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 (you must use your Andrew email address, so we can give you participation credit this semester!),
  2. sign up for Piazza and Discord,
  3. read carefully through the Course Description and Grading Policy, and
  4. 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.)

49 thoughts on “Assignment -1: Favorite Formula”

  1. $$V – E + F = 2$$
    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.”

  2. 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.

  3. $$x \cdot y = \Vert x \Vert \Vert y \Vert \cos \theta$$
    makes comp geo substantially more fun when you can do trig-related things staying in integers/with few operations.

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

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

  5. 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.

  6. 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

  7. Linear/Angular Momentum Balance in Continuum Mechanics
    $\nabla \cdot \mathbf{\sigma}+\rho\mathbf{b}=\rho\mathbf{a}$

  8. Holder’s inequality:

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

  9. 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 $v_x = – u_y$.

  10. \begin{equation}
    \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)}

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