Dimension
How to find the relationship between two quantities? For example, what is the dimensional relationship between length and mass.
Plank constant: \(\mathrm{ \hbar \sim [Energy]\cdot [Time] \sim [Mass]\cdot [Length]^2 \cdot [Time]^{-1} }\)
* Speed of light in vacuum: \(\mathrm{ c\sim [Length]\cdot [Time]^{-1} }\)
* Gravitational constant: \(\mathrm{ G \sim [Length]^3\cdot [Mass]^{-1} \cdot [Time]^{-2} }\)
Then it is easy to find that a combination of \(c/\hbar\) cancels the dimension of mass and leaves the inverse of length. That is
\[[\mathrm{Length}]^2 = \frac{\hbar G}{c^3}\]
Most Wonderful Equations That Should Never Be Forgotten
Electrodynamics
Maxwell Equations
\[\begin{split}\nabla\times\vec E&=&-\partial_t \vec B \\
\nabla\times\vec H&=&\vec J+\partial_t \vec D \\
\nabla\cdot \vec D&=&\rho \\
\nabla\cdot \vec B&=&0\end{split}\]
For linear meterials,
\[\begin{split}\begin{eqnarray}
\vec D&=&\epsilon \vec E \\
\vec B&=&\mu \vec H \\
\vec J&=& \sigma \vec E
\end{eqnarray}\end{split}\]
Dynamics
Hamilton conanical equations
\[\begin{split}\begin{eqnarray}
\dot q_i &=& \frac{\partial H}{\partial p_i} \\
\dot p_i &=& - \frac{\partial H}{\partial q_i}
\end{eqnarray}\end{split}\]
Thermodynamics and Statistical Physics
Liouville’s Law
\[\begin{eqnarray}
\frac{\mathrm d \rho}{\mathrm d t}\equiv \frac{\partial \rho}{\partial t} + \sum_i \left[ \frac{\partial \rho}{\partial q_i}\dot q_i + \frac{\partial \rho}{\partial p_i}\dot p_i \right] = 0
\end{eqnarray}\]