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<h1>Basic Mathematics</h1>
<div class="keepws">
(<span class="bold">notes</span>
(date 2025-06-04)
(category <a href="../../notes.html">culture</a>)
(status <span class="status-growing">growing</span>)
(confidence certain)
(importance 6)
(tags methodology personal learning))
</div>
<p class="preview">a roadmap for a firm foundation in basic principles of mathematics</p>
<div class="prose">
<p>Basic Mathematics is a roadmap for a firm foundation in basic principles of mathematics. It covers the following topics:</p>
<ul>
<li><strong>History</strong></li>
<li><strong>Mathematical Writing</strong></li>
<li><strong>Arithmetic</strong></li>
<li><strong>Trigonometry</strong></li>
<li><strong>Precalculus</strong></li>
<li><strong>Differential Calculus</strong></li>
<li><strong>Integral Calculus</strong></li>
<li><strong>Linear Algebra</strong></li>
<li><strong>Multivariable Calculus</strong></li>
<li><strong>Differential Equations</strong></li>
<li><strong>Proof Writing</strong></li>
<li><strong>Real Analysis</strong></li>
</ul>
<p>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 <a href="https://en.wikipedia.org/wiki/How_to_Read_a_Book">Synoptic Reading</a> 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</p>
<p>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.</p>
<blockquote>
To learn and not to do is really not to learn. To know and not to do is really not to know.<br>
<br>
-- Stephen R. Covey<br>
</blockquote>
<h3>On the Mathematical Life</h3>
<p>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 <a href="/blog/2025/toward-a-mathematical">Toward a Mathematical Life</a>. A guide into coaxing the unconcious mind into a state of mathematical thought through enviornmental factors, and
intention.</p>
<h3>Aspects of Modularity in Structure</h3>
<p>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.</p>
<h3>Mathematical History</h3>
<p>Generalized Mathematics History can leave a lot out</p>
<h3>Mathematical Writing</h3>
<p>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.</p>
<p><a href="https://terrytao.wordpress.com/advice-on-writing-papers/">On Writing</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/write-in-your-own-voice/">Write in your own voice</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/use-the-introduction-to-%E2%80%9Csell%E2%80%9D-the-key-points-of-your-paper/">Use the Introduction to "Sell" the key points of your paper</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/describe-the-results-accurately/">Describe the Results Accurately</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/organise-the-paper/">Organise the paper</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/motivate-the-paper/">Motivate the paper</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/use-good-notation/">Use good notation</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/give-appropriate-amounts-of-detail/">Give appropriate amounts of detail</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/dont-overoptimise/">Don't overoptimise</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/create-lemmas/">Create lemmas</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/write-a-rapid-prototype-first/">Write a rapid prototype first</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/write-professionally/">Write Professionally</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/take-advantage-of-the-english-language/">Take Advantage of the English Language</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/maximising-the-results-to-effort-ratio/">Maximising the results-to-effort ratio</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/proofread-and-double-check-your-paper-before-submission/">Proofread and double-check your paper before submission</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/submit-a-final-draft-not-a-first-draft/">Submit a final draft, not a first draft</a>
<a href="https://terrytao.wordpress.com/advice-on-writing-papers/submit-to-an-appropriate-journal/">Submit to an appropriate journal</a></p>
<p>Further Exposition on Mathematical Writing</p>
<ul>
<li><a href="https://webhomes.maths.ed.ac.uk/~cbarwick/papers/style.pdf">Notes on mathematical writing</a></li>
<li><a href="https://www.math.ucla.edu/~bertozzi/advice-papers.html">Advice for Writing Research Papers and Dissertations</a></li>
<li><a href="https://cohn.mit.edu/advice/">Advice for amateur mathematicians on writing and publishing papers</a></li>
<li><a href="https://kconrad.math.uconn.edu/blurbs/proofs/writingtips.pdf">Advice on Mathematical Writing</a></li>
<li><a href="https://www.wisdom.weizmann.ac.il/%7Eoded/writing.html">How to not write a paper</a> by Oded Goldreich</li>
<li><a href="https://gowers.wordpress.com/2007/10/19/my-favourite-pedagogical-principle-examples-first/">Pegagogical Principle Examples</a> by Timothy gowers</li>
<li><a href="https://gowers.wordpress.com/2007/10/24/examples-first-ii/">Pedagogical Principle Examples II</a></li>
<li><a href="https://www.amazon.com/How-Write-Mathematics-Steenrod/dp/0821800558">How to Write Mathematics</a></li>
<li><a href="https://mirror.gutenberg-asso.fr/tex.loria.fr/">A LaTeX Encyclopedia</a></li>
<li><a href="https://rjlipton.com/2011/05/06/navigating-cities-and-understanding-proofs/">Navigating Cities and Understanding Proofs</a> by The Godel Letter</li>
<li><a href="https://www.jmilne.org/math/tips.html">Tips for Authors</a> by J.S. Milne</li>
<li><a href="https://igorpak.wordpress.com/2017/07/12/how-to-write-math-papers-clearly/">How to write math papers clearly</a> by Igor Pak</li>
<li><a href="https://web.mit.edu/jrickert/www/mathadvice.html">Writing a Research Paper in Mathematics</a></li>
<li><a href="https://www.dailymotion.com/video/xf88b5">Writing Mathematics Badly</a> by Jean-Pierre Serre</li>
</ul>
<p>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.</p>
<blockquote>
In mathematics, the art of proposing a question must be held of higher value than solving.<br>
<br>
-- David Hilbert<br>
</blockquote>
<h3>Foundations of Basic Mathematics</h3>
<div class="max-w-5xl mx-auto my-12">
</div>
<p>For this section I would like to make a few additions particularly pertaining to a deeper understanding of cyclical functions.</p>
<ul>
<li><a href="https://krisyotam.com/references/trig-without-tears">Trig Without Tears</a></li>
<li>[]</li>
<li>[]</li>
<li>[]</li>
</ul>
<p>Here are a number of problem books to practice the various skills learned in this phase.</p>
<ul>
<li><a href="">Prealgebra by Chris McMullen</a></li>
<li><a href="">Algebra by Chris McMullen</a></li>
<li><a href="">Basic Linear Graphing by Chris McMullen</a></li>
<li><a href="">50 Challenging Algebra Problems</a></li>
<li><a href="">Trigonometry by Chris McMullen</a></li>
<li><a href="">50 Challenging Algebra Problems</a></li>
<li><a href="">50 Challenging Algebra Problems</a></li>
</ul>
<h3>Discrete Mathematics, Proof Writing, Naive Set Theory, Formal Logic</h3>
<div class="max-w-5xl mx-auto my-12">
</div>
<h3>Combinatorics, Graph Theory, and Number Theory</h3>
<div class="max-w-5xl mx-auto my-12">
</div>
<h3>Calculus, Linear Algebra, Abstract Algebra</h3>
<div class="w-screen relative left-[calc(-50vw+50%)] right-[calc(-50vw+50%)] my-12">
</div>
<h3>Differential Equations and Dynamical Systems</h3>
<div class="max-w-5xl mx-auto my-12">
</div>
<h3>Real Analysis, and Metric Spaces</h3>
<div class="max-w-5xl mx-auto my-12">
</div>
<p>a^2 + b^2 = c^2</p>
<p> ### Component Overview</p>
<ul>
<li><strong>Box</strong>: A simple container with styling for highlighted content</li>
<li><strong>Collapse</strong>: An expandable/collapsible section (like this one!)</li>
<li><strong>Excerpt</strong>: For displaying quotes with title, author, and optional version/year</li>
<li><strong>Quote</strong>: For displaying simpler quotes with just the author</li>
<li><strong>Book</strong>: For displaying book recommendations with cover image</li>
</ul>
<div class="flex justify-center">
<p> \begin{tikzpicture}
% Nodes
\node[circle,draw] (A) at (0,0) {A};
\node[circle,draw] (B) at (2,1) {B};
\node[circle,draw] (C) at (2,-1) {C};
\node[circle,draw] (D) at (4,0) {D};</p>
<p> % Edges
\draw[->] (A) -- (B);
\draw[->] (A) -- (C);
\draw[->] (B) -- (D);
\draw[->] (C) -- (D);
\end{tikzpicture}</p>
</div>
<div class="flex justify-center">
<p>\usetikzlibrary{decorations.pathmorphing, backgrounds}</p>
<p>\begin{tikzpicture}[
node/.style={circle, draw, fill=white, minimum size=8mm},
edge/.style={->, semithick, shorten >=2pt},
every loop/.style={min distance=10mm, looseness=10},
>=Stealth
]
% Center coordinate
\coordinate (C) at (0,0);</p>
<p> % Place five nodes on a circle
\foreach \i/\name in {90/1, 18/2, -54/3, -126/4, -198/5} {
\node[node] (N\name) at (\i:3cm) {\name};
}</p>
<p> % Radial edges from center
\foreach \name in {1,2,3,4,5}
\draw[edge] (C) -- (N\name);</p>
<p> % Circular "ring" edges
\foreach \a/\b in {1/2,2/3,3/4,4/5,5/1}
\draw[edge, red, bend left=20] (N\a) to (N\b);</p>
<p> % Diagonal chords
\draw[edge, blue, dashed] (N1) to (N3);
\draw[edge, green, decorate, decoration={snake, amplitude=1pt, segment length=4pt}] (N2) to[bend right=30] (N5);</p>
<p> % Loop at the center
\draw[edge, purple, thick] (C) edge[loop above] ();</p>
<p> % Background highlight
\begin{scope}[on background layer]
\fill[yellow, opacity=0.1] (C) circle (3.2cm);
\end{scope}</p>
<p> % Title label
\node[below=1.5cm] at (C) {\itshape Complex Five-Node Graph};</p>
<p>\end{tikzpicture}</p>
</div>
<h3>Classic Literature Collection</h3>
<div class="w-screen relative left-[calc(-50vw+50%)] right-[calc(-50vw+50%)] my-12">
</div>
<h2>On the Beauty of Mathematical Notation</h2>
<p>Mathematics is not merely a language—it is a symphony of symbols, a choreography of logic and form. To test the robustness of our <code>Math</code> component, we present a flurry of expressions both elementary and esoteric.</p>
<p>Consider the <strong>Pythagorean Theorem</strong> in its most basic form:
a^2 + b^2 = c^2</p>
<h3>Triple Integrals and Volume</h3>
<p>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:</p>
<p>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:</p>
<h3>Summation and Limits</h3>
<p>The Basel problem, solved by Euler, explores the value of the infinite sum of reciprocals of squares:</p>
<p>Nested summations can test vertical alignment and large limits:</p>
<p>Or even:</p>
<h3>Stirling's Approximation and Limits</h3>
<p>The behavior of factorials at infinity:</p>
<h3>Fourier Transform</h3>
<p>A classic in signal analysis:</p>
<p>And the inverse:</p>
<h3>Dirac Delta Function</h3>
<p>In distribution theory:</p>
<h3>Gaussian Integral</h3>
<p>A gem of analysis:</p>
<h3>Tensor Notation and Einstein Summation</h3>
<p>Stress test with indices:</p>
<p>Or an expression involving the Levi-Civita symbol:</p>
<h3>Conclusion</h3>
<p>This post serves both as an ode to the expressive power of mathematics and a rigorous stress test for symbol rendering.</p>
<p>\blacksquare
Following our <a href="/2020/next-for-vercel">Series B last December</a>, I'm happy to announce we've raised a <a href="https://techcrunch.com/2021/06/23/vercel-raises-102m-series-c-for-its-next-js-based-front-end-development-platform"><strong>$102M Series C</strong></a> from existing and new investors.</p>
<p>Our vision of the Web is a global realtime medium for both creators and consumers,
where all friction and latency are eliminated.</p>
<p>We'll use this investment to:</p>
<ol>
<li><a href="#next-momentum"><strong>Build the SDK for Web</strong></a></li>
<li><a href="#realtime"><strong>Lower the barrier of entry</strong></a></li>
<li><a href="#real"><strong>Focus on the end-user</strong></a></li>
</ol>
<h3>1. Build the SDK for the Web [#next-momentum]</h3>
<p> Next.js continues to grow organically as developers <a href="https://twitter.com/vincentchw/status/1392775589094838277">explain it</a> and <a href="https://twitter.com/vadim_kravcenko/status/1401937274972037122">recommend it</a> to one another</p>
</span>}
<p>/></p>
<p>Unlike other software development platforms, the Web never came with an SDK or Standard Development Kit.</p>
<p>While new browser APIs are constantly being added, the responsibility still falls on the developer to assemble them into an _engine_ for every new project.</p>
<p>The resulting diversity of tools and choices makes the Web the innovative and dynamic
medium that it is, but it can also be a major obstacle, as companies rally
to reinvent themselves in the digital medium.</p>
<p>React, <a href="https://reactjs.org/blog/2013/06/05/why-react.html">now 8 years old</a>, has proven to be a durable abstraction and <a href="https://twitter.com/rauchg/status/1068183829737664512?lang=en">building block</a>, a narrative violation of the common wisdom that "frontend frameworks are changed every week."</p>
<p>Next.js builds on this formidable foundation to give developers and companies
a meaningful starting point to build great pages, <a href="https://nextjs.org/blog/next-11#conformance">open sourcing the lessons</a> learned from the giants of the Web.</p>
<h3>2. Lower the barrier of entry [#realtime]</h3>
<p>Vercel <a href="https://twitter.com/bobfmuller/status/1404795808525004805">has</a> <a href="https://twitter.com/erikras/status/1382420067485417477">given</a> <a href="https://twitter.com/otpfy/status/1380458535356669953">developers</a> access to global infrastructure to
deploy and scale frontend projects. But significant challenges remain to
frictionless collaboration:</p>
<ul>
<li>Setting up and maintaining your developer environment</li>
<li>Integrating it into Git for continuous integration</li>
<li>Manually pushing and waiting to share progress with your team</li>
</ul>
<p>Perhaps the most incredible aspect of the Web is that it's not a read-only medium.
Each Web browser is simultaneously the <a href="https://twitter.com/sulco/status/1177559150563344384">consumption and creation mechanism</a>.</p>
<p>With <a href="https://nextjs.org/live">Next.js Live</a>, we want to build a Web that
everyone can contribute to. Whether it's to pitch in an idea or an edit,
by an experienced developer or a new designer, from a local editor or the
browser itself.</p>
<h3>3. Focus on the end-user [#real]</h3>
<p>A particularly interesting challenge of building a platform and tooling
company is that you have <strong>two users</strong> (whether it's acknowledged or not):</p>
<ol>
<li>The user of the tool</li>
<li>The user of the output of the tool</li>
</ol>
<p>No matter how many downloads Next.js gets, or how delightful and
realtime the development experience, a universal truth remains: <strong><a href="https://en.wikipedia.org/wiki/The_customer_is_always_right">the customer is king</a></strong>.</p>
<p>Performance for the end-user has been baked in the design of everything
we make. To name some examples:</p>
<ul>
<li>Next.js was born out of the insight that React in its initial Single Page Application presentation was putting the rendering burden on the user's device that should have been on the <a href="https://reactjs.org/blog/2020/12/21/data-fetching-with-react-server-components.html">server</a> instead.</li>
<li><a href="https://vercel.com/analytics">Vercel Analytics</a> prioritized giving you a Real Experience Score calculated by each device, rather than just listing traffic stats or TTFB latencies.</li>
<li>Pre-rendering at build time or via <a href="https://www.smashingmagazine.com/2021/04/incremental-static-regeneration-nextjs/">Incremental Static Regeneration</a> is all about <a href="/2020/static-hoisting#hoist-to-the-edge">taking the CPU cycles entirely out</a> of the page serving equation, removing latency for the end user.</li>
</ul>
<p>Our primary fitness function is your success for your customers. Not
every visitor might understand fully how every page is built, but if the
experience is delightful and fast, they ought to think it was powered by Vercel.</p>
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