python5.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![питон](python.png)\n",
    "\n",
    "# numpy\n",
    "\n",
    "Пакет `numpy` предоставляет $n$-мерные однородные массивы (все элементы одного типа); в них нельзя вставить или удалить элемент в произвольном месте. В `numpy` реализовано много операций над массивами в целом. Если задачу можно решить, произведя некоторую последовательность операций над массивами, то это будет столь же эффективно, как в `C` или `matlab` - львиная доля времени тратится в библиотечных функциях, написанных на `C`.\n",
    "\n",
    "## Одномерные массивы"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from numpy import (array,zeros,ones,arange,linspace,logspace,\n",
    "                   float64,int64,sin,cos,pi,exp,log,sqrt,abs,\n",
    "                   nan,inf,any,all,sort,hstack,vstack,hsplit,\n",
    "                   delete,insert,append,eye,fromfunction,\n",
    "                   trace,diag,average,std,outer,meshgrid)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Можно преобразовать список в массив."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(array([0, 2, 1]), numpy.ndarray)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a=array([0,2,1])\n",
    "a,type(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`print` печатает массивы в удобной форме."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 2 1]\n"
     ]
    }
   ],
   "source": [
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Класс `ndarray` имеет много методов."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'T',\n",
       " '__abs__',\n",
       " '__add__',\n",
       " '__and__',\n",
       " '__array__',\n",
       " '__array_finalize__',\n",
       " '__array_interface__',\n",
       " '__array_prepare__',\n",
       " '__array_priority__',\n",
       " '__array_struct__',\n",
       " '__array_wrap__',\n",
       " '__bool__',\n",
       " '__contains__',\n",
       " '__copy__',\n",
       " '__deepcopy__',\n",
       " '__delitem__',\n",
       " '__divmod__',\n",
       " '__float__',\n",
       " '__floordiv__',\n",
       " '__getitem__',\n",
       " '__iadd__',\n",
       " '__iand__',\n",
       " '__ifloordiv__',\n",
       " '__ilshift__',\n",
       " '__imatmul__',\n",
       " '__imod__',\n",
       " '__imul__',\n",
       " '__index__',\n",
       " '__int__',\n",
       " '__invert__',\n",
       " '__ior__',\n",
       " '__ipow__',\n",
       " '__irshift__',\n",
       " '__isub__',\n",
       " '__iter__',\n",
       " '__itruediv__',\n",
       " '__ixor__',\n",
       " '__len__',\n",
       " '__lshift__',\n",
       " '__matmul__',\n",
       " '__mod__',\n",
       " '__mul__',\n",
       " '__neg__',\n",
       " '__or__',\n",
       " '__pos__',\n",
       " '__pow__',\n",
       " '__radd__',\n",
       " '__rand__',\n",
       " '__rdivmod__',\n",
       " '__rfloordiv__',\n",
       " '__rlshift__',\n",
       " '__rmatmul__',\n",
       " '__rmod__',\n",
       " '__rmul__',\n",
       " '__ror__',\n",
       " '__rpow__',\n",
       " '__rrshift__',\n",
       " '__rshift__',\n",
       " '__rsub__',\n",
       " '__rtruediv__',\n",
       " '__rxor__',\n",
       " '__setitem__',\n",
       " '__setstate__',\n",
       " '__sub__',\n",
       " '__truediv__',\n",
       " '__xor__',\n",
       " 'all',\n",
       " 'any',\n",
       " 'argmax',\n",
       " 'argmin',\n",
       " 'argpartition',\n",
       " 'argsort',\n",
       " 'astype',\n",
       " 'base',\n",
       " 'byteswap',\n",
       " 'choose',\n",
       " 'clip',\n",
       " 'compress',\n",
       " 'conj',\n",
       " 'conjugate',\n",
       " 'copy',\n",
       " 'ctypes',\n",
       " 'cumprod',\n",
       " 'cumsum',\n",
       " 'data',\n",
       " 'diagonal',\n",
       " 'dot',\n",
       " 'dtype',\n",
       " 'dump',\n",
       " 'dumps',\n",
       " 'fill',\n",
       " 'flags',\n",
       " 'flat',\n",
       " 'flatten',\n",
       " 'getfield',\n",
       " 'imag',\n",
       " 'item',\n",
       " 'itemset',\n",
       " 'itemsize',\n",
       " 'max',\n",
       " 'mean',\n",
       " 'min',\n",
       " 'nbytes',\n",
       " 'ndim',\n",
       " 'newbyteorder',\n",
       " 'nonzero',\n",
       " 'partition',\n",
       " 'prod',\n",
       " 'ptp',\n",
       " 'put',\n",
       " 'ravel',\n",
       " 'real',\n",
       " 'repeat',\n",
       " 'reshape',\n",
       " 'resize',\n",
       " 'round',\n",
       " 'searchsorted',\n",
       " 'setfield',\n",
       " 'setflags',\n",
       " 'shape',\n",
       " 'size',\n",
       " 'sort',\n",
       " 'squeeze',\n",
       " 'std',\n",
       " 'strides',\n",
       " 'sum',\n",
       " 'swapaxes',\n",
       " 'take',\n",
       " 'tobytes',\n",
       " 'tofile',\n",
       " 'tolist',\n",
       " 'tostring',\n",
       " 'trace',\n",
       " 'transpose',\n",
       " 'var',\n",
       " 'view'}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "set(dir(a))-set(dir(object))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Наш массив одномерный."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.ndim"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "В $n$-мерном случае возвращается кортеж размеров по каждой координате."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3,)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`size` - это полное число элементов в массиве; `len` - размер по первой координате (в 1-мерном случае это то же самое)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 3)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(a),a.size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`numpy` предоставляет несколько типов для целых (`int16`, `int32`, `int64`) и чисел с плавающей точкой (`float32`, `float64`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(dtype('int64'), 'int64', 8)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.dtype,a.dtype.name,a.itemsize"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Индексировать массив можно обычным образом."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Массивы - изменяемые объекты."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 3 1]\n"
     ]
    }
   ],
   "source": [
    "a[1]=3\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Массивы, разумеется, можно использовать в `for` циклах. Но при этом теряется главное преимущество `numpy` - быстродействие. Всегда, когда это возможно, лучше использовать операции над массивами как едиными целыми."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0\n",
      "3\n",
      "1\n"
     ]
    }
   ],
   "source": [
    "for i in a:\n",
    "    print(i)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Массив чисел с плавающей точкой."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dtype('float64')"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b=array([0.,2,1])\n",
    "b.dtype"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Точно такой же массив."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  2.  1.]\n"
     ]
    }
   ],
   "source": [
    "c=array([0,2,1],dtype=float64)\n",
    "print(c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Массив, значения которого вычисляются функцией. Функции передаётся массив. Так что в ней можно использовать только такие операции, которые применимы к массивам."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 1 2 3 4]\n",
      "[ 0  1  4  9 16]\n"
     ]
    }
   ],
   "source": [
    "def f(i):\n",
    "    print(i)\n",
    "    return i**2\n",
    "a=fromfunction(f,(5,),dtype=int64)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a=fromfunction(f,(5,),dtype=float64)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Массивы, заполненные нулями или единицами. Часто лучше сначала создать такой массив, а потом присваивать значения его элементам."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  0.  0.]\n"
     ]
    }
   ],
   "source": [
    "a=zeros(3)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1 1 1]\n"
     ]
    }
   ],
   "source": [
    "b=ones(3,dtype=int64)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Функция `arange` подобна `range`. Аргументы могут быть с плавающей точкой. Следует избегать ситуаций, когда $(конец-начало)/шаг$ - целое число, потому что в этом случае включение последнего элемента зависит от ошибок округления. Лучше, чтобы конец диапазона был где-то посредине шага."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 2 4 6 8]\n"
     ]
    }
   ],
   "source": [
    "a=arange(0,9,2)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  2.  4.  6.  8.]\n"
     ]
    }
   ],
   "source": [
    "b=arange(0.,9,2)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Последовательности чисел с постоянным шагом можно также создавать функцией `linspace`. Начало и конец диапазона включаются; последний аргумент - число точек."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  2.  4.  6.  8.]\n"
     ]
    }
   ],
   "source": [
    "a=linspace(0,8,5)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Последовательность чисел с постоянным шагом по логарифмической шкале от $10^0$ до $10^1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  1.           1.77827941   3.16227766   5.62341325  10.        ]\n"
     ]
    }
   ],
   "source": [
    "b=logspace(0,1,5)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Массив случайных чисел."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.41772321  0.93106624  0.10164623  0.52871506  0.90588913]\n"
     ]
    }
   ],
   "source": [
    "from numpy.random import random,normal\n",
    "print(random(5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Случайные числа с нормальным (гауссовым) распределением (среднее `0`, среднеквадратичное отклонение `1`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.39930386  0.88814365 -0.63928799 -0.46487922  0.63126724]\n"
     ]
    }
   ],
   "source": [
    "print(normal(size=5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Операции над одномерными массивами\n",
    "\n",
    "Арифметические операции проводятся поэлементно."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  1.           3.77827941   7.16227766  11.62341325  18.        ]\n"
     ]
    }
   ],
   "source": [
    "print(a+b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-1.          0.22172059  0.83772234  0.37658675 -2.        ]\n"
     ]
    }
   ],
   "source": [
    "print(a-b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.           3.55655882  12.64911064  33.74047951  80.        ]\n"
     ]
    }
   ],
   "source": [
    "print(a*b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Скалярное произведение"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "129.9461489721723"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a@b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.          1.12468265  1.26491106  1.06696765  0.8       ]\n"
     ]
    }
   ],
   "source": [
    "print(a/b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.   4.  16.  36.  64.]\n"
     ]
    }
   ],
   "source": [
    "print(a**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Когда операнды разных типов, они пиводятся к большему типу."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  3.  5.  7.  9.]\n"
     ]
    }
   ],
   "source": [
    "i=ones(5,dtype=int64)\n",
    "print(a+i)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`numpy` содержит элементарные функции, которые тоже применяются к массивам поэлементно. Они называются универсальными функциями (`ufunc`)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(<ufunc 'sin'>, numpy.ufunc)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin,type(sin)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.          0.90929743 -0.7568025  -0.2794155   0.98935825]\n"
     ]
    }
   ],
   "source": [
    "print(sin(a))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Один из операндов может быть скаляром, а не массивом."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  3.  5.  7.  9.]\n"
     ]
    }
   ],
   "source": [
    "print(a+1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.   4.   8.  12.  16.]\n"
     ]
    }
   ],
   "source": [
    "print(2*a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Сравнения дают булевы массивы."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False  True  True  True False]\n"
     ]
    }
   ],
   "source": [
    "print(a>b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False False False False False]\n"
     ]
    }
   ],
   "source": [
    "print(a==b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False False False  True  True]\n"
     ]
    }
   ],
   "source": [
    "c=a>5\n",
    "print(c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Кванторы \"существует\" и \"для всех\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, False)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "any(c),all(c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Модификация на месте."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  3.  5.  7.  9.]\n"
     ]
    }
   ],
   "source": [
    "a+=1\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.           3.55655882   6.32455532  11.2468265   20.        ]\n"
     ]
    }
   ],
   "source": [
    "b*=2\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 2.          1.18551961  1.26491106  1.6066895   2.22222222]\n"
     ]
    }
   ],
   "source": [
    "b/=a\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Так делать можно."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a+=i"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "А так нельзя."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "Cannot cast ufunc add output from dtype('float64') to dtype('int64') with casting rule 'same_kind'",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[1;32m<ipython-input-42-300abee39dd1>\u001b[0m in \u001b[0;36m<module>\u001b[1;34m()\u001b[0m\n\u001b[1;32m----> 1\u001b[1;33m \u001b[0mi\u001b[0m\u001b[1;33m+=\u001b[0m\u001b[0ma\u001b[0m\u001b[1;33m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m: Cannot cast ufunc add output from dtype('float64') to dtype('int64') with casting rule 'same_kind'"
     ]
    }
   ],
   "source": [
    "i+=a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "При выполнении операций над массивами деление на 0 не возбуждает исключения, а даёт значения `np.nan` или `np.inf`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.  nan  inf -inf]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/lib64/python3.5/site-packages/ipykernel/__main__.py:1: RuntimeWarning: divide by zero encountered in true_divide\n",
      "  if __name__ == '__main__':\n",
      "/usr/lib64/python3.5/site-packages/ipykernel/__main__.py:1: RuntimeWarning: invalid value encountered in true_divide\n",
      "  if __name__ == '__main__':\n"
     ]
    }
   ],
   "source": [
    "print(array([0.0,0.0,1.0,-1.0])/array([1.0,0.0,0.0,0.0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(nan, inf, nan, 0.0)"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nan+1,inf+1,inf*0,1./inf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Сумма и произведение всех элементов массива; максимальный и минимальный элемент; среднее и среднеквадратичное отклонение."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(8.2793423935260435,\n",
       " 10.708241812210389,\n",
       " 2.2222222222222223,\n",
       " 1.1855196066926152,\n",
       " 1.6558684787052087,\n",
       " 0.40390033426607452)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b.sum(),b.prod(),b.max(),b.min(),b.mean(),b.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.0080841995242346542, 0.99010116358700062)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x=normal(size=1000)\n",
    "x.mean(),x.std()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Функция `sort` возвращает отсортированную копию, метод `sort` сортирует на месте."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.18551961  1.26491106  1.6066895   2.          2.22222222]\n",
      "[ 2.          1.18551961  1.26491106  1.6066895   2.22222222]\n"
     ]
    }
   ],
   "source": [
    "print(sort(b))\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.18551961  1.26491106  1.6066895   2.          2.22222222]\n"
     ]
    }
   ],
   "source": [
    "b.sort()\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Объединение массивов."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.           4.           6.           8.          10.           1.18551961\n",
      "   1.26491106   1.6066895    2.           2.22222222]\n"
     ]
    }
   ],
   "source": [
    "a=hstack((a,b))\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Расщепление массива в позициях 3 и 6."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[array([ 2.,  4.,  6.]),\n",
       " array([  8.        ,  10.        ,   1.18551961]),\n",
       " array([ 1.26491106,  1.6066895 ,  2.        ,  2.22222222])]"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hsplit(a,[3,6])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Функции `delete`, `insert` и `append` не меняют массив на месте, а возвращают новый массив, в котором удалены, вставлены в середину или добавлены в конец какие-то элементы."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.           4.           6.           8.          10.           1.26491106\n",
      "   2.           2.22222222]\n"
     ]
    }
   ],
   "source": [
    "a=delete(a,[5,7])\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.           4.           0.           0.           6.           8.          10.\n",
      "   1.26491106   2.           2.22222222]\n"
     ]
    }
   ],
   "source": [
    "a=insert(a,2,[0,0])\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  2.           4.           0.           0.           6.           8.          10.\n",
      "   1.26491106   2.           2.22222222   1.           2.           3.        ]\n"
     ]
    }
   ],
   "source": [
    "a=append(a,[1,2,3])\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Есть несколько способов индексации массива. Вот обычный индекс."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.   0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1. ]\n"
     ]
    }
   ],
   "source": [
    "a=linspace(0,1,11)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2\n"
     ]
    }
   ],
   "source": [
    "b=a[2]\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Диапазон индексов. Создаётся новый заголовок массива, указывающий на те же данные. Изменения, сделанные через такой массив, видны и в исходном массиве."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.2  0.3  0.4  0.5]\n"
     ]
    }
   ],
   "source": [
    "b=a[2:6]\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.2  0.3  0.4  0.5]\n"
     ]
    }
   ],
   "source": [
    "b[0]=-0.2\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.   0.1 -0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1. ]\n"
     ]
    }
   ],
   "source": [
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Диапазон с шагом 2."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.1  0.3  0.5  0.7  0.9]\n"
     ]
    }
   ],
   "source": [
    "b=a[1:10:2]\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  -0.1 -0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1. ]\n"
     ]
    }
   ],
   "source": [
    "b[0]=-0.1\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Массив в обратном порядке."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.   0.9  0.8  0.7  0.6  0.5  0.4  0.3 -0.2 -0.1]\n"
     ]
    }
   ],
   "source": [
    "b=a[len(a):0:-1]\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Подмассиву можно присвоить значение - массив правильного размера или скаляр."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.   0.  -0.2  0.3  0.   0.5  0.6  0.   0.8  0.9  1. ]\n"
     ]
    }
   ],
   "source": [
    "a[1:10:3]=0\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Тут опять создаётся только новый заголовок, указывающий на те же данные."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.   0.1 -0.2  0.3  0.   0.5  0.6  0.   0.8  0.9  1. ]\n"
     ]
    }
   ],
   "source": [
    "b=a[:]\n",
    "b[1]=0.1\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Чтобы скопировать и данные массива, нужно использовать метод `copy`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.   0.1  0.   0.3  0.   0.5  0.6  0.   0.8  0.9  1. ]\n",
      "[ 0.   0.1 -0.2  0.3  0.   0.5  0.6  0.   0.8  0.9  1. ]\n"
     ]
    }
   ],
   "source": [
    "b=a.copy()\n",
    "b[2]=0\n",
    "print(b)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Можно задать список индексов."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.2  0.3  0.5]\n"
     ]
    }
   ],
   "source": [
    "print(a[[2,3,5]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Можно задать булев массив той же величины."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[False  True False  True False  True  True False  True  True  True]\n"
     ]
    }
   ],
   "source": [
    "b=a>0\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.1  0.3  0.5  0.6  0.8  0.9  1. ]\n"
     ]
    }
   ],
   "source": [
    "print(a[b])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## 2-мерные массивы"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.  1.]\n",
      " [-1.  0.]]\n"
     ]
    }
   ],
   "source": [
    "a=array([[0.0,1.0],[-1.0,0.0]])\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.ndim"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 2)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2, 4)"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(a),a.size"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.0"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[1,0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Атрибуту `shape` можно присвоить новое значение - кортеж размеров по всем координатам. Получится новый заголовок массива; его данные не изменятся."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  1.  2.  3.]\n"
     ]
    }
   ],
   "source": [
    "b=linspace(0,3,4)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(4,)"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.  1.]\n",
      " [ 2.  3.]]\n"
     ]
    }
   ],
   "source": [
    "b.shape=2,2\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Поэлементное и матричное умножение."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.  1.]\n",
      " [-2.  0.]]\n"
     ]
    }
   ],
   "source": [
    "print(a*b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2.  3.]\n",
      " [ 0. -1.]]\n"
     ]
    }
   ],
   "source": [
    "print(a@b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-1.  0.]\n",
      " [-3.  2.]]\n"
     ]
    }
   ],
   "source": [
    "print(b@a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Умножение матрицы на вектор."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-1. -1.]\n"
     ]
    }
   ],
   "source": [
    "v=array([1,-1],dtype=float64)\n",
    "print(b@v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-2. -2.]\n"
     ]
    }
   ],
   "source": [
    "print(v@b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Внешнее произведение $a_{ij}=u_i v_j$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  2.]\n",
      "[ 2.  3.  4.]\n"
     ]
    }
   ],
   "source": [
    "u=linspace(1,2,2)\n",
    "v=linspace(2,4,3)\n",
    "print(u)\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 2.  3.  4.]\n",
      " [ 4.  6.  8.]]\n"
     ]
    }
   ],
   "source": [
    "a=outer(u,v)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Двумерные массивы, зависящие только от одного индекса: $x_{ij}=u_j$, $y_{ij}=v_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  2.]\n",
      " [ 1.  2.]\n",
      " [ 1.  2.]]\n",
      "[[ 2.  2.]\n",
      " [ 3.  3.]\n",
      " [ 4.  4.]]\n"
     ]
    }
   ],
   "source": [
    "x,y=meshgrid(u,v)\n",
    "print(x)\n",
    "print(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Единичная матрица."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  0.  0.  0.]\n",
      " [ 0.  1.  0.  0.]\n",
      " [ 0.  0.  1.  0.]\n",
      " [ 0.  0.  0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "I=eye(4)\n",
    "print(I)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Метод `reshape` делает то же самое, что присваивание атрибуту `shape`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  0.  0.  0.  0.  1.  0.  0.  0.  0.  1.  0.  0.  0.  0.  1.]\n"
     ]
    }
   ],
   "source": [
    "print(I.reshape(16))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  0.  0.  0.  0.  1.  0.  0.]\n",
      " [ 0.  0.  1.  0.  0.  0.  0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "print(I.reshape(2,8))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Строка."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  1.  0.  0.]\n"
     ]
    }
   ],
   "source": [
    "print(I[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Цикл по строкам."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.  0.  0.  0.]\n",
      "[ 0.  1.  0.  0.]\n",
      "[ 0.  0.  1.  0.]\n",
      "[ 0.  0.  0.  1.]\n"
     ]
    }
   ],
   "source": [
    "for row in I:\n",
    "    print(row)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Столбец."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  0.  1.  0.]\n"
     ]
    }
   ],
   "source": [
    "print(I[:,2])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Подматрица."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.  0.]\n",
      " [ 1.  0.]]\n"
     ]
    }
   ],
   "source": [
    "print(I[0:2,1:3])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Можно построить двумерный массив из функции."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 0 0 0]\n",
      " [1 1 1 1]\n",
      " [2 2 2 2]\n",
      " [3 3 3 3]]\n",
      "[[0 1 2 3]\n",
      " [0 1 2 3]\n",
      " [0 1 2 3]\n",
      " [0 1 2 3]]\n",
      "[[ 0  1  2  3]\n",
      " [10 11 12 13]\n",
      " [20 21 22 23]\n",
      " [30 31 32 33]]\n"
     ]
    }
   ],
   "source": [
    "def f(i,j):\n",
    "    print(i)\n",
    "    print(j)\n",
    "    return 10*i+j\n",
    "print(fromfunction(f,(4,4),dtype=int64))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Транспонированная матрица."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 0.  2.]\n",
      " [ 1.  3.]]\n"
     ]
    }
   ],
   "source": [
    "print(b.T)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Соединение матриц по горизонтали и по вертикали."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1]\n",
      " [2 3]]\n",
      "[[4 5 6]\n",
      " [7 8 9]]\n",
      "[[4 5]\n",
      " [6 7]\n",
      " [8 9]]\n"
     ]
    }
   ],
   "source": [
    "a=array([[0,1],[2,3]])\n",
    "b=array([[4,5,6],[7,8,9]])\n",
    "c=array([[4,5],[6,7],[8,9]])\n",
    "print(a)\n",
    "print(b)\n",
    "print(c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1 4 5 6]\n",
      " [2 3 7 8 9]]\n"
     ]
    }
   ],
   "source": [
    "print(hstack((a,b)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[0 1]\n",
      " [2 3]\n",
      " [4 5]\n",
      " [6 7]\n",
      " [8 9]]\n"
     ]
    }
   ],
   "source": [
    "print(vstack((a,c)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Сумма всех элементов; суммы столбцов; суммы строк."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "39\n",
      "[11 13 15]\n",
      "[15 24]\n"
     ]
    }
   ],
   "source": [
    "print(b.sum())\n",
    "print(b.sum(0))\n",
    "print(b.sum(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Аналогично работают `prod`, `max`, `min` и т.д."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[7 8 9]\n",
      "[4 7]\n"
     ]
    }
   ],
   "source": [
    "print(b.max(0))\n",
    "print(b.min(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "След - сумма диагональных элементов."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trace(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Линейная алгебра"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-2.0"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from numpy.linalg import det,inv,solve,eig\n",
    "det(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Обратная матрица."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-1.5  0.5]\n",
      " [ 1.   0. ]]\n"
     ]
    }
   ],
   "source": [
    "a1=inv(a)\n",
    "print(a1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.  0.]\n",
      " [ 0.  1.]]\n",
      "[[ 1.  0.]\n",
      " [ 0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "print(a@a1)\n",
    "print(a1@a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Решение линейной системы $au=v$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.5  0. ]\n"
     ]
    }
   ],
   "source": [
    "v=array([0,1],dtype=float64)\n",
    "print(a1@v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.5  0. ]\n"
     ]
    }
   ],
   "source": [
    "u=solve(a,v)\n",
    "print(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Проверим."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  0.]\n"
     ]
    }
   ],
   "source": [
    "print(a@u-v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Собственные значения и собственные векторы: $a u_i = \\lambda_i u_i$. `l` - одномерный массив собственных значений $\\lambda_i$, столбцы матрицы $u$ - собственные векторы $u_i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.56155281  3.56155281]\n"
     ]
    }
   ],
   "source": [
    "l,u=eig(a)\n",
    "print(l)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.87192821 -0.27032301]\n",
      " [ 0.48963374 -0.96276969]]\n"
     ]
    }
   ],
   "source": [
    "print(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Проверим."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.00000000e+00   1.66533454e-16]\n",
      "[  1.11022302e-16   0.00000000e+00]\n"
     ]
    }
   ],
   "source": [
    "for i in range(2):\n",
    "    print(a@u[:,i]-l[i]*u[:,i])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Функция `diag` от одномерного массива строит диагональную матрицу; от квадратной матрицы - возвращает одномерный массив её диагональных элементов."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[-0.56155281  0.        ]\n",
      " [ 0.          3.56155281]]\n",
      "[-0.56155281  3.56155281]\n"
     ]
    }
   ],
   "source": [
    "L=diag(l)\n",
    "print(L)\n",
    "print(diag(L))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Все уравнения $a u_i = \\lambda_i u_i$ можно собрать в одно матричное уравнение $a u = u \\Lambda$, где $\\Lambda$ - диагональная матрица с собственными значениями $\\lambda_i$ по диагонали."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  0.00000000e+00   1.11022302e-16]\n",
      " [  1.66533454e-16   0.00000000e+00]]\n"
     ]
    }
   ],
   "source": [
    "print(a@u-u@L)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Поэтому $u^{-1} a u = \\Lambda$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ -5.61552813e-01   0.00000000e+00]\n",
      " [ -2.22044605e-16   3.56155281e+00]]\n"
     ]
    }
   ],
   "source": [
    "print(inv(u)@a@u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Найдём теперь левые собственные векторы $v_i a = \\lambda_i v_i$ (собственные значения $\\lambda_i$ те же самые)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.56155281  3.56155281]\n",
      "[[-0.96276969 -0.48963374]\n",
      " [ 0.27032301 -0.87192821]]\n"
     ]
    }
   ],
   "source": [
    "l,v=eig(a.T)\n",
    "print(l)\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Собственные векторы нормированы на 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[ 1.         -0.23570226]\n",
      " [-0.23570226  1.        ]]\n",
      "[[ 1.          0.23570226]\n",
      " [ 0.23570226  1.        ]]\n"
     ]
    }
   ],
   "source": [
    "print(u.T@u)\n",
    "print(v.T@v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Левые и правые собственные векторы, соответствующие разным собственным значениям, ортогональны, потому что $v_i a u_j = \\lambda_i v_i u_j = \\lambda_j v_i u_j$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[  9.71825316e-01   0.00000000e+00]\n",
      " [ -5.55111512e-17   9.71825316e-01]]\n"
     ]
    }
   ],
   "source": [
    "print(v.T@u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Преобразование Фурье"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.   0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1. ]\n"
     ]
    }
   ],
   "source": [
    "a=linspace(0,1,11)\n",
    "print(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 5.50+0.j         -0.55+1.87312798j -0.55+0.85581671j -0.55+0.47657771j\n",
      " -0.55+0.25117658j -0.55+0.07907806j -0.55-0.07907806j -0.55-0.25117658j\n",
      " -0.55-0.47657771j -0.55-0.85581671j -0.55-1.87312798j]\n"
     ]
    }
   ],
   "source": [
    "from numpy.fft import fft,ifft\n",
    "b=fft(a)\n",
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Обратное преобразование Фурье."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  1.61486985e-15+0.j   1.00000000e-01+0.j   2.00000000e-01+0.j\n",
      "   3.00000000e-01+0.j   4.00000000e-01+0.j   5.00000000e-01+0.j\n",
      "   6.00000000e-01+0.j   7.00000000e-01+0.j   8.00000000e-01+0.j\n",
      "   9.00000000e-01+0.j   1.00000000e+00+0.j]\n"
     ]
    }
   ],
   "source": [
    "print(ifft(b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Интегрирование"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import quad,odeint\n",
    "from scipy.special import erf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return exp(-x**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Адаптивное численное интегрирование (может быть до бесконечности). `err` - оценка ошибки."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.886226925453 0.8862269254527579 7.101318390472462e-09\n"
     ]
    }
   ],
   "source": [
    "res,err=quad(f,0,inf)\n",
    "print(sqrt(pi)/2,res,err)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.746824132812 0.7468241328124271 8.291413475940725e-15\n"
     ]
    }
   ],
   "source": [
    "res,err=quad(f,0,1)\n",
    "print(sqrt(pi)/2*erf(1),res,err)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Дифференциальные уравнения"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Уравнение осциллятора с затуханием $\\ddot{x} + 2 a \\dot{x} + x = 0$. Перепишем его в виде системы уравнений первого порядка для $x$, $v=\\dot{x}$:\n",
    "\n",
    "$\\frac{d}{dt} \\begin{pmatrix}x\\\\v\\end{pmatrix} = \\begin{pmatrix}v\\\\-2av-x\\end{pmatrix}$\n",
    "\n",
    "Решим эту систему численно при $a=0.2$ с начальным условием\n",
    "$\\begin{pmatrix}x\\\\v\\end{pmatrix}=\\begin{pmatrix}1\\\\0\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a=0.2\n",
    "def f(x,t):\n",
    "    global a\n",
    "    return [x[1],-x[0]-2*a*x[1]]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "t=linspace(0,10,1000)\n",
    "x=odeint(f,[1,0],t)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Графики координаты и скорости."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f34a8015d30>,\n",
       " <matplotlib.lines.Line2D at 0x7f34a8015f28>]"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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pUrZuzaBBKTv+9wu/M2P7DP5d49+eDcxLjIGuXaFJE/s+KKVSzi+T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\nkCZ/pZQKQpr8lVIqCP0/Thmr7gje5CkAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f34b04783c8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from matplotlib.pyplot import plot\n",
    "%matplotlib inline\n",
    "plot(t,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Точное решение для координаты."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "b=sqrt(1-a**2)\n",
    "x0=exp(-a*t)*(cos(b*t)+a/b*sin(b*t))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Максимальное отличие численного решения от точного."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7.4104573116740013e-08"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "abs(x[:,0]-x0).max()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{eqnarray}\n",
       "\\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n",
       "\\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n",
       "\\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n",
       "\\nabla \\cdot \\vec{\\mathbf{B}} & = 0 \n",
       "\\end{eqnarray}"
      ],
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Latex\n",
    "Latex(r\"\"\"\\begin{eqnarray}\n",
    "\\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n",
    "\\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n",
    "\\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n",
    "\\nabla \\cdot \\vec{\\mathbf{B}} & = 0 \n",
    "\\end{eqnarray}\"\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "%%latex\n",
    "\\begin{align}\n",
    "\\nabla \\times \\vec{\\mathbf{B}} -\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{E}}}{\\partial t} & = \\frac{4\\pi}{c}\\vec{\\mathbf{j}} \\\\\n",
    "\\nabla \\cdot \\vec{\\mathbf{E}} & = 4 \\pi \\rho \\\\\n",
    "\\nabla \\times \\vec{\\mathbf{E}}\\, +\\, \\frac1c\\, \\frac{\\partial\\vec{\\mathbf{B}}}{\\partial t} & = \\vec{\\mathbf{0}} \\\\\n",
    "\\nabla \\cdot \\vec{\\mathbf{B}} & = 0\n",
    "\\end{align}"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python [Root]",
   "language": "python",
   "name": "Python [Root]"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}