a roadmap for a firm foundation in basic principles of mathematics
Basic Mathematics is a roadmap for a firm foundation in basic principles of mathematics. It covers the following topics:
This core was specifically designed for myself, or rather the avatar of "Undergrad Pure Mathematics Student with a Poor Foundation". If you feel that you are truly struggling in mathematics, and are looking for something slower paced this might not be for you. If you are struggling, and still looking to persevere through a more rigirous learning path filled with Synoptic Reading on each of the various subjects you are in the right place. Every level here provides just about the most rigirous standards for each discipline that myself and the community at large seem to agree upon. If you believe that statement is incorrect for any of the various entries I would love to hear your inquiry, and can be reached at krisyotam@protonmail.com
This is a original roadmpap that took months to create. It is a work in progress, and some insertions, deletions, and changes may be made in the future.
To learn and not to do is really not to learn. To know and not to do is really not to know.
-- Stephen R. Covey
It is to the suprise of no one that has put serious effort into the meticulous study of mathematics that the subject requires the balancing of a great deal of temperance, and creative thought. I have found that even obsession to a reasonable degree is not enough. One must become immersed in the subject, wholly engaging both mind and faculties in tandem. Therefore, and for that cause I recommend my essay Toward a Mathematical Life. A guide into coaxing the unconcious mind into a state of mathematical thought through enviornmental factors, and intention.
As mentioned above in "Who is this for?". This roadmap was designed for the Undergrad Mathematics Major. So there is to a extend a assumption of natural talent, giftedness, perseverance, and/or passion. I suggest that most work through this list linearly as those that are naturally quick to mathematical understanding will move through this fairly quickly. I do understand that this is not possible for everyone as you may be in the middle of a semester struggling to keep up with your current work. In that case, the structure of this Syllabis was designed in such a modular way that allows for the simultaneous learning of several areas. If your goal is to fill in most of the gaps that a poor elementary, middle, and highschool education left you with. Then this is the recommended way.
Generalized Mathematics History can leave a lot out
I first and foremost reccomend that each one of you start a blog. Mathematical Exposition is not in of itself more complex than most other types of exposition. It is the dull expositional insticts of one that lend to the difficulty of more formal writing. For that reason I suggest first the reading of a series of essays by Terrance Tao with a list of additional essays he reccomends prior to our main curricula here. I will do my best to present them in the most logical fashion.
On Writing Write in your own voice Use the Introduction to "Sell" the key points of your paper Describe the Results Accurately Organise the paper Motivate the paper Use good notation Give appropriate amounts of detail Don't overoptimise Create lemmas Write a rapid prototype first Write Professionally Take Advantage of the English Language Maximising the results-to-effort ratio Proofread and double-check your paper before submission Submit a final draft, not a first draft Submit to an appropriate journal
Further Exposition on Mathematical Writing
The contents of Mathematical Writing, Like Mathematical History or quite unique in the fashion that they have no prerequisite requirements for there completion. Do not worry about having something "meaningful", or "Impactful" to write. That will come later. Just write.
In mathematics, the art of proposing a question must be held of higher value than solving.
-- David Hilbert
For this section I would like to make a few additions particularly pertaining to a deeper understanding of cyclical functions.
Here are a number of problem books to practice the various skills learned in this phase.
a^2 + b^2 = c^2
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Mathematics is not merely a language—it is a symphony of symbols, a choreography of logic and form. To test the robustness of our Math component, we present a flurry of expressions both elementary and esoteric.
Consider the Pythagorean Theorem in its most basic form: a^2 + b^2 = c^2
A fundamental result in multivariable calculus is the computation of volume via triple integrals. For a region E in 3D space, the volume is given by:
In spherical coordinates, where x = \rho\sin\phi\cos\theta, y = \rho\sin\phi\sin\theta, and z = \rho\cos\phi, the Jacobian determinant introduces an extra factor of \rho^2\sin\phi. Thus:
The Basel problem, solved by Euler, explores the value of the infinite sum of reciprocals of squares:
Nested summations can test vertical alignment and large limits:
Or even:
The behavior of factorials at infinity:
A classic in signal analysis:
And the inverse:
In distribution theory:
A gem of analysis:
Stress test with indices:
Or an expression involving the Levi-Civita symbol:
This post serves both as an ode to the expressive power of mathematics and a rigorous stress test for symbol rendering.
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