What is the Best Order to Learn Mathematics and Physics for Deep Understanding?

[theoristo]

For someone who want to understand physics concepts as deeply as possible with all the mathematics with it,what 's the order in which you should learn Mathematics and physics from zero,starting from from the most elementary? And at what pace?

 

[brocks]

Presumably, universities want their students to understand physics concepts as deeply as possible, and most of them have settled on approximately the same order of courses, which is laid out in their online catalogs. So just look at the curricula of some universities you would like to attend, and see what they do.

 

[theoristo]

Thanks.

 

[Tsunoyukami]

After completing the majority of my physics undergrad I wish I had taken my subjects in a different order. I find that, in general, I am introduced to mathematical concepts in my physics classes before being taught them in my math classes. This amounts to simply applying the procedure for solving a problem in physics without really understanding the underlying mathematics (for example, much of vector calculus was needed in my electricity and magnetism class prior to it being taught in multivariable & vector calculus class).You may want to note that my experience is mostly computational (which is often useful for physics, as you are often asked to solve integrals etc.), but I wish I had been exposed to a more rigourous treatment earlier. If you have free time I would recommend something like this - note that I've divided it up into a "regular" school year division which you may be able to compress depending on time and your own prior knowledge of the subjects).

First Year:
-Calculus 1
-Linear Algebra 1
-Introduction to Proofs
-Intro to Physics 1
-Computer Programming

-Calculus 2
-Ordinary Differential Equations
-Introduction to Proofs
-Intro to Physics 2
-Computer Programming

First Summer
-Calculus 3 & 4 (Multivariable and Vector Calculus)
-A proof-based rigourous calculus/analysis course
-Computer programming

At this point you should have developed a strong background with computational math in calculus, linear algebra, and differential equations, all of which are very useful to know prior to taking the typical "second year" physics classes. (For example, partial derivatives are common in thermodynamics; vector calculus is common in E&M; ODEs are common in mechanics; linear algebra is useful in QM). Furthermore, you will have started to appreciate calculus at a 'higher' level which will greatly deepen you understanding of math.

Second Year
-Complex Variables
-Thermodynamics
-Mechanics 1
-Proof-based multivariable calculus
-Practical physics 1 (computer programming)

-Electricity and Magnetism 1
-Quantum Mechanics 1
-Proof-based multivariable calculus
-(Partial Differential Equations)
-practical physics 2 (computer programming)

Second Summer
-Partial Differential Equations
-Computer programmingAt this point you should have a very strong understanding of all lower-level undergraduate physics and math concepts and have had the mathematical idea presented prior to their appearance in your physics classes. You should also be comfortable with many of the ideas presented in the more rigourous calculus classes.

At this point you should be entering your third year of study and the outline presented by your school likely presents topics in a useful manner. I recommended computer programming even though its not officially part of the physics program at my university because I feel it's a very important and useful skill.