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:

- create an account (you
**must**use your Andrew email address, so we can give you participation credit this semester!), - sign up for Piazza and Discord,
- read carefully through the Course Description and Grading Policy, and
- 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.)
$e^{i\pi} + 1 = 0$

$$Fib(n) = \frac{\phi^n – (-\frac{1}{\phi})^n}{\sqrt{5}}$$

$\sin^2(\theta) + \cos^2(\theta) = 1$

$$ \sum_{i=1}^{\infty} \frac{1}{i^2} = \frac{\pi^2}{6} $$

Change of Variables

\[

\int_{g(E)} f(y) dy = \int_E f(g(x)) |\text{det} J_g (x)| dx

\]

$L_0(p, w_0) = L_e(p, w_0) + \int_{H^2} \! f_r(p, w_i \rightarrow w_0)L_i(p, w_i)cos(\theta) \, \mathrm{d}w_i$

$$V – E + F = 2$$

i.e. a polyhedron has 2 fewer edges than vertices and faces. Or more generally

$$V – E + F = \chi$$

This defines the Euler characteristic, a topological invariant which can be broadly generalized in fascinating ways to prove seemingly unrelated theorems, such as “you can’t comb flat the hair on a coconut, but you could do the same for a torus.”

$$\lim_{n\rightarrow \infty} Pr(|X_n – \mu| > \epsilon) = 0$$

Inspired by the Euler Characteristic mentioned above, here is another form:

$\mathcal{X}(V_{\centerdot})=\sum_{i}(-1)^{i}dim(V_i)=\sum_{i}(-1)^{i}dim(H_{i}(V_{\centerdot}))$ where $V_{\centerdot}$ is a complex of finite-dimensional vector spaces and $H_i$ is homology group.

$a^2 + b^2 = (a+b)^2$

It’s true in a ring of characteristic 2!

Triangle Inequality:

$ ||x+y|| \le ||x|| + ||y|| $

$\|A\|_F^2 = \|A(I-B^+B)\|_F^2 + \|AB^+B\|_F^2$

$$ \iiint_E \textrm{div}(\vec{F})\textrm{d}V = \iint_{\partial E} \vec{F} \cdot \textrm{d}A $$

$$u(x) = \int_{\partial\Omega} \frac{dG(x,y)}{dn} u(y) – \frac{du(y)}{dn} G(x,y) dy$$

$$(\lambda x.M[x])\to(\lambda y.M[y])$$

$\int_{- \infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$

$$x \cdot y = \Vert x \Vert \Vert y \Vert \cos \theta$$

makes comp geo substantially more fun when you can do trig-related things staying in integers/with few operations.

\[\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}\]

Hardy-Weinberg equilibrium: $p^2 + 2pq + q^2 = 1$

Pythagorean Theorem:

For a right triangle,

\[a^2+b^2=c^2\]

where \(a\) = side of right triangle,

\(b\) = side of right triangle,

\(c\) = hypotenuse

\[ x_{n+1} = rx_{n}(1 – x_{n}) \]

$1 + 2 + … + n = \frac{n(n+1)}{2}$

Euler’s Formula: $$e^{ix} = cos(x) + isin(x)$$

\[1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + … = \infty \]

$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$

Einstein’s field equation:

$$R_{\mu \nu} – \frac{1}{2}Rg_{\mu \nu} = \frac{8\pi G}{c^4}T_{\mu \nu}$$ . Metric rules!

$$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2)}$$

$$(x+y)^n = \sum\limits_{k=0}^n\binom{n}{k}x^ky^{n-k}$$

Not sure it’s my favorite, but I don’t think anyone’s used it and I’m fond of the Gaussian integral:

$$ \int_{-\infty}^\infty e^{-x^2}dx = \sqrt{\pi} $$

Fourier Transform:

$X(f) = \int_{}{} x(t) \times e^{-i2 \pi ft} \,dt$

Not like I use it a lot (if at all) on a daily basis but give my lack of knowledge of complex numbers I find it pretty interesting that we can use imaginary numbers and such in practical applications.

Mass-energy equivalence:

$E = mc^2$

The number of edges in a plane triangulation (planar graph with the most possible edges) with $n>2$ vertices is $3n-6$

It seems sort of related to triangle mesh representations of solids

\[ \int_{M}^{ } \,dw = int_{dM}^{ } w\, \]

Linear/Angular Momentum Balance in Continuum Mechanics

$\nabla \cdot \mathbf{\sigma}+\rho\mathbf{b}=\rho\mathbf{a}$

$\mathbf{\sigma}=\mathbf{\sigma}^T$

For a vector $V$ it’s Euclidean length $l$ can be found as $l=\sqrt{\sum_{s\in V}^{}s^2}$

Holder’s inequality:

For $p, q \in [1, \infty)$ with $1/p + 1/q = 1$, and real vectors $f, g$, $$||\langle f, g \rangle ||_1\leq ||f||_p \cdot ||g||_q$$

Pythagorean Theorem: simple but I use it often

$a^2 + b^2 = c^2$

For $u + iv = f(z)$, where $z=x+ iy$, to be a analytical function the partial derivatives condition is given by Cauchy and Riemann eqns:

$u_x = v_y$ and $v_x = – u_y$.

\begin{equation}

\begin{aligned}

\frac{\partial\mathcal{D}}{\partial t} \quad & = \quad \nabla\times\mathcal{H}, & \quad \text{(Loi de Faraday)} \\[5pt]

\frac{\partial\mathcal{B}}{\partial t} \quad & = \quad -\nabla\times\mathcal{E}, & \quad \text{(Loi d’Ampère)} \\[5pt]

\nabla\cdot\mathcal{B} \quad & = \quad 0, & \quad \text{(Loi de Gauss)} \\[5pt]

\nabla\cdot\mathcal{D} \quad & = \quad 0. & \quad \text{(Loi de Colomb)}

\end{aligned}

\end{equation}

“Let there be light”

\begin{equation}

L_o = L_e + \int\limits_\Omega L_i f_r \cos\theta d\omega

\end{equation}

The Euler-Lagrange Equation:

$$\frac{\partial \cal{L}}{\partial q_i} – \frac{d}{dt}\frac{\partial \cal{L}}{\partial \dot{q_i}} = 0$$

Parkinson’s Law

$\Delta t_\textsf{taken to do work} \leq \Delta t_\textsf{allotted for its completion}$

$$\sum_{i=0}^\infty r^i = {1 \over 1 – r}, \vert r \vert \lt 1$$

$\sum_{k=0}^n {n \choose k}^2 = {2n \choose n}$

$$ \sum_{i=0}^{n} \binom{i}{r} = \binom{n+1}{r+1} $$

$$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$$

1-(1/2)+(1/3)-…+… = ln(2)

$$\sum_{i=0}^\infty 2^i= -1$$