mgttt/mymath.md

## mymath.md

      
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              mymath.md
            
          
    approximation function for natural logarithm
# ln=(x,a=2**32)=>a*x**(1/a)-a
#ln=lambda x,a=1<<32:a*x**(1/a)-a
ln=lambda x,a=1<<32:a*(x**(1/a)-1)

>>> numpy.log(400) - ln(400)
-6.776018235399306e-08


经典（牛顿引力）
$F = G \frac{m_1 m_2}{r^2}$
广义相对论
$G_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu}$
$G_{\mu\nu}$ 爱因斯坦张量
$T_{\mu\nu}$ 能量-动量张量
G 是牛顿的引力常数
c 是光速

$$T = \begin{pmatrix}
T_{00} & T_{01} & T_{02} & T_{03} \\\
T_{10} & T_{11} & T_{12} & T_{13} \\\
T_{20} & T_{21} & T_{22} & T_{23} \\\
T_{30} & T_{31} & T_{32} & T_{33}
\end{pmatrix}$$
T = \begin{pmatrix}
T_{00} & T_{01} & T_{02} & T_{03} \\
T_{10} & T_{11} & T_{12} & T_{13} \\
T_{20} & T_{21} & T_{22} & T_{23} \\
T_{30} & T_{31} & T_{32} & T_{33}
\end{pmatrix}


$$
\begin{aligned}
& \phi(x,y) = \phi \left(\sum_{i=1}^n x_ie_i, \sum_{j=1}^n y_je_j \right)
= \sum_{i=1}^n \sum_{j=1}^n x_i y_j \phi(e_i, e_j) = \\
& (x_1, \ldots, x_n) \left( \begin{array}{ccc}
\phi(e_1, e_1) & \cdots & \phi(e_1, e_n) \\
\vdots & \ddots & \vdots \\
\phi(e_n, e_1) & \cdots & \phi(e_n, e_n)
\end{array} \right)
\left( \begin{array}{c}
y_1 \\
\vdots \\
y_n
\end{array} \right)
\end{aligned}
$$
$$
\begin{aligned}
  & \phi(x,y) = \phi \left(\sum_{i=1}^n x_ie_i, \sum_{j=1}^n y_je_j \right)
  = \sum_{i=1}^n \sum_{j=1}^n x_i y_j \phi(e_i, e_j) = \\
  & (x_1, \ldots, x_n) \left( \begin{array}{ccc}
      \phi(e_1, e_1) & \cdots & \phi(e_1, e_n) \\
      \vdots & \ddots & \vdots \\
      \phi(e_n, e_1) & \cdots & \phi(e_n, e_n)
    \end{array} \right)
  \left( \begin{array}{c}
      y_1 \\
      \vdots \\
      y_n
    \end{array} \right)
\end{aligned}
$$