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.)

38 thoughts on “Assignment -1: Favorite Formula”

  1. Geodesic equation:
    \[ \frac{\partial^2 x_k}{\partial t^2} + {\sum_{i j }\Gamma_{ij}^{k} \frac{\mathrm{d} x_i}{\mathrm{d} t}\frac{\mathrm{d} x_j}{\mathrm{d} t}} = 0\]

  2. Divergence/Stokes

    \[
    \iiint_V \left(\nabla \cdot \mathbf{F}\right) dV = \iint_S \left(\mathbf{F}\cdot \vec{n}\right) dS
    \]
    \[
    \iint_S \left(\nabla \times \mathbf{F}\right) dS = \int_C \mathbf{F}\cdot dC
    \]

  3. ELBO

    \begin{equation}
    \text{ELBO} = \log P(x) – \text{KL}(q_x || P(z | x) ) &= \int_z q_x(z) \log \frac{P(x, z)}{q_x(z)} \,dz \\
    &= E_{z \sim q_x} \bigg[ \log \frac{P(x, z)}{q_x(z)} \bigg]
    \end{equation}

  4. ELBO

    \begin{equation}
    \text{ELBO} = \log P(x) – \text{KL}(q_x || P(z | x) ) = \int_z q_x(z) \log \frac{P(x, z)}{q_x(z)} \,dz = E_{z \sim q_x} \bigg[ \log \frac{P(x, z)}{q_x(z)} \bigg]
    \end{equation}

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