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, 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.)
Let $f$ be the function defined by $f(x) = 3x + 7$, and
let $a$ be a positive real number.
That was a test post and I couldn’t delete it whatsoever….
Taylor Series
\[\sum \limits_{n=0}^\infty \frac {{f^{(n)}} (a)}{n!} (x-a)^n\]
$latex e^{i\pi}+1=0$
If function $f$ is integrable on interval $[a,b],$ and there exists differentiable function $F$ such that $F’ = f$, then $\int_a^b f(x) dx = F(b) – F(a)$
$\lim_{h\to 0} \frac{(x+h)^2 – x^2}{h}=2x$
$V – E + F =2$, where $V$ denotes to vertex, $E$ denotes to edge, and $F$ denotes to face
Discrete Fourier transform：
$ X_k=\sum_{n=0}^{N-1} x_n \cdot e^{-\frac{i 2 \pi}{N} k n} $
$6\times 7=42$
$ f'(x) = \lim_{h\to 0} \frac{f(x+h) – f(x)}{h} $
Hanner’s inequality:
$||f + g||_p^p + ||f-g||_p^p \geq (||f||_p + ||g||_p)^p + \Big| ||f||_p – ||g||_p \Big|^p$
where $f, g \in L^p$ and $1 \leq p \leq 2$. The inequality is flipped for $p \geq 2$.
$v$ = $M$ * $\Gamma$(n) * $\kappa$
where $v$ denotes to grain boundary velocity, $M$ denotes to grain boundary mobility,
$\Gamma$(n) denotes to interface stiffness tensor and $\kappa$ denotes to grain boundary curvature matrix
Euler’s Identity
$ latex e^{i \pi} + 1 = 0 $
$e^{i \pi} + 1 = 0$
$\Delta \phi = \star d \star d\phi$
$ latex V = (\pi r^2)(2\pi R) $ 🍩
🍩, $V = (\pi 𝑟^2)(2\pi 𝑅)$
Fermat’s Principle:
$\delta T = \delta \int \frac{ds}{n_r} = 0
$ \delta T = \delta \int \frac{ds}{n_r} = 0 $
$\beta$-reduction! $(\lambda x.\ \varphi) \psi = \varphi[x \mapsto \psi]$
$$ A = \frac{1}{2} b\cdot h$$ for a triangle
Any planar graph is $4+3-2-1$-colorable.
Stokes’ theorem:
$\int_{\Omega} d\alpha = \int_{\partial \Omega} \alpha$
Shannon entropy: $H(X) = – \sum_{x \in \mathcal{X}} p(x) \log p(x)$
big fan of tupper’s self-referential formula $$\frac{1]{2} < \lfloor \mod\left(\lfloor \frac{y}{17} \rfloor 2^{-17 \lfloor x \rfloor – \mod(\lfloor y \rfloor, 17)}, 2\right ) \rfloor$$
$test$
apparently that’s not how you do latex on here haha
attempt #2:
$latex \frac{1]{2} < \lfloor \mod\left(\lfloor \frac{y}{17} \rfloor 2^{-17 \lfloor x \rfloor – \mod(\lfloor y \rfloor, 17)}, 2\right ) \rfloor$
this is so sad
third time’s the charm?
$\frac{1}{2} < \left \lfloor \textrm{mod} \left( \left\lfloor \frac{y}{17} \right\rfloor 2^{-17 \lfloor x \rfloor – \textrm{mod}(\lfloor y \rfloor, 17)}, 2\right ) \right\rfloor$
$$1 + 1 = 2$$
$a ^2+ b^2 = c^2$ – One of the very first formulas that got me interested in geometry.
$$|\mathbf{a}+\mathbf{b}|\leq|\mathbf{a}|+|\mathbf{b}|$$
$$|(\mathbf{a},\mathbf{b})|\leq\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert$$
Quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$
The prime factorization of 2023 is $7 × 17^2$. Happy 2023!
$latex \lvert\langle\mathbf{u},\mathbf{v}\rangle\rvert^2\leq\langle\mathbf{u},\mathbf{u} \rangle\cdot\langle\mathbf{v},\mathbf{v}\rangle $
Cholesky decomposition
$A = LL^{*}$
The manipulator equation, a general form EOM for rigidbody robots with control input ($u$) and constraints/contact forces ($\lambda$).
$M(q)\ddot{q} +C(q, \dot{q})\dot(q) = \tau_g(q) + Bu + J^T\lambda$
$M(q)$ is the mass matrix, $C(q, \dot{q})\dot{q}$ contains Coriolis forces, and $\tau_g(q)$ is the gravitational forces.
Oops, small error in the Latex:
$M(q)\ddot{q} +C(q, \dot{q})\dot{q} = \tau_g(q) + Bu + J^T\lambda$
The volume of the \(n\)-dimensional unit cube is
\[ r^n \]
while the volume of the $n$-dimensional unit sphere is
$$\frac{\pi^{n/2}}{\Gamma(\tfrac{n}2 + 1)}$$
The Binomial Theorem is
\[ (x+y)^n = \sum_{k=1}^{n} \binom{n}{k} x^{n-k} y^k = \sum_{k=1}^{n} \binom{n}{k} x^k y^{n-k} \]
I don’t wanna pick an obvious one but I’d have to choose Euler’s identity:
$latex \displaystyle e^{i \pi} + 1 = 0$
$\displaystyle\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt\pi$
$e^{i\pi} +1 = 0 $ Corrected
The action of a system described by a Lagrangian $L$ from time $t_a$ to time $t_b$ is $S = \int_{t_a}^{t_b} L(\dot x,x,t)dt$.
The n-th prime, ${p_{n}=1+\sum _{i=1}^{2^{n}}\left\lfloor \left({\frac {n}{\sum _{j=1}^{i}\left\lfloor \left(\cos {\frac {(j-1)!+1}{j}}\pi \right)^{2}\right\rfloor }}\right)^{1/n}\right\rfloor }$
Navier Stokes Equations
$\frac{du}{dt} + (u \cdot \nabla)u = – \nabla p + \frac{1}{Re}\nabla^{2}u$
$\nabla \cdot u = 0$
where $u$ is fluid velocity, $p$ is pressure, and $Re$ is the Reynolds #.
$\int_{\Omega} |\nabla u|= \int_{-\infty}^{\infty} H^{n-1}\{u>t\} dt$
Area of a triangle projected onto a sphere, where $\alpha_{1}$, $\alpha_{2}$, and $\alpha_{3}$ are the interior angles of the spherical triangle.
$A = \alpha_{1} + \alpha_{2} + \alpha_{3} – \pi$
Woah, that’s cool.
$9+10=21$
Since Shannon Entropy is taken, I’ll go with Renyi entropy. For $X \sim f$:
$h_p(X) = \frac{1}{1-p} \log \left(\int_{\mathbb{R}^d} f^p(x) dx \right)$
where $p \in (0,1) \cup (1,\infty)$, with $p=1$ recovering Shannon entropy via L’Hopital.
$\frac{\sin x}{n}=\textrm{six}$
$$L(\mathbf{x}, \vec{\omega}) = \int_{0}^{\infty} T_r(\mathbf{x}, \mathbf{x}_t) \sigma(\mathbf{x}_t) L_e(\mathbf{x}_t, \vec{\omega}) \mathrm{d} t$$
$\rho \dot{\mathbf v} = \nabla \cdot \mathbf \sigma + \rho \mathbf b$
— Banach’s Fixed Point Theorem —
For any complete metric space $X$ and contraction mapping $T: X \rightarrow X$
$$
\exists x^\star \in X: \: T(x^\star) = x^\star \text{ and } \forall x \in X, \lim_{n\rightarrow\infty} T^n(x) = x^\star
$$
Where $T^n(x) = T(T^{n-1}(x))$
Complementary variables, hell yeah:
$\sigma_x \sigma_p \geq \frac{\hbar}{2}$
Most popular in quantum mechanics, it’s a fundamental limit on accuracy (or is it???) for measuring certain variables concurrently…
Kirchhoff’s Matrix Tree Theorem: used for counting the number of spanning trees in a graph.
\[t(G) = \frac{1}{n} \prod_1^{n-1} \lambda_i\]
I wanna go for Schrödinger equation:
\[i\hbar\frac{\partial}{\partial t}|\Psi(t)\rangle = \hat{H}|\Psi(t)\rangle\]
$E = mc^2$
Maxwell’s Equations to describe electromagnetic fields and how it interacts with matters:
1. Gauss’s law for electric fields: $$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$
2. Gauss’s law for magnetic fields: $$\nabla \cdot \mathbf{B} = 0$$
3. Faraday’s law of induction: $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$
4. Ampere’s law with Maxwell’s correction: $$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$
References: https://en.wikipedia.org/wiki/Maxwell%27s_equations
Equation of equilibrium in cylindrical coordinates.
\begin{align}
&\frac{\partial \sigma_{r}}{\partial r}+\frac{\partial \tau_{r z}}{\partial z}+\frac{\sigma_{r}-\sigma_{\theta}}{r}+\rho \omega^{2} r=0, \\
&\frac{\partial \tau_{r z}}{\partial r}+\frac{\partial \sigma_{z}}{\partial z}+\frac{\tau_{r z}}{r}=0
\end{align}
$$0^{0^0}=1$$
$$| \textbf{Aut}(S_6) |= 1440$$
\textbf{T} \cdot \textbf{D} + div\textbf{q} + \rho r = \rho \dot{\epsilon}
$\textbf{T} \cdot \textbf{D} + div\textbf{q} + \rho r = \rho \dot{\epsilon}$
$$\frac{\partial \rho}{\partial t} + \nabla \cdot(\rho \mathbf{u})=0$$
Probability is one of my favorite subjects, so I can choose Bayes’ rule
$ P(A|B) = \frac{P(B|A)P(A)}{P(B)} $
I like this triple product that can be interpreted to describe the volume of a parallelepiped, yet swapping the order of vectors will change the sign to mean this volume in a different direction.
$\textbf{u} \times (\textbf{v} \times \textbf{w}) = – \textbf{u} \times (\textbf{w} \times \textbf{v})$
$$X_{ffd} = \sum_{i=0}^l \binom{l}{i} (1-s)^{l-i}s^i \left[ \sum_{j=0}^m \binom{m}{j} (1-t)^{m-j}t^j \left[\sum_{k=0}^n \binom{n}{k} (1-u)^{n-k}u^k \mathbf{P_{i,j,k}} \right]\right]$$
A Free Form Deformation of 3D objects. Given object points $(s,t,u)$ within the frame of the control points $\mathbf{P}$, the output is the deformed coordinates of the inputs according to the control points. Fun shapes arise from this 🙂 https://people.eecs.berkeley.edu/~sequin/CS285/PAPERS/Sederberg_Parry.pdf
I find the overlapping of architecture and mathematics really fascinating.
[ x (\theta) = r (\theta) \cos(\theta) ]
[ y (\theta) =r (\theta) \sin(\theta) ]
\hat{g}(f) = \int_{t1}^{t2} g(t)e^{-2\pi ift
\begin{align}
V-E+F=2
\end{align}
Peaucellier-Lipkin linkage: If you take two equal-length sides connected at one edge, and ten connect a rhombus to their other edges, then if you rotate one vertex of the rhombus along a circular path, its opposite vertex in the rhombus will move along a straight line. This mechanism is used to translate rotational motion into linear motion.