Self studying the MIT applied math curriculum (2019)
If anyone wants to attempt this, that&#x27;s awesome.  It&#x27;s a lot of hard work, but I think it opens a lot of opportunities for personally and professionally.  I&#x27;ve a Ph.D in Applied Mathematics from a traditional program, so I wanted to chime in based on some of the comments I&#x27;m seeing.  As a note, this is an opinion and others may feel differently and strongly at that.<p>To me, applied mathematics is the art of transforming something into a linear system, which is something that we can tangibly solve on a computer.  There are lot&#x27;s of ways to do this such as Taylor series and Galerkin methods, so a lot of the field is understanding how, when, and why each method can be used.  This is coupled with mastery over linear solvers, which includes direct methods, iterative methods, preconditioners, etc.<p>I wanted to write this comment, though, to focus on certain areas that may end up blocking what I view as appropriate progression in the field.  These are things that I believe are necessary to understand advanced topics, but don&#x27;t necessarily fall under applied mathematics.  First, you really do need mastery of real analysis.  It&#x27;s necessary because it covers formally topics such as differentiation, integration, and series, which are required to understand theorems and algorithms.  In my opinion, calculus books are not sufficient.  Rudin&#x27;s Principle&#x27;s of Mathematical Analysis is the most concise, well written book that contains enough.  Second, enough functional analysis to understand Hilbert Spaces is required.  This prerequisite to this is the real analysis above.  The issue here is that algorithms for things like differential equations require function spaces to do properly.  Certainly, you can go really deep in this regard, but Hilbert Spaces are generally enough for practical algorithms.  This also affects optimization theory, which impacts machine learning.  Technically, you can do optimization theory with only real analysis, but the theory is cleaner in Hilbert Space.  Questions that need to be answered are things like does the infimum exist and can it be obtained?  Working with a general inner product is also a valuable tool for parallelization as well as a modeling tool.  Third, some integration or measure theory is required.  It depends on what you&#x27;re doing, so I don&#x27;t think mastery is strictly necessary, but spaces like L2 don&#x27;t make a lot of sense unless you know what a Lebesgue integral is.  Even if you want to just work with spaces that are Riemann integrable, measure theory helps understand when this is possible and the ramifications of it.  And, to be clear, this is important outside of differential equations.  If you want to understand optimization theory in a Hilbert Space, the inner products used will require some understand of measure theory.<p>Anyway, these are some random thoughts and ideas about the field.  I do believe strongly that any amount of study is beneficial as most engineering fields benefit from applied mathematics.