Function transforms (providing a broader picture of Laplace Transforms)

Posted on Wed 12 May 2021 in Mathematics • Tagged with Mathematics

I'll begin this article by brushing up a few definitions:

A function is a mapping between two sets: the domain D and the codomain C.

function definition

It's very important to note here that the function is the mapping itself, and not an element in the codomain or the domain. The function …


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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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