Discrete derivative operator Let a function \(f: \mathbb{R} \mapsto \mathbb{R}\). On discretizing the domain of \(f\) into quanta \(h\) centred at \(a_0\), \(f: \mathbb{A} \mapsto \mathbb{A}\) where \(\mathbb{A} = \left\{ kh+a_0 : k \in \mathbb{Z}, a_0 \in \mathbb{R} \right\}\), the derivative operator is replaced by the discrete derivative ope...

A sweet problem The classic statement Imagine you have a cake. How can you slice it into \(8\) pieces in exactly \(3\) steps? Well, you divide the cake into two, three times, so that the number of pieces compounds to \(2^3 = 8\). This can be done by cutting the cake along different planes, in the following manner: How mathematicians probably...

A story without a film Consider a gravitational system comprising the earth and an apple. The apple is released from a certain height and it plummets to the ground. How would the evolution of this system proceed, if, instead, time ran backwards? Our intuition tells us that if time runs backwards, the apple should, as if by definition, rise up ...

A covector basis \(\left\{ \pmb{\theta}^i \right\}\) is defined to act on the corresponding vector basis \(\left\{ \pmb{e}_j \right\}\) in the manner, [\pmb{\theta}^i \left( \pmb{e}j \right) = \delta^i{\phantom{i} j}] Where \(\delta^i_{\phantom{i} j}\) represents the Kronecker delta. But where does the above definition even come from? Well, tu...

Welcome to Part 2 of ‘Algebra Done Tensorially’. If you haven’t already done so, make sure to check out the previous post, Part 1 (Bilinear Products) before reading this post :) I will start right from where we stopped in Part 1. Parts Topics Part 1 (Bilinear Products) tensors, bilinear products ...

Welcome to this five-part series of posts: Parts Topics Part 1 (Bilinear Products) tensors, bilinear products Part 2 (Algebras Over Fields) linear maps, algebra, degrees of freedom Part 3 (Complex Numbers and Quaternions) complex numbers, quaternions, g...

The war (Disclaimer: All events narrated in the ‘bar’ are fictional. Any historical inaccuracies are intentional.) Two men walk into a bar. They enter just in time, for it starts raining heavily. One man is rather small. “I’m a point Charge”, says he. The other looks athletic. Almost sprinting, he shouts, “I’m Current”. The two men then glar...

What is the gamma function about? About 300 years ago, the influential mathematician Leonhard Euler solved the problem of extending the factorial function to non-integers. He originally found an infinite product representation, which he soon expressed in the integral form, [\displaystyle{ z! = \int_0^1 \left( - \ln u \right)^z du }] By the su...