Understanding Jacobians

Posted on Fri 29 January 2021 in Mathematics • Tagged with Mathematics

$\newcommand{\pdv}[2]{\frac{\partial{#1}}{\partial{#2}}}$ $\newcommand{\ah}{\pmb{a} + \pmb{h}}$ $\newcommand{\a}{\pmb{a}}$ $\newcommand{\h}{\pmb{h}}$

The Jacobian Matrix

Consider a function that maps reals to reals, $f:\Bbb{R} \to \Bbb{R}$. The linear approximation of this function is given by $$f(a …

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Setting up MathNotes - an online math repository

Posted on Fri 16 October 2020 in Mathematics • Tagged with Mathematics, Programming

Just after 10th and as I was beginning my JEE Preparation, I felt the need for an extensible note-taking apparatus. Online notes would be too much trouble and would keep me hooked to the computer. Notebooks were also difficult, as I wanted my notes to be extensible; adding pages in …

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A Good JEE Main (September) problem

Posted on Fri 11 September 2020 in Mathematics • Tagged with Mathematics, JEE

This beauty came in the 2nd September shift 2 paper:

Let $A = \{ X = (x, y, z)^T : PX = 0 \text{ and } x^2+y^2+z^2=1 \}$, where $$P = \left[ \begin{array}{l l l}1&2&1 \\ -2&3&-4 \\ 1&9&-1 \end{array}\right]$$ then the …

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Math Notes: Lagrange interpolation

Posted on Fri 14 August 2020 in Mathematics • Tagged with Mathematics, Math Notes

This is a small set of posts that come into the category of "notes": self-explanations of concepts that I have recently picked up and found interesting.

Lagrange Interpolation

Lagrange Interpolation is a concept that allows us to find a polynomial of least degree passing through a given set of points …

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Is the shortest solution to a problem the best?

Posted on Sun 26 July 2020 in Mathematics • Tagged with Mathematics

My old man always used to say that there are two ways to solve a problem: there's the horse way, and there's the donkey way. The donkey way involves tedious calculations, whereas the horse way 'cuts' through the problem. Take this problem as an example:

For each positive integer $n …

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ISC 2020 Analysis: An application of Data Science and Statistics

Posted on Sat 11 July 2020 in Mathematics • Tagged with Mathematics, ISC, Data Science

The ISC Exam results were released on 10th June, 3 PM IST. In this article, I'll be analyzing the results of the 117 students in the science stream of my school and showing how they performed. I'll also weave a story with the data and point out things that could …

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Solving the African Integral (from YG file cover)

Posted on Thu 09 July 2020 in Mathematics • Tagged with Mathematics

A long time ago, I gave my friend (Let's call him C) an integral to solve. He came back to me a few days later and the conversation went something like this:

C: Debu, I couldn't solve that African integral you gave me.
Me: Ok, but why are you calling …

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A Compilation of hard limits

Posted on Fri 05 June 2020 in Mathematics • Tagged with Mathematics

This list consists of the limits that I found most challenging.

  1. $$\lim_{n \to \infty} \left( \frac{n!}{n^n} \right) ^\frac 1n$$
  2. $$\lim_{x \to 0} \left( \frac{1}{\ln(x + \sqrt{x^2+1})} - \frac 1{\ln(x+1)} \right)$$
  3. $$\lim_{n \to \infty} \frac{n + n^2 …

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Limit involving higher order infinitesimals

Posted on Fri 05 June 2020 in Mathematics • Tagged with Mathematics

Simple limit problems consist of the form $\lim_{x \to 0}\frac{O_1(x)O_2(x)..}{O_a(x)O_b(x)..}$, such as $\lim_{x \to 0} \frac{\sin 3x \tan 2x \tan^{-1} 5x}{x^2 \ln(1+x)}$. Here, the infinitesimals are well defined and cancel out easily. Some …

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COVID-19 USA Analysis: effects of the lack of lockdown

Posted on Fri 29 May 2020 in Mathematics • Tagged with Mathematics, COVID 19

The USA currently stands at 1.76 million COVID-19 cases. That's more than the next 5 nations combined. A large number of these cases are due to government inaction against the virus. The lack of a concerted lockdown across the country is also to blame. Here's a graph showing the …

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