topic(s) you choose and we could provide additional references if you know or when you figure it out.
Regards -A