Assignment -1: Favorite Formula

Part of your course grade is determined by participation, which can include both in-class participation as well as discussion here on the course webpage.  Therefore, your first assignment is to:

  1. create an account, and
  2. leave a comment on this post containing your favorite mathematical formula (see below).
To make things interesting, your comment should include a description of your favorite mathematical formula typeset in $\LaTeX$.  If you don’t know how to use $\LaTeX$ this is a great opportunity to learn — a very basic introduction can be found here.  (And if you don’t have a favorite mathematical formula, this is a great time to pick one!)
(P.S. Anyone interested in hearing about some cool “favorite theorems” should check out this podcast.)

58 thoughts on “Assignment -1: Favorite Formula”

  1. Guess I liked it because I just learnt it in Set Theory yesterday, but…
    The recursive set encoding of natural numbers

    $0 = \emptyset $
    $n + 1 = \cup \{ n , \{ n \} \}$

  2. Not exactly math, but the classic rendering equation!

    $$ L_o(\textbf{x},\omega_o,\lambda,t) = L_e(\textbf{x},\omega_o,\lambda,t) + \int_{\Omega} f_r(\textbf{x},\omega_i,\lambda,t)L_i(\textbf{x},\omega_i,\lambda,t)(\omega_i \cdot \textbf{n})d\omega_i $$

  3. An integral producing the GCD!
    \[ \int_0^{\pi/2}\ln{\lvert\sin(mx)\rvert}\cdot \ln{\lvert\sin(nx)\rvert}\, dx=\frac{\pi^3}{24}\frac{\gcd^2(m,n)}{mn}+\frac{\pi\ln^2(2)}{2} \]

    1. I should probably use this account instead. So here’s a nice fourier series used in the derivation:

      \[ \sum_{n=1}^\infty \frac{\cos(kx)}{k}=-\ln\left(\bigg\lvert \sin\left(\frac{x}{2}\right) \bigg\rvert\right) \]

  4. fresh out of 21-801 Asymptotic Convex Geometry. this is what baby geometers play with.

    it is the unit ball in $(\mathbb{R}^n,||\cdot||_p)$ – that is: $\mathbb{R}^n$ equipped with the p-norm $||\cdot||_p$

    B_{p}^{n} = \left\{x \in \mathbb{R}^n \,:\, ||x||_p \leq 1\right\}

    1. actual favorite: slicing with hyperplanes, the volume of a slice

      given a centred convex body $K\in \mathbb{R}^n$ of volume 1 and a direction $\theta \in \mathbb{S}^{n-1}$ consider the function of the (n-1)-dimensional volume of the slice of $K$ with a hyperplane orthogonal to $\theta$ passing through $t\theta$ for $t \in \mathbb{R}.$

      f(t) = |K \cap t\theta + \theta^\bot|

  5. I took my first PDE class that was more theoretical last semester and used Divergence Theorem a lot

    $$\int_{\Omega} \nabla \cdot F dV = \int_{\partial \Omega} F \cdot \nu dS$$

  6. My current favorite formula is the KdV (Soliton) equation:

    $$\frac{\partial u}{\partial t}+6 u \frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial x^3}=0$$

    because 1) It is solvable even though it is nonlinear and third order, 2) Has really cool pattern formation (traveling stable wave) solutions, and 3) a number of applications/relationships

  7. Taylor series: great generalization of function approximation methods. Fundamental view of dealing with non-linear systems. Inspiring idea of function decomposition.

  8. The Rendering Equation, because I feel like I need to review rendering.

    \[L_o(\mathbf{x},\omega_o,\lambda,t)=L_e(\mathbf{x},\omega_o,\lambda,t)+\int_{\Omega}f_r(\mathbf{x},\omega_i,\omega_o,\lambda,t)L_i(\mathbf{x},\omega_o,\lambda,t)(w_i\ \cdot \ \mathbf{n})d\omega_i\]

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