# Assignment 0

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
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!)

## 56 thoughts on “Assignment 0”

1. Ray Li says:

$\int_0^1dx = 1$

1. Keenan says:

Ah yes, how true. The length of the unit interval is 1.

2. LB419545068 says:

Hello world

1. Keenan says:

Hmm… I don’t know if “Hello world” counts as a mathematical formula per se, but it certainly is a classic!

1. haocheng LIU says:

Hmm…, sorry prof i just realize that we need to type a mathematical formula….
Here is my favorite one :
$e^{i\pi} + 1 = 0$

1. haocheng LIU says:

eh, it should be $e^{i\pi} + 1 = 0$ Need more study on LATEX

3. Stokes2BeHere says:

Bonjour

1. Keenan says:

Salut! Voulez-vous d’écrire une formule mathématique?

1. Stokes2BeHere says:

Je l’ai fait!

2. Stokes2BeHere says:

Oh, I didn’t see that we had to include our favorite mathematical formula. Here you go:

$e^{i\theta} = \cos(\theta) + i\sin(\theta)$

1. Keenan says:

Euler’s identity—beautiful. This fact will have important geometric meaning for us, especially in our study of surface parameterization. Basically it lets us represent rotations via unit complex numbers, and hence interpret multiplication by any complex number as a “stretch-rotation” (rotation composed with uniform scaling).

4. aksahu says:

The discrete version of the celebrated Hamilton-Jacobi-Bellman equation

$V(x_{0})=\max_{a_{0}}{F(x_{0}, a_{0})+bV(x_{1})}$

1. aksahu says:

Don’t know relevant this would be for the course, but this is the Bellman equation. To be specific it’s the discrete version of the H-J-B equation.

2. Keenan says:

Great. Discretizing variational principles can have some very nice computational benefits, such as conservation of momental in numerical time integrators.

5. Stokes2BeHere says:

So I guess wordpress won’t let us edit comments?

6. shushman says:

$\mathbb{P}[A|B] = \frac{\mathbb{P}[A] \mathbb{P}[B|A]}{\mathbb{P}[B]}$

I know it is probably not too relevant to this course, but I’ve always been fascinated by the myriad implications behind this otherwise unassuming formula! Oh I also adore Euler’s identity.

7. kborst says:

This appears to be a popular one, but mine is also $e^{\pi i}+1=0$

1. Keenan says:

Definitely a classic.

8. lotusword says:

$\sin^2x+\cos^2 x=1$

1. Keenan says:

Indeed—points on the unit circle have unit norm!

9. Khryl says:

$$3^2 + 4^2 = 5^2$$

1. Keenan says:

…and the area is 6.

10. Liam says:

$\int_M d\omega = \int_{\partial M} \omega$

1. Keenan says:

Stokes’ Theorem—fantastic. We will be making good use of this theorem in our class. In fact, it is the basic idea behind how we go from smooth to discrete differential forms.

11. The Lagrangian of the Standard Model of Particle Physics is pretty neat.
http://nuclear.ucdavis.edu/~tgutierr/files/stmL1.html

$-\frac{1}{2}\partial_{\nu}g^{a}_{\mu}\partial_{\nu}g^{a}_{\mu} -g_{s}f^{abc}\partial_{\mu}g^{a}_{\nu}g^{b}_{\mu}g^{c}_{\nu} -\frac{1}{4}g^{2}_{s}f^{abc}f^{ade}g^{b}_{\mu}g^{c}_{\nu}g^{d}_{\mu}g^{e}_{\nu} +\frac{1}{2}ig^{2}_{s}(\bar{q}^{\sigma}_{i}\gamma^{\mu}q^{\sigma}_{j})g^{a}_{\mu} +\bar{G}^{a}\partial^{2}G^{a}+g_{s}f^{abc}\partial_{\mu}\bar{G}^{a}G^{b}g^{c}_{\mu} -\partial_{\nu}W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-M^{2}W^{+}_{\mu}W^{-}_{\mu} -\frac{1}{2}\partial_{\nu}Z^{0}_{\mu}\partial_{\nu}Z^{0}_{\mu}-\frac{1}{2c^{2}_{w}} M^{2}Z^{0}_{\mu}Z^{0}_{\mu} -\frac{1}{2}\partial_{\mu}A_{\nu}\partial_{\mu}A_{\nu} -\frac{1}{2}\partial_{\mu}H\partial_{\mu}H-\frac{1}{2}m^{2}_{h}H^{2} -\partial_{\mu}\phi^{+}\partial_{\mu}\phi^{-}-M^{2}\phi^{+}\phi^{-} -\frac{1}{2}\partial_{\mu}\phi^{0}\partial_{\mu}\phi^{0}-\frac{1}{2c^{2}_{w}}M\phi^{0}\phi^{0} -\beta_{h}[\frac{2M^{2}}{g^{2}}+\frac{2M}{g}H+\frac{1}{2}(H^{2}+\phi^{0}\phi^{0}+2\phi^{+}\phi^{-%%@ })]+\frac{2M^{4}}{g^{2}}\alpha_{h} -igc_{w}[\partial_{\nu}Z^{0}_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu}) -Z^{0}_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu}) +Z^{0}_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})] -igs_{w}[\partial_{\nu}A_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu}) -A_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu}) +A_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})] -\frac{1}{2}g^{2}W^{+}_{\mu}W^{-}_{\mu}W^{+}_{\nu}W^{-}_{\nu}+\frac{1}{2}g^{2} W^{+}_{\mu}W^{-}_{\nu}W^{+}_{\mu}W^{-}_{\nu} +g^2c^{2}_{w}(Z^{0}_{\mu}W^{+}_{\mu}Z^{0}_{\nu}W^{-}_{\nu}-Z^{0}_{\mu}Z^{0}_{\mu}W^{+}_{\nu} W^{-}_{\nu}) +g^2s^{2}_{w}(A_{\mu}W^{+}_{\mu}A_{\nu}W^{-}_{\nu}-A_{\mu}A_{\mu}W^{+}_{\nu} W^{-}_{\nu}) +g^{2}s_{w}c_{w}[A_{\mu}Z^{0}_{\nu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu})-%%@ 2A_{\mu}Z^{0}_{\mu}W^{+}_{\nu}W^{-}_{\nu}] -g\alpha[H^3+H\phi^{0}\phi^{0}+2H\phi^{+}\phi^{-}] -\frac{1}{8}g^{2}\alpha_{h}[H^4+(\phi^{0})^{4}+4(\phi^{+}\phi^{-})^{2}+4(\phi^{0})^{2} \phi^{+}\phi^{-}+4H^{2}\phi^{+}\phi^{-}+2(\phi^{0})^{2}H^{2}] -gMW^{+}_{\mu}W^{-}_{\mu}H-\frac{1}{2}g\frac{M}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}H -\frac{1}{2}ig[W^{+}_{\mu}(\phi^{0}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{0}) -W^{-}_{\mu}(\phi^{0}\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}\phi^{0})] +\frac{1}{2}g[W^{+}_{\mu}(H\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}H) -W^{-}_{\mu}(H\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}H)] +\frac{1}{2}g\frac{1}{c_{w}}(Z^{0}_{\mu}(H\partial_{\mu}\phi^{0}-\phi^{0}\partial_{\mu}H) -ig\frac{s^{2}_{w}}{c_{w}}MZ^{0}_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) +igs_{w}MA_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) -ig\frac{1-2c^{2}_{w}}{2c_{w}}Z^{0}_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-%%@ }\partial_{\mu}\phi^{+}) +igs_{w}A_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{+}) -\frac{1}{4}g^{2}W^{+}_{\mu}W^{-}_{\mu}[H^{2}+(\phi^{0})^{2}+2\phi^{+}\phi^{-}] -\frac{1}{4}g^{2}\frac{1}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}[H^{2}+(\phi^{0})^{2}+2(2s^{2}_{w}-%%@ 1)^{2}\phi^{+}\phi^{-}] -\frac{1}{2}g^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-%%@ }_{\mu}\phi^{+}) -\frac{1}{2}ig^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) +\frac{1}{2}g^{2}s_{w}A_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-}_{\mu}\phi^{+}) +\frac{1}{2}ig^{2}s_{w}A_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) -g^{2}\frac{s_{w}}{c_{w}}(2c^{2}_{w}-1)Z^{0}_{\mu}A_{\mu}\phi^{+}\phi^{-}-%%@ g^{1}s^{2}_{w}A_{\mu}A_{\mu}\phi^{+}\phi^{-} -\bar{e}^{\lambda}(\gamma\partial+m^{\lambda}_{e})e^{\lambda} -\bar{\nu}^{\lambda}\gamma\partial\nu^{\lambda} -\bar{u}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{u})u^{\lambda}_{j} -\bar{d}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{d})d^{\lambda}_{j} +igs_{w}A_{\mu}[-(\bar{e}^{\lambda}\gamma^{\mu} e^{\lambda})+\frac{2}{3}(\bar{u}^{\lambda}_{j}\gamma^{\mu} %%@ u^{\lambda}_{j})-\frac{1}{3}(\bar{d}^{\lambda}_{j}\gamma^{\mu} d^{\lambda}_{j})] +\frac{ig}{4c_{w}}Z^{0}_{\mu} [(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda})+ (\bar{e}^{\lambda}\gamma^{\mu}(4s^{2}_{w}-1-\gamma^{5})e^{\lambda})+ (\bar{u}^{\lambda}_{j}\gamma^{\mu}(\frac{4}{3}s^{2}_{w}-1-\gamma^{5})u^{\lambda}_{j})+ (\bar{d}^{\lambda}_{j}\gamma^{\mu}(1-\frac{8}{3}s^{2}_{w}-\gamma^{5})d^{\lambda}_{j})] +\frac{ig}{2\sqrt{2}}W^{+}_{\mu}[(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})e^{\lambda}) +(\bar{u}^{\lambda}_{j}\gamma^{\mu}(1+\gamma^{5})C_{\lambda\kappa}d^{\kappa}_{j})] +\frac{ig}{2\sqrt{2}}W^{-}_{\mu}[(\bar{e}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda}) +(\bar{d}^{\kappa}_{j}C^{\dagger}_{\lambda\kappa}\gamma^{\mu}(1+\gamma^{5})u^{\lambda}_{j})] +\frac{ig}{2\sqrt{2}}\frac{m^{\lambda}_{e}}{M} [-\phi^{+}(\bar{\nu}^{\lambda}(1-\gamma^{5})e^{\lambda}) +\phi^{-}(\bar{e}^{\lambda}(1+\gamma^{5})\nu^{\lambda})] -\frac{g}{2}\frac{m^{\lambda}_{e}}{M}[H(\bar{e}^{\lambda}e^{\lambda}) +i\phi^{0}(\bar{e}^{\lambda}\gamma^{5}e^{\lambda})] +\frac{ig}{2M\sqrt{2}}\phi^{+} [-m^{\kappa}_{d}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1-\gamma^{5})d^{\kappa}_{j}) +m^{\lambda}_{u}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1+\gamma^{5})d^{\kappa}_{j}] +\frac{ig}{2M\sqrt{2}}\phi^{-} [m^{\lambda}_{d}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1+\gamma^{5})u^{\kappa}_{j}) -m^{\kappa}_{u}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1-\gamma^{5})u^{\kappa}_{j}] -\frac{g}{2}\frac{m^{\lambda}_{u}}{M}H(\bar{u}^{\lambda}_{j}u^{\lambda}_{j}) -\frac{g}{2}\frac{m^{\lambda}_{d}}{M}H(\bar{d}^{\lambda}_{j}d^{\lambda}_{j}) +\frac{ig}{2}\frac{m^{\lambda}_{u}}{M}\phi^{0}(\bar{u}^{\lambda}_{j}\gamma^{5}u^{\lambda}_{j}) -\frac{ig}{2}\frac{m^{\lambda}_{d}}{M}\phi^{0}(\bar{d}^{\lambda}_{j}\gamma^{5}d^{\lambda}_{j}) +\bar{X}^{+}(\partial^{2}-M^{2})X^{+}+\bar{X}^{-}(\partial^{2}-M^{2})X^{-} +\bar{X}^{0}(\partial^{2}-\frac{M^{2}}{c^{2}_{w}})X^{0}+\bar{Y}\partial^{2}Y +igc_{w}W^{+}_{\mu}(\partial_{\mu}\bar{X}^{0}X^{-}-\partial_{\mu}\bar{X}^{+}X^{0}) +igs_{w}W^{+}_{\mu}(\partial_{\mu}\bar{Y}X^{-}-\partial_{\mu}\bar{X}^{+}Y) +igc_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}X^{0}-\partial_{\mu}\bar{X}^{0}X^{+}) +igs_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}Y-\partial_{\mu}\bar{Y}X^{+}) +igc_{w}Z^{0}_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-}) +igs_{w}A_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-}) -\frac{1}{2}gM[\bar{X}^{+}X^{+}H+\bar{X}^{-}X^{-}H+\frac{1}{c^{2}_{w}}\bar{X}^{0}X^{0}H] +\frac{1-2c^{2}_{w}}{2c_{w}}igM[\bar{X}^{+}X^{0}\phi^{+}-\bar{X}^{-}X^{0}\phi^{-}] +\frac{1}{2c_{w}}igM[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}] +igMs_{w}[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}] +\frac{1}{2}igM[\bar{X}^{+}X^{+}\phi^{0}-\bar{X}^{-}X^{-}\phi^{0}]$

1. I hope this one works.
$-\frac{1}{2}\partial_{\nu}g^{a}_{\mu}\partial_{\nu}g^{a}_{\mu} -g_{s}f^{abc}\partial_{\mu}g^{a}_{\nu}g^{b}_{\mu}g^{c}_{\nu} -\frac{1}{4}g^{2}_{s}f^{abc}f^{ade}g^{b}_{\mu}g^{c}_{\nu}g^{d}_{\mu}g^{e}_{\nu} +\frac{1}{2}ig^{2}_{s}(\bar{q}^{\sigma}_{i}\gamma^{\mu}q^{\sigma}_{j})g^{a}_{\mu} +\bar{G}^{a}\partial^{2}G^{a}+g_{s}f^{abc}\partial_{\mu}\bar{G}^{a}G^{b}g^{c}_{\mu} -\partial_{\nu}W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-M^{2}W^{+}_{\mu}W^{-}_{\mu} -\frac{1}{2}\partial_{\nu}Z^{0}_{\mu}\partial_{\nu}Z^{0}_{\mu}-\frac{1}{2c^{2}_{w}} M^{2}Z^{0}_{\mu}Z^{0}_{\mu} -\frac{1}{2}\partial_{\mu}A_{\nu}\partial_{\mu}A_{\nu} -\frac{1}{2}\partial_{\mu}H\partial_{\mu}H-\frac{1}{2}m^{2}_{h}H^{2} -\partial_{\mu}\phi^{+}\partial_{\mu}\phi^{-}-M^{2}\phi^{+}\phi^{-} -\frac{1}{2}\partial_{\mu}\phi^{0}\partial_{\mu}\phi^{0}-\frac{1}{2c^{2}_{w}}M\phi^{0}\phi^{0} -\beta_{h}[\frac{2M^{2}}{g^{2}}+\frac{2M}{g}H+\frac{1}{2}(H^{2}+\phi^{0}\phi^{0}+2\phi^{+}\phi^{-%%@ })]+\frac{2M^{4}}{g^{2}}\alpha_{h} -igc_{w}[\partial_{\nu}Z^{0}_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu}) -Z^{0}_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu}) +Z^{0}_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})] -igs_{w}[\partial_{\nu}A_{\mu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu}) -A_{\nu}(W^{+}_{\mu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\mu}\partial_{\nu}W^{+}_{\mu}) +A_{\mu}(W^{+}_{\nu}\partial_{\nu}W^{-}_{\mu}-W^{-}_{\nu}\partial_{\nu}W^{+}_{\mu})] -\frac{1}{2}g^{2}W^{+}_{\mu}W^{-}_{\mu}W^{+}_{\nu}W^{-}_{\nu}+\frac{1}{2}g^{2} W^{+}_{\mu}W^{-}_{\nu}W^{+}_{\mu}W^{-}_{\nu} +g^2c^{2}_{w}(Z^{0}_{\mu}W^{+}_{\mu}Z^{0}_{\nu}W^{-}_{\nu}-Z^{0}_{\mu}Z^{0}_{\mu}W^{+}_{\nu} W^{-}_{\nu}) +g^2s^{2}_{w}(A_{\mu}W^{+}_{\mu}A_{\nu}W^{-}_{\nu}-A_{\mu}A_{\mu}W^{+}_{\nu} W^{-}_{\nu}) +g^{2}s_{w}c_{w}[A_{\mu}Z^{0}_{\nu}(W^{+}_{\mu}W^{-}_{\nu}-W^{+}_{\nu}W^{-}_{\mu})-%%@ 2A_{\mu}Z^{0}_{\mu}W^{+}_{\nu}W^{-}_{\nu}] -g\alpha[H^3+H\phi^{0}\phi^{0}+2H\phi^{+}\phi^{-}] -\frac{1}{8}g^{2}\alpha_{h}[H^4+(\phi^{0})^{4}+4(\phi^{+}\phi^{-})^{2}+4(\phi^{0})^{2} \phi^{+}\phi^{-}+4H^{2}\phi^{+}\phi^{-}+2(\phi^{0})^{2}H^{2}] -gMW^{+}_{\mu}W^{-}_{\mu}H-\frac{1}{2}g\frac{M}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}H -\frac{1}{2}ig[W^{+}_{\mu}(\phi^{0}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{0}) -W^{-}_{\mu}(\phi^{0}\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}\phi^{0})] +\frac{1}{2}g[W^{+}_{\mu}(H\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}H) -W^{-}_{\mu}(H\partial_{\mu}\phi^{+}-\phi^{+}\partial_{\mu}H)] +\frac{1}{2}g\frac{1}{c_{w}}(Z^{0}_{\mu}(H\partial_{\mu}\phi^{0}-\phi^{0}\partial_{\mu}H) -ig\frac{s^{2}_{w}}{c_{w}}MZ^{0}_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) +igs_{w}MA_{\mu}(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) -ig\frac{1-2c^{2}_{w}}{2c_{w}}Z^{0}_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-%%@ }\partial_{\mu}\phi^{+}) +igs_{w}A_{\mu}(\phi^{+}\partial_{\mu}\phi^{-}-\phi^{-}\partial_{\mu}\phi^{+}) -\frac{1}{4}g^{2}W^{+}_{\mu}W^{-}_{\mu}[H^{2}+(\phi^{0})^{2}+2\phi^{+}\phi^{-}] -\frac{1}{4}g^{2}\frac{1}{c^{2}_{w}}Z^{0}_{\mu}Z^{0}_{\mu}[H^{2}+(\phi^{0})^{2}+2(2s^{2}_{w}-%%@ 1)^{2}\phi^{+}\phi^{-}] -\frac{1}{2}g^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-%%@ }_{\mu}\phi^{+}) -\frac{1}{2}ig^{2}\frac{s^{2}_{w}}{c_{w}}Z^{0}_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) +\frac{1}{2}g^{2}s_{w}A_{\mu}\phi^{0}(W^{+}_{\mu}\phi^{-}+W^{-}_{\mu}\phi^{+}) +\frac{1}{2}ig^{2}s_{w}A_{\mu}H(W^{+}_{\mu}\phi^{-}-W^{-}_{\mu}\phi^{+}) -g^{2}\frac{s_{w}}{c_{w}}(2c^{2}_{w}-1)Z^{0}_{\mu}A_{\mu}\phi^{+}\phi^{-}-%%@ g^{1}s^{2}_{w}A_{\mu}A_{\mu}\phi^{+}\phi^{-} -\bar{e}^{\lambda}(\gamma\partial+m^{\lambda}_{e})e^{\lambda} -\bar{\nu}^{\lambda}\gamma\partial\nu^{\lambda} -\bar{u}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{u})u^{\lambda}_{j} -\bar{d}^{\lambda}_{j}(\gamma\partial+m^{\lambda}_{d})d^{\lambda}_{j} +igs_{w}A_{\mu}[-(\bar{e}^{\lambda}\gamma^{\mu} e^{\lambda})+\frac{2}{3}(\bar{u}^{\lambda}_{j}\gamma^{\mu} %%@ u^{\lambda}_{j})-\frac{1}{3}(\bar{d}^{\lambda}_{j}\gamma^{\mu} d^{\lambda}_{j})] +\frac{ig}{4c_{w}}Z^{0}_{\mu} [(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda})+ (\bar{e}^{\lambda}\gamma^{\mu}(4s^{2}_{w}-1-\gamma^{5})e^{\lambda})+ (\bar{u}^{\lambda}_{j}\gamma^{\mu}(\frac{4}{3}s^{2}_{w}-1-\gamma^{5})u^{\lambda}_{j})+ (\bar{d}^{\lambda}_{j}\gamma^{\mu}(1-\frac{8}{3}s^{2}_{w}-\gamma^{5})d^{\lambda}_{j})] +\frac{ig}{2\sqrt{2}}W^{+}_{\mu}[(\bar{\nu}^{\lambda}\gamma^{\mu}(1+\gamma^{5})e^{\lambda}) +(\bar{u}^{\lambda}_{j}\gamma^{\mu}(1+\gamma^{5})C_{\lambda\kappa}d^{\kappa}_{j})] +\frac{ig}{2\sqrt{2}}W^{-}_{\mu}[(\bar{e}^{\lambda}\gamma^{\mu}(1+\gamma^{5})\nu^{\lambda}) +(\bar{d}^{\kappa}_{j}C^{\dagger}_{\lambda\kappa}\gamma^{\mu}(1+\gamma^{5})u^{\lambda}_{j})] +\frac{ig}{2\sqrt{2}}\frac{m^{\lambda}_{e}}{M} [-\phi^{+}(\bar{\nu}^{\lambda}(1-\gamma^{5})e^{\lambda}) +\phi^{-}(\bar{e}^{\lambda}(1+\gamma^{5})\nu^{\lambda})] -\frac{g}{2}\frac{m^{\lambda}_{e}}{M}[H(\bar{e}^{\lambda}e^{\lambda}) +i\phi^{0}(\bar{e}^{\lambda}\gamma^{5}e^{\lambda})] +\frac{ig}{2M\sqrt{2}}\phi^{+} [-m^{\kappa}_{d}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1-\gamma^{5})d^{\kappa}_{j}) +m^{\lambda}_{u}(\bar{u}^{\lambda}_{j}C_{\lambda\kappa}(1+\gamma^{5})d^{\kappa}_{j}] +\frac{ig}{2M\sqrt{2}}\phi^{-} [m^{\lambda}_{d}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1+\gamma^{5})u^{\kappa}_{j}) -m^{\kappa}_{u}(\bar{d}^{\lambda}_{j}C^{\dagger}_{\lambda\kappa}(1-\gamma^{5})u^{\kappa}_{j}] -\frac{g}{2}\frac{m^{\lambda}_{u}}{M}H(\bar{u}^{\lambda}_{j}u^{\lambda}_{j}) -\frac{g}{2}\frac{m^{\lambda}_{d}}{M}H(\bar{d}^{\lambda}_{j}d^{\lambda}_{j}) +\frac{ig}{2}\frac{m^{\lambda}_{u}}{M}\phi^{0}(\bar{u}^{\lambda}_{j}\gamma^{5}u^{\lambda}_{j}) -\frac{ig}{2}\frac{m^{\lambda}_{d}}{M}\phi^{0}(\bar{d}^{\lambda}_{j}\gamma^{5}d^{\lambda}_{j}) +\bar{X}^{+}(\partial^{2}-M^{2})X^{+}+\bar{X}^{-}(\partial^{2}-M^{2})X^{-} +\bar{X}^{0}(\partial^{2}-\frac{M^{2}}{c^{2}_{w}})X^{0}+\bar{Y}\partial^{2}Y +igc_{w}W^{+}_{\mu}(\partial_{\mu}\bar{X}^{0}X^{-}-\partial_{\mu}\bar{X}^{+}X^{0}) +igs_{w}W^{+}_{\mu}(\partial_{\mu}\bar{Y}X^{-}-\partial_{\mu}\bar{X}^{+}Y) +igc_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}X^{0}-\partial_{\mu}\bar{X}^{0}X^{+}) +igs_{w}W^{-}_{\mu}(\partial_{\mu}\bar{X}^{-}Y-\partial_{\mu}\bar{Y}X^{+}) +igc_{w}Z^{0}_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-}) +igs_{w}A_{\mu}(\partial_{\mu}\bar{X}^{+}X^{+}-\partial_{\mu}\bar{X}^{-}X^{-}) -\frac{1}{2}gM[\bar{X}^{+}X^{+}H+\bar{X}^{-}X^{-}H+\frac{1}{c^{2}_{w}}\bar{X}^{0}X^{0}H] +\frac{1-2c^{2}_{w}}{2c_{w}}igM[\bar{X}^{+}X^{0}\phi^{+}-\bar{X}^{-}X^{0}\phi^{-}] +\frac{1}{2c_{w}}igM[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}] +igMs_{w}[\bar{X}^{0}X^{-}\phi^{+}-\bar{X}^{0}X^{+}\phi^{-}] +\frac{1}{2}igM[\bar{X}^{+}X^{+}\phi^{0}-\bar{X}^{-}X^{-}\phi^{0}]$

2. Keenan says:

Great example of an equation where notation masks the underlying (potentially beautiful…) meaning!

12. Derek says:

$w- \mathrm{\ln}|ap^{2}H+e^{ar}|=\mathrm{\ln}|y|-\mathrm{ln}|N|$

1. Keenan says:

Interesting—what is it?

1. Derek says:

$\begin{array}{rcl} w- \mathrm{ln}|ap^{2}H+e^{ar}|&=&\mathrm{ln}|y|-\mathrm{ln}|N|\\ w &=&\mathrm{ln}|(ap^{2}H+e^{ar})\cdot \frac{y}{N}|\\ e^{w}&=&(ap^{2}H+e^{ar}) \cdot \frac{y}{N}\\ e^{w} \cdot N &=& ap^{2}Hy+e^{ar}y\\ ap^{2}Hy &=& e^{w}N-e^{ar}y\\ Happy &=& Ne^{w} – ye^{ar}\\ \end{array}$

1. Keenan says:

Aha. And to you as well.

13. Oliver says:

$2(\mathbf{r} \cdot x) \mathbf{r} = x + \mathbf{r} x \mathbf{r}$

Requires some assumptions for what $\mathbf{r}$ and $x$ are, but I like it.

14. junanita says:

Complex $i^2 = -1$

1. Keenan says:

Great. Geometrically: two quarter-turns in the counter-clockwise direction are equivalent to a 180-degree turn.

15. YeHan says:

stress vector $\mathbf{T^{(n)}}$ across an imaginary surface perpendicular to unit vector $\mathbf{n}$, $\mathbf{T^{(n)} = n \cdot \sigma}$ where $\mathbf{\sigma}$ is the Cauchy stress tensor.

1. Keenan says:

Excellent. Many standard objects and terms from mechanics have an equivalent expression in geometry—for instance, the “left Cauchy strain tensor” in solid mechanics is effectively the “induced Riemannian metric” in geometry. Learning to speak and translate between these two languages can be quite valuable, especially when it comes to leveraging various numerical algorithms for computation.

16. zshearer says:

$\int_R K dA + \int_{\partial R} k_g ds + \sum_{\theta \in \{ \textrm{ext angles of R} \} }\theta = 2\pi$
(For a simple region R on a manifold)

1. Keenan says:

Great. You will play around with a discrete version of Gauss-Bonnet in your first assignment.

17. eayvali says:

$\frac{d^2x^k}{dt^2} + \Gamma_{ij}^k\frac{xi^i}{dt}\frac{x^j}{dt}=0$ (geodesic equation)

1. Keenan says:

Nice. We will look at a very different way to compute geodesics later on in the term.

1. Stokes2BeHere says:

Will it involve the heat equation?

18. vidya says:

Euler characteristic $\chi = V- E +F$

1. Keenan says:

Great—you’ll need this for your next assignment!

19. yousufsoliman says:

$\chi(K)=\sum_{i=1}^n(-1)^i b_i(K)$, where $b_i(K)$ is the $i^{\text{th}}$ Betti number of $K$.

1. Keenan says:

Hey great—you’ll use a simpler version of this formula in your homework.

20. jbern says:

$$u(t)=k_pe(t)+k_i\int_{0}^{t}e(\tau)d\tau + k_d\frac{de}{dt}$$ (PID control)

1. Keenan says:

Totally out of control.

21. ckaffine says:

$F(j \omega) = \int_{-\infty}^{\infty} f(t) e^{-j \omega t} dt$

1. Keenan says:

Cool. Later in class we’ll see how to do spectral transformations of data on meshes.

22. jbrakensiek says:

$\displaystyle \zeta(2) = \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$

1. Keenan says:

Yeah, but where does it equal zero?

23. Chris Yu says:

$$C = \left[0,1\right] \setminus \bigcup_{m=1}^{\infty} \bigcup_{k=0}^{3^{m-1} – 1} \left(\frac{3k+1}{3^m}, \frac{3k+2}{3^m}\right)$$

The middle-thirds Cantor set, which is a great source of counterexamples.

1. Keenan says:

Great. This is also a good counter example:

24. avinash says:

$I \dot{\omega} + \omega \times I \omega = M$
(Euler’s rigid body dynamics equation)

1. Keenan says:

Nice. Interestingly enough, these same dynamics govern the dynamics of fluids! If you adopt the right perspective (called the Euler-Arnold equation). Terence Tao has a blog post that talks about this relationship.