basic_tools:symbols
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basic_tools:symbols [2018/03/28 16:00] jakobadmin |
basic_tools:symbols [2018/04/15 12:20] (current) ida |
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| ====== Symbols ====== | ====== Symbols ====== | ||
| + | |||
| + | * Derivatives with respect to the four-vector $x^{\mu}=(ct,\vec{x})$ are denoted by | ||
| + | \begin{eqnarray} | ||
| + | \partial_{\mu}\equiv {\partial\over \partial x^{\mu}} | ||
| + | =\left({1\over c}{\partial\over\partial t},\vec{\nabla}\right). | ||
| + | \end{eqnarray} | ||
| + | * Space-time indices are labelled by Greek letters ($\mu,\nu,\ldots=0,1,2,3$) | ||
| + | * Latin indices are used for spatial directions ($i,j,\ldots=1,2,3$). | ||
| + | * Moreover, $\sigma^{\mu}=(\mathbf{1},\sigma^{i})$ where $\sigma^{i}$ are the Pauli matrices $$ | ||
| + | \sigma^{1}=\left( | ||
| + | \begin{array}{rr} | ||
| + | 0 & 1 \\ | ||
| + | 1 & 0 | ||
| + | \end{array} | ||
| + | \right), \quad \sigma^{2}=\left( | ||
| + | \begin{array}{rr} | ||
| + | 0 & -i \\ | ||
| + | i & 0 | ||
| + | \end{array} | ||
| + | \right), \quad | ||
| + | \sigma^{3}=\left( | ||
| + | \begin{array}{rr} | ||
| + | 1 & 0 \\ | ||
| + | 0 & -1 | ||
| + | \end{array} | ||
| + | \right).$$ | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | **Math Symbols** | ||
| \begin{align} | \begin{align} | ||
| &\mathbb{N} = \{0, 1, 2, 3, \ldots\} \\ | &\mathbb{N} = \{0, 1, 2, 3, \ldots\} \\ | ||
| Line 8: | Line 39: | ||
| &\mathbb{C} = \rm Complex \; Numbers \it \\ | &\mathbb{C} = \rm Complex \; Numbers \it \\ | ||
| &\mathbb{Z}_n = \mathbb{Z} \; \mod \; n \\ | &\mathbb{Z}_n = \mathbb{Z} \; \mod \; n \\ | ||
| - | &\Rightarrow &\rm \; is \; read \; ``implies"\\ | + | &\Rightarrow \rm \; is \; read \; ``implies"\\ |
| &\rm iff \; is \; read \; ``if\; and \; only \; if" \\ | &\rm iff \; is \; read \; ``if\; and \; only \; if" \\ | ||
| &\forall \rm \; is \; read \; ``for \; every" \\ | &\forall \rm \; is \; read \; ``for \; every" \\ | ||
| Line 17: | Line 48: | ||
| &\subset \; \rm is \; ``subset\; of" \\ | &\subset \; \rm is \; ``subset\; of" \\ | ||
| &\equiv \; \rm is\;``defined\; as" \\ | &\equiv \; \rm is\;``defined\; as" \\ | ||
| + | &= \quad \text{exactly equal} \notag \\ | ||
| + | &\propto \quad \text{equality except perhaps for a factor with dimension} \notag\\ | ||
| + | &\sim \quad \text{equality except perhaps for a factor without dimensions} \notag \\ | ||
| + | &\approx \quad \text{equality except perhaps for a factor close to 1} | ||
| \end{align} | \end{align} | ||
| - | There are several species of equality: | ||
| - | \begin{align} | ||
| - | = & \quad \text{exactly equal} \notag \\ | ||
| - | \propto & \quad \text{equality except perhaps for a factor with dimension} \notag\\ | ||
| - | \sim & \quad \text{equality except perhaps for a factor without dimensions} \notag \\ | ||
| - | \approx & \quad \text{equality except perhaps for a factor close to 1} | ||
| - | \end{align} | ||
| - | |||
| - | Source: page 6 in Street Fighting Mathematics by Mahajan | ||
| ---- | ---- | ||
basic_tools/symbols.1522245621.txt.gz ยท Last modified: 2018/03/28 14:00 (external edit)
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