Untitled.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from numpy import sin,cos,sqrt\n",
    "#z1就是r，z2就是theta\n",
    "\n",
    "def metric_KRZ(spin,   d1,   z1,   z2):\n",
    "    r = z1;\n",
    "    sqr = r*r;\n",
    "    cuber = r*sqr;\n",
    "    fourthr = sqr*sqr;\n",
    "\n",
    "    th = z2;\n",
    "\n",
    "    sinth = sin(th);\n",
    "    costh = cos(th);\n",
    "    sqsinth = sinth*sinth;\n",
    "    sqcosth = costh*costh;\n",
    "\n",
    "    sqspin = spin*spin;\n",
    "    fourthspin = sqspin*sqspin;\n",
    "    r0 = 1 + sqrt(1 - sqspin);\n",
    "    sqr0 = r0*r0;\n",
    "    cuber0 = sqr0*r0;\n",
    "    fourthr0 = sqr0*sqr0;\n",
    "\n",
    "    a20 = (2 * sqspin) / cuber0;\n",
    "    a21 = -fourthspin / fourthr0 + 0;\n",
    "    e0 = (2 - r0) / r0;\n",
    "    k00 = sqspin / sqr0;\n",
    "    k21 = fourthspin / fourthr0 - 2 * sqspin / cuber0 - 0;\n",
    "    w00 = 2 * spin / sqr0;\n",
    "\n",
    "    k22 = -sqspin / sqr0;\n",
    "    k23 = sqspin / sqr0;\n",
    "\n",
    "    N2 = (1 - r0 / r)*(1 - e0*r0 / r + (k00 - e0)*sqr0 / sqr + d1*cuber0 / cuber)+(a20*cuber0/cuber+a21*fourthr0/fourthr+k21*cuber0/cuber/(1+k22*(1-r0/r)/(1+k23*(1-r0/r))))*sqcosth;\n",
    "    B = 1 + 0 * sqr0 / sqr + 0 * sqr0*sqcosth / sqr;\n",
    "    Sigma = 1 + sqspin*sqcosth / sqr;\n",
    "    W = (w00*sqr0 / sqr + 0 * cuber0 / cuber + 0 * cuber0*sqcosth / cuber) / Sigma;\n",
    "    K2 = 1 + spin*W / r + (k00*sqr0 / sqr + k21*cuber0*sqcosth / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r)))) / Sigma;\n",
    "\n",
    "\n",
    "    mn=np.zeros((4,4))\n",
    "    mn[0][0] = -(N2 - W*W*sqsinth) / K2;\n",
    "    mn[0][3] = -W*r*sqsinth;\n",
    "    mn[1][1] = Sigma*B*B / N2;\n",
    "    mn[2][2] = Sigma*sqr;\n",
    "    mn[3][0] = mn[0][3];\n",
    "    mn[3][3] = K2*sqr*sqsinth;\n",
    "\n",
    "    return mn\n",
    "\n",
    "def metric_KRZ_rderivatives_num(  spin,   d1,   z1,   z2):\n",
    "\n",
    "\n",
    "    r = z1;\n",
    "    theta = z2;\n",
    "    dr = 0.001*r;\n",
    "\n",
    "\n",
    "    mnm=metric_KRZ(spin, d1, r - dr, theta);\n",
    "    mnp=metric_KRZ(spin, d1, r + dr, theta);\n",
    "    \n",
    "    rdmn=np.zeros((4,4))\n",
    "    rdmn[0][0] = (mnp[0][0] - mnm[0][0])*0.5 / dr;\n",
    "    rdmn[0][3] = (mnp[0][3] - mnm[0][3])*0.5 / dr;\n",
    "    rdmn[1][1] = (mnp[1][1] - mnm[1][1])*0.5 / dr;\n",
    "    rdmn[2][2] = (mnp[2][2] - mnm[2][2])*0.5 / dr;\n",
    "    rdmn[3][0] = rdmn[0][3];\n",
    "    rdmn[3][3] = (mnp[3][3] - mnm[3][3])*0.5 / dr;\n",
    "\n",
    "    return rdmn\n",
    " \n",
    "\n",
    "def metric_KRZ_thderivatives_num(  spin,   d1,   z1,   z2):\n",
    "\n",
    "\n",
    "    r = z1;\n",
    "    theta = z2;\n",
    "    dtheta = 0.01;\n",
    "\n",
    "\n",
    "    metric_KRZ(spin, d1, r, theta - dtheta, mnm);\n",
    "    metric_KRZ(spin, d1, r, theta + dtheta, mnp);\n",
    "    \n",
    "    thdmn=np.zeros((4,4))\n",
    "    thdmn[0][0] = (mnp[0][0] - mnm[0][0])*0.5 / dtheta;\n",
    "    thdmn[0][3] = (mnp[0][3] - mnm[0][3])*0.5 / dtheta;\n",
    "    thdmn[1][1] = (mnp[1][1] - mnm[1][1])*0.5 / dtheta;\n",
    "    thdmn[2][2] = (mnp[2][2] - mnm[2][2])*0.5 / dtheta;\n",
    "    thdmn[3][0] = thdmn[0][3];\n",
    "    thdmn[3][3] = (mnp[3][3] - mnm[3][3])*0.5 / dtheta;\n",
    "\n",
    "    return thdmn\n",
    " \n",
    "\n",
    "def metric_KRZ_rderivatives(  spin,   d1,   z1,   z2) :\n",
    "\n",
    "    r = z1;\n",
    "    th = z2;\n",
    "\n",
    "    sqr = r*r;\n",
    "    cuber = r*sqr;\n",
    "    fourthr = sqr*sqr;\n",
    "\n",
    "    sinth = sin(th);\n",
    "    costh = cos(th);\n",
    "    sqsinth = sinth*sinth;\n",
    "    sqcosth = costh*costh;\n",
    "\n",
    "    sqspin = spin*spin;\n",
    "    cubespin = sqspin*spin;\n",
    "    fourthspin = sqspin*sqspin;\n",
    "\n",
    "    r0 = 1 + sqrt(1 - sqspin);\n",
    "    sqr0 = r0*r0;\n",
    "    cuber0 = sqr0*r0;\n",
    "    fourthr0 = sqr0*sqr0;\n",
    "\n",
    "    a20 = (2 * sqspin) / cuber0;\n",
    "    a21 = -fourthspin / fourthr0 + 0;\n",
    "    e0 = (2 - r0) / r0;\n",
    "    k00 = sqspin / sqr0;\n",
    "    k21 = fourthspin / fourthr0 - 2 * sqspin / cuber0 - 0;\n",
    "    w00 = 2 * spin / sqr0;\n",
    "\n",
    "    k22 = -sqspin / sqr0;\n",
    "    k23 = sqspin / sqr0;\n",
    "\n",
    "    N2 = (1 - r0 / r)*(1 - e0*r0 / r + (k00 - e0)*sqr0 / sqr + d1*cuber0 / cuber) + (a20*cuber0 / cuber + a21*fourthr0 / fourthr + k21*cuber0 / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r))))*sqcosth;\n",
    "    B = 1 + 0 * sqr0 / sqr + 0 * sqr0*sqcosth / sqr;\n",
    "    Sigma = 1 + sqspin*sqcosth / sqr;\n",
    "    W = (w00*sqr0 / sqr + 0 * cuber0 / cuber + 0 * cuber0*sqcosth / cuber) / Sigma;\n",
    "    K2 = 1 + spin*W / r + (k00*sqr0 / sqr + k21*cuber0*sqcosth / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r)))) / Sigma;\n",
    "    #2018-4-4\n",
    "    #the oringinal N2 was wrong, see the correct KRZ metric\n",
    "\n",
    "    rderN2 = (1 - r0 / r)*((2 - r0) / sqr - (2 * (-e0 + k00)*sqr0) / cuber - (3 * d1*cuber0) / fourthr) +\\\n",
    "    (r0*(1 - (2 - r0) / r + ((-e0 + k00)*sqr0) / sqr + (d1*cuber0) / cuber)) / sqr\\\n",
    "    +sqcosth*(-3*a20*cuber0/fourthr-4*a21*fourthr0/fourthr/r-3*k21*cuber0/fourthr/ (   1    +      k22*(1-r0/r)/(1+k23*(1-r0/r))  )- \\\n",
    "    k21*cuber0/cuber/  (1+  k22*(1-r0/r)/(1+k23*(1-r0/r ))  ) / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r))  )  *  ( k22*r0/sqr/(1+  k23*(1-r0/r)  ) - k22*(1-r0/r) / (1 + k23* (1 - r0 / r)) / (1 + k23* (1 - r0 / r)) *k23*r0/sqr  )\\\n",
    "    )\n",
    "    #2018-4-4 modified with the correct metric expression\n",
    "\n",
    "    '''+\n",
    "    ((-3 * (-0 + fourthspin / fourthr0)*cuber0) / fourthr - (4 * a21*fourthr0) / np.power(r, 5))*sqcosth''';\n",
    "    rderB = (-2 * 0 * sqr0) / cuber - (2 * 0 * sqr0*sqcosth) / cuber;\n",
    "    rderSigma = (-2 * sqcosth*sqspin) / cuber;\n",
    "    rderW = (2 * sqcosth*((0 * np.power(r0, 3)) / np.power(r, 3) + (2 * spin) / np.power(r, 2) + (0 * np.power(r0, 3)*sqcosth) / np.power(r, 3))*sqspin) /\\\n",
    "    (np.power(r, 3)*np.power(1 + (sqcosth*sqspin) / np.power(r, 2), 2)) +\\\n",
    "    ((-3 * 0 * np.power(r0, 3)) / np.power(r, 4) - (4 * spin) / np.power(r, 3) - (3 * 0 * np.power(r0, 3)*sqcosth) / np.power(r, 4)) /\\\n",
    "    (1 + (sqcosth*sqspin) / np.power(r, 2));\n",
    "    '''\t  rderK2 = (2 * cubespin*sqcosth*((0*cuber0) / cuber + (2 * spin) / sqr + (0*cuber0*sqcosth) / cuber)) /\n",
    "    (fourthr*np.power(1 + (sqcosth*sqspin) / sqr, 2)) +\n",
    "    (2 * sqcosth*sqspin*((k21*cuber0*sqcosth) / cuber + sqspin / sqr)) /\n",
    "    (cuber*np.power(1 + (sqcosth*sqspin) / sqr, 2)) +\n",
    "    (spin*((-3 *  0*cuber0) / fourthr - (4 * spin) / cuber - (3 * 0*cuber0*sqcosth) / fourthr)) /\n",
    "    (r*(1 + (sqcosth*sqspin) / sqr)) - (spin*\n",
    "    ((0*cuber0) / cuber + (2 * spin) / sqr + (0*cuber0*sqcosth) / cuber)) /\n",
    "    (sqr*(1 + (sqcosth*sqspin) / sqr)) +\n",
    "    ((-3 * k21*cuber0*sqcosth) / fourthr - (2 * sqspin) / cuber) / (1 + (sqcosth*sqspin) / sqr);'''\n",
    "    rderK2 = spin / r*rderW - spin*W / sqr - rderSigma / Sigma / Sigma * (k00*sqr0 / sqr + k21*cuber0*sqcosth / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r))))\\\n",
    "    + (-2 * k00 * sqr0 / cuber - 3 * k21 * cuber0 * sqcosth / fourthr / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r)))\\\n",
    "    - k21 * sqcosth * cuber0 / cuber * (r0 * k22 / sqr / (1 + k23*(1 - r0 / r)) / (1 + k23*(1 - r0 / r))) / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r))) / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r)))) / Sigma;#2017-10-26\n",
    "\n",
    "    rdmn=np.zeros((4,4))\n",
    "    rdmn[0][0] = (-rderN2 + 2 * rderW*sqsinth*W) / K2 - (rderK2*(-N2 + sqsinth*np.power(W, 2))) / np.power(K2, 2);\n",
    "\n",
    "    rdmn[1][1] = (np.power(B, 2)*rderSigma) / N2 + (2 * B*rderB*Sigma) / N2 - (np.power(B, 2)*rderN2*Sigma) / np.power(N2, 2);\n",
    "\n",
    "    rdmn[2][2] = np.power(r, 2)*rderSigma + 2 * r*Sigma;\n",
    "\n",
    "    rdmn[3][3] = 2 * K2*r*sqsinth + np.power(r, 2)*rderK2*sqsinth;\n",
    "\n",
    "    rdmn[0][3] = -(r*rderW*sqsinth) - sqsinth*W;\n",
    "\n",
    "    rdmn[3][0] = rdmn[0][3];\n",
    "\n",
    "    return rdmn\n",
    "\n",
    "\n",
    "def metric_KRZ_thderivatives(  spin,   d1,   z1,   z2) :\n",
    "    r = z1;\n",
    "    th = z2;\n",
    "\n",
    "    sqr = r*r;\n",
    "    cuber = r*sqr;\n",
    "    fourthr = sqr*sqr;\n",
    "\n",
    "    sinth = sin(th);\n",
    "    costh = cos(th); #2018-8-19 之前算cosine居然是开根号。。。不知道以前的raytracing是不是也出过这种问题\n",
    "    sqsinth = sinth*sinth;\n",
    "    sqcosth = costh*costh;\n",
    "\n",
    "    sqspin = spin*spin;\n",
    "    cubespin = sqspin*spin;\n",
    "    fourthspin = sqspin*sqspin;\n",
    "\n",
    "    r0 = 1 + sqrt(1 - sqspin);\n",
    "    sqr0 = r0*r0;\n",
    "    cuber0 = sqr0*r0;\n",
    "    fourthr0 = sqr0*sqr0;\n",
    "\n",
    "    a20 = (2 * sqspin) / cuber0;\n",
    "    a21 = -fourthspin / fourthr0 + 0;\n",
    "    e0 = (2 - r0) / r0;\n",
    "    k00 = sqspin / sqr0;\n",
    "    k21 = fourthspin / fourthr0 - 2 * sqspin / cuber0 - 0;\n",
    "    w00 = 2 * spin / sqr0;\n",
    "\n",
    "    k22 = -sqspin / sqr0;#2017-10-26\n",
    "    k23 = sqspin / sqr0;#2017-10-26\n",
    "\n",
    "    N2 = (1 - r0 / r)*(1 - e0*r0 / r + (k00 - e0)*sqr0 / sqr + d1*cuber0 / cuber) + (a20*cuber0 / cuber + a21*fourthr0 / fourthr + k21*cuber0 / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r))))*sqcosth;\n",
    "    B = 1 + 0 * sqr0 / sqr + 0 * sqr0*sqcosth / sqr;\n",
    "    Sigma = 1 + sqspin*sqcosth / sqr;\n",
    "    W = (w00*sqr0 / sqr + 0 * cuber0 / cuber + 0 * cuber0*sqcosth / cuber) / Sigma;\n",
    "    K2 = 1 + spin*W / r + (k00*sqr0 / sqr + k21*cuber0*sqcosth / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r)))) / Sigma;\n",
    "    #2018-4-4\n",
    "\n",
    "    thderN2 = -2 * costh*sinth* (a20*cuber0 / cuber + a21*fourthr0 / fourthr + k21*cuber0 / cuber / (1 + k22* (1 - r0 / r) / (1 + k23 * (1 - r0 / r))));\n",
    "    #2018-4-4 modified with the correct metric expression\n",
    "    ''' -2 * costh*(((-0 + fourthspin / fourthr0)*cuber0) / cuber + (a21*fourthr0) / fourthr)*sinth ''';\n",
    "    thderB = (-2 * costh * 0 * sqr0*sinth) / sqr;\n",
    "    thderSigma = (-2 * costh*sinth*sqspin) / sqr;\n",
    "    thderW = (2 * costh*sinth*((0 * cuber0) / cuber + (sqcosth * 0 * cuber0) / cuber + (2 * spin) / sqr)*\\\n",
    "    sqspin) / (sqr*np.power(1+(sqcosth*sqspin)/sqr, 2)) -\\\n",
    "    (2 * costh * 0 * cuber0*sinth) / (cuber*(1 + (sqcosth*sqspin) / sqr));\n",
    "    '''  thderK2 = (2 * costh*cubespin*sinth*((0*cuber0) / cuber + (sqcosth*0*cuber0) / cuber +\n",
    "    (2 * spin) / sqr)) / (cuber*np.power(1 + (sqcosth*sqspin) / sqr, 2)) +\n",
    "    (2 * costh*sinth*sqspin*((sqcosth*k21*cuber0) / cuber + sqspin / sqr)) /\n",
    "    (sqr*np.power(1 + (sqcosth*sqspin) / sqr, 2)) -\n",
    "    (2 * costh*k21*cuber0*sinth) / (cuber*(1 + (sqcosth*sqspin) / sqr)) -\n",
    "    (2 * costh*0*cuber0*sinth*spin) / (fourthr*(1 + (sqcosth*sqspin) / sqr));'''\n",
    "\n",
    "    thderK2 = spin / r * thderW - thderSigma / Sigma / Sigma  * (k00*sqr0 / sqr + k21*cuber0*sqcosth / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r))))\\\n",
    "    + (-2* costh * sinth *  k21 * cuber0 / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r)))) / Sigma;#2017-10-26\n",
    "\n",
    "    thdmn=np.zeros((4,4))\n",
    "    \n",
    "    thdmn[0][0] = (-thderN2 + 2 * np.power(sinth, 2)*thderW*W + 2 * costh*sinth*np.power(W, 2)) / K2 -\\\n",
    "    (thderK2*(-N2 + np.power(sinth, 2)*np.power(W, 2))) / np.power(K2, 2);\n",
    "    thdmn[1][1] = (2 * B*Sigma*thderB) / N2 - (np.power(B, 2)*Sigma*thderN2) / np.power(N2, 2) + (np.power(B, 2)*thderSigma) / N2;\n",
    "    thdmn[2][2] = np.power(r, 2)*thderSigma;\n",
    "    thdmn[3][3] = 2 * costh*K2*np.power(r, 2)*sinth + np.power(r, 2)*np.power(sinth, 2)*thderK2;\n",
    "    thdmn[0][3] = -(r*np.power(sinth, 2)*thderW) - 2 * costh*r*sinth*W;\n",
    "    thdmn[3][0] = thdmn[0][3];\n",
    "\n",
    "\n",
    "    return thdmn\n",
    "\n",
    "\n",
    "def metric_KRZ_inverse(  spin,   d1,   z1,   z2) :\n",
    "    r = z1;\n",
    "    th = z2;\n",
    "\n",
    "    sqr = r*r;\n",
    "    cuber = r*sqr;\n",
    "    fourthr = sqr*sqr;\n",
    "\n",
    "    sinth = sin(th);\n",
    "    costh = sqrt(1 - sinth*sinth);\n",
    "    sqsinth = sinth*sinth;\n",
    "    sqcosth = costh*costh;\n",
    "\n",
    "    sqspin = spin*spin;\n",
    "    cubespin = sqspin*spin;\n",
    "    fourthspin = sqspin*sqspin;\n",
    "\n",
    "    r0 = 1 + sqrt(1 - sqspin);\n",
    "    sqr0 = r0*r0;\n",
    "    cuber0 = sqr0*r0;\n",
    "    fourthr0 = sqr0*sqr0;\n",
    "\n",
    "    a20 = (2 * sqspin) / cuber0;\n",
    "    a21 = -fourthspin / fourthr0 + 0;\n",
    "    e0 = (2 - r0) / r0;\n",
    "    k00 = sqspin / sqr0;\n",
    "    k21 = fourthspin / fourthr0 - 2 * sqspin / cuber0 - 0;\n",
    "    w00 = 2 * spin / sqr0;\n",
    "\n",
    "    k22 = -sqspin / sqr0;#2017-10-26\n",
    "    k23 = sqspin / sqr0;#2017-10-26\n",
    "    '''\n",
    "    N2 = (1 - r0 / r)*(1 - e0*r0 / r + (k00 - e0)*sqr0 / sqr + d1*cuber0 / cuber) + ((k21 + a20)*cuber0 / cuber + a21*fourthr0 / fourthr)*sqcosth;\n",
    "    B = 1 + 0 * sqr0 / sqr + 0 * sqr0*sqcosth / sqr;\n",
    "    Sigma = 1 + sqspin*sqcosth / sqr;\n",
    "    W = (w00*sqr0 / sqr + 0 * cuber0 / cuber + 0 * cuber0*sqcosth / cuber) / Sigma;\n",
    "    K2 = 1 + spin*W / r + (k00*sqr0 / sqr + k21*cuber0*sqcosth / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r)))) / Sigma;#2017-10-26\n",
    "    '''\n",
    "\n",
    "    N2 = (1 - r0 / r)*(1 - e0*r0 / r + (k00 - e0)*sqr0 / sqr + d1*cuber0 / cuber) + (a20*cuber0 / cuber + a21*fourthr0 / fourthr + k21*cuber0 / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r))))*sqcosth;\n",
    "    B = 1 + 0 * sqr0 / sqr + 0 * sqr0*sqcosth / sqr;\n",
    "    Sigma = 1 + sqspin*sqcosth / sqr;\n",
    "    W = (w00*sqr0 / sqr + 0 * cuber0 / cuber + 0 * cuber0*sqcosth / cuber) / Sigma;\n",
    "    K2 = 1 + spin*W / r + (k00*sqr0 / sqr + k21*cuber0*sqcosth / cuber / (1 + k22*(1 - r0 / r) / (1 + k23*(1 - r0 / r)))) / Sigma;#2017-10-26\n",
    "\n",
    "\n",
    "    invg=np.zeros((4,4))\n",
    "    invg[0][0] = -(K2 / N2);\n",
    "    invg[1][1] = N2 / (np.power(B, 2)*Sigma);\n",
    "    invg[2][2] = 1 / (sqr*Sigma);\n",
    "    invg[3][3] = (-np.power(W, 2) + N2 / np.power(sin(th), 2)) / (K2*N2*sqr);\n",
    "    invg[0][3] = -(W / (N2*r));\n",
    "    invg[3][0] = invg[0][3];\n",
    "\n",
    "\n",
    "    '''  g[4][4], gg;\n",
    "    metric_KRZ(spin, d1, z1, z2, g);\n",
    "    #metric_KRZ_inverse(spin, d1,  w1, w2, invg);\n",
    "    gg = g[0][0] * g[3][3] - g[0][3] * g[0][3];\n",
    "    if (std::fabs(gg) > 1e10) :\n",
    "    invg[0][0] = g[3][3] / gg;\n",
    "    invg[0][3] = -g[0][3] / gg;\n",
    "    invg[1][1] = 1 / g[1][1];\n",
    "    invg[2][2] = 1 / g[2][2];\n",
    "    invg[3][0] = invg[0][3];\n",
    "    invg[3][3] = g[0][0] / gg;\n",
    "    '''\n",
    "    return invg\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "invg=metric(0.5,0,)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "ename": "SyntaxError",
     "evalue": "invalid syntax (<ipython-input-2-6b6962ebf81e>, line 1)",
     "output_type": "error",
     "traceback": [
      "\u001b[1;36m  File \u001b[1;32m\"<ipython-input-2-6b6962ebf81e>\"\u001b[1;36m, line \u001b[1;32m1\u001b[0m\n\u001b[1;33m    40\t66.7 \t80.0\u001b[0m\n\u001b[1;37m      \t   ^\u001b[0m\n\u001b[1;31mSyntaxError\u001b[0m\u001b[1;31m:\u001b[0m invalid syntax\n"
     ]
    }
   ],
   "source": [
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
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   "pygments_lexer": "ipython3",
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