{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\n",
    "\\newcommand{\\norm}[1]{{\\left\\|#1\\right\\|}}\n",
    "\\newcommand{\\abs}[1]{{\\left\\vert#1\\right\\vert}}\n",
    "\\newcommand{\\ip}[2]{{\\left\\langle#1,#2\\right\\rangle}}\n",
    "\\newcommand{\\R}{\\mathbb{R}}\n",
    "\\newcommand{\\p}{\\ensuremath{\\partial}}$\n",
    "\n",
    "\n",
    "# Plotting Direction Fields \n",
    "\n",
    "Our primary computational tool will be Python using Jupyter notebooks. I'll point out at the beginning that this is not a \n",
    "programming class and I will often sacrifice good programming principles to make the things more clear. \n",
    "\n",
    "## Direction Fields\n",
    "A **direction field** is a particular type of **vector field**. Recall that a two dimensional vector field on $R^2$ is a \n",
    "function that assigns to each point $(t,x)$ in $\\R^2$ a vector denoted, for example, by $\\ip{u(t,x)}{v(t,x)}$. In the \n",
    "context of ODEs, the vector field is constructed in the following way: If $(t,\\varphi(t))$ is a solution to the ODE $x' = \n",
    "f(t,x)$, then the vector field is :\n",
    "$$\n",
    "\\ip{u(t,x)}{v(t,x)} \n",
    "= \\ip{1}{\\varphi'(t)} \n",
    "= \\ip{t}{f(t,x)}.\n",
    "$$\n",
    "\n",
    "Of course, when drawing a vector field using python (or any software) we're not going to draw a vector at every point of \n",
    "$\\R^2$. Instead, we'll do it at a regurlarly spaced set of points over a rectangular grid $[t_0,t_1]\\times[x_0,x_1]$. \n",
    "Decisions like what rectangular grid to choose and how many points to sample the direction field at are domain specific. \n",
    "For example, if $x(t)$ represents the population of field mice, we wouldn't want to include any negative values of \n",
    "$x$ in the sampling. \n",
    "\n",
    "The way numpy does this is using meshgrids. First, we define two arrays: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "t = np.arange(0,10,1)\n",
    "x = np.arange(0,2,.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To run a block of code, press Shift+Enter. The ```import numpy``` allows us to use the functionality of numpy. If this is\n",
    "all we did, then we would have to write \n",
    "things like ```numpy.arange(0,10,1)```. The ```as np``` allows us to refer to ```numpy``` as ```np```. These import \n",
    "statements usually go at the top of the Jupyter notebook, and in the future this is what we will do. We will have other \n",
    "import statements below that would usually go at the top as well. \n",
    "\n",
    "To see the value of a variable you can use the print command:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "t=[0 1 2 3 4 5 6 7 8 9]\n",
      "x=[0.  0.5 1.  1.5]\n"
     ]
    }
   ],
   "source": [
    "print (f\"t={t}\")\n",
    "print(f\"x={x}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expression ```np.arange(0,10,1)``` gives an array from 0 to 10 (not including 10) that is evenly spaced \n",
    "with step size 1. In general ```np.arange(a,b,h)``` gives an evenly spaced array from ```a``` to ```b``` (but not including \n",
    "```b```) with step size equal to ```h```.\n",
    "\n",
    "The array ```t``` has length $10$ and the array ```x``` has length $4$. The two arrays together encode $4\\times 10 = 40$ \n",
    "ordered pairs in the $(t,x)$ plane at which to draw a vector field. The way that numpy organizes this data is via a \n",
    "\"Meshgrid\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "T, X = np.meshgrid(t, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This assigns to ```T``` and ```X``` a $4\\times 10$ matrix. Each **row** of the ```T``` matrix is the vector ```t```\n",
    "and each **column** of the ```X``` matrix is the vector ```x```. That is: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "T = \n",
      "[[0 1 2 3 4 5 6 7 8 9]\n",
      " [0 1 2 3 4 5 6 7 8 9]\n",
      " [0 1 2 3 4 5 6 7 8 9]\n",
      " [0 1 2 3 4 5 6 7 8 9]]\n",
      "X = \n",
      "[[0.  0.  0.  0.  0.  0.  0.  0.  0.  0. ]\n",
      " [0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5]\n",
      " [1.  1.  1.  1.  1.  1.  1.  1.  1.  1. ]\n",
      " [1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5]]\n"
     ]
    }
   ],
   "source": [
    "print(f\"T = \\n{T}\")\n",
    "print(f\"X = \\n{X}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Observe that we have used lower caseletters - ```t``` and ```x``` - for the arrays and uppercase letters -\n",
    "```T``` and ```X``` - for the corresponding vectors. This is a good convention to use. \n",
    "\n",
    "If $T_{kj}$ represents the the $(k,j)$ entry of $T$ and similarly for $X_{kj}$, then as $k$ runs through $0$ to $3$ and \n",
    "$j$ runs through $0$ to $9$, the ordered pairs $(T_{kj}, X_{kj})$ run through all $40$ sample points."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At this point we have an efficient way to store the coordinates in the $(t,x)$ plane at which we will draw our \n",
    "direction field. We now need to tell numpy which vectors are the ones to be drawn and then we need to get python to draw \n",
    "them. \n",
    "\n",
    "We'll assume that we are dealing with ODEs of the form: \n",
    "$$\n",
    "\\frac{dx}{dt} = f(t,x). \n",
    "$$\n",
    "\n",
    "And we will use the specific example: \n",
    "$$\n",
    "\\frac{dv}{dt} \n",
    "= -v - 9.8.\n",
    "$$\n",
    "\n",
    "We first define a function that we can use (this isn't necssary but it makes things much easier). The \n",
    "```#...``` is a comment. The Python interpreter doesn't look at this when running the code. It is just for us to \n",
    "give a short description of what the code is doing. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# This represents the right hand side of the ODE\n",
    "def f(t,x): \n",
    "    return -9.8-x/5"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that we are using the variable ```x``` instead of ```v```. This is because I prefer to keep things standardized \n",
    "throughout different problems (as a form of abstraction.) But this is mostly a personal choice and in your own code, \n",
    "you could have done: \n",
    "```\n",
    "def f(t,v): \n",
    "    return -v-9.8.\n",
    "```\n",
    "And instead of using ```x``` and ```X``` you can use ```v``` and ```V```. However, ```u```, ```U```, ```v```, ```V``` are \n",
    "used for the vector field as you will see below. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "dXdT = f(T,X)\n",
    "U = (1 / (1 + dXdT**2)**.5)*np.ones(T.shape)\n",
    "V = (1 / (1 + dXdT**2)**.5)*dXdT"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is a lot going on here. Let's look at ```dXdT```: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "dXdT = \n",
      "[[ -9.8  -9.8  -9.8  -9.8  -9.8  -9.8  -9.8  -9.8  -9.8  -9.8]\n",
      " [ -9.9  -9.9  -9.9  -9.9  -9.9  -9.9  -9.9  -9.9  -9.9  -9.9]\n",
      " [-10.  -10.  -10.  -10.  -10.  -10.  -10.  -10.  -10.  -10. ]\n",
      " [-10.1 -10.1 -10.1 -10.1 -10.1 -10.1 -10.1 -10.1 -10.1 -10.1]]\n"
     ]
    }
   ],
   "source": [
    "print(f\"dXdT = \\n{dXdT}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a $4 \\times 10 $ matrix. The $(k,j)$ entry is: \n",
    "$$\n",
    "f(T_{kj}, X_{kj}).\n",
    "$$\n",
    "This is why the rows are constant (because $f(t,x)$ really only depends on $x$ in the sense that \n",
    "$\\frac{\\partial f}{\\partial t}\\equiv 0$). The fact that this works is a minor miracle and is both liberating and somewhat\n",
    "alarming for those who are familiar with a language like C. \n",
    "\n",
    "To explain the lines: \n",
    "```\n",
    "U = (1 / (1 + dXdT**2)**.5)*np.ones(T.shape)\n",
    "V = (1 / (1 + dXdT**2)**.5)*dXdT\n",
    "```\n",
    "recall that our vector field is $\\ip{1}{f(t,x)}$. We actually don't care about the lengths too much, and especially for \n",
    "display purposes we want to pick lengths that make things easier to see. So instead of plotting this vector field, \n",
    "we will plot a normalized version: \n",
    "$$\n",
    "\\frac{1}{\\norm{\\ip{1}{f(t,x)}}} \\ip{1}{f(t,x)}\n",
    "=\\ip{\\frac{1}{\\sqrt{1 + f(t,x)^2}}}{\\frac{f(t,x)}{\\sqrt{1 + f(t,x)^2}}}. \n",
    "$$\n",
    "\n",
    "```U``` and ```V``` will both be $4\\times 10$ matrices and the vector drawn at the point $(T_{kj}, X_{kj})$ will be \n",
    "$\\ip{U(k,j)}{V(k,j)}$. Note that this means dimensions need to be consistent among ```T```, ```X```, ```dXdT```, ```U```, \n",
    "and ```V```. \n",
    "\n",
    "The expression ```T.shape``` returns a $2\\times 1$ array containg the dimensions of ```T```:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(4, 10)\n"
     ]
    }
   ],
   "source": [
    "print(T.shape)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expression ```np.ones(T.shape)``` creates a matrix of dimension equal to ```T.shape```:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[[1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n",
      " [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n",
      " [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]\n",
      " [1. 1. 1. 1. 1. 1. 1. 1. 1. 1.]]\n"
     ]
    }
   ],
   "source": [
    "print(np.ones(T.shape))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In python, the expression ```a**b``` raises ```a``` to the power of ```b```:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8\n"
     ]
    }
   ],
   "source": [
    "print(2**3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So multplying ```np.ones(T.shape)``` by ```(1 / (1 + dXdT**2)**.5)``` has the effect of dividing\n",
    "```np.ones(T.shape)``` by $\\norm{\\ip{1}{f(t,x)}}$. Similarly, the expression ```V = (1 / (1 + dXdT**2)**.5)*dXdT```\n",
    "divides the matrix representing $f(t,x)$ by $\\norm{\\ip{1}{f(t,x)}}$. Overall, as discussed above, $(U,V)$ represents \n",
    "the vector field: \n",
    "$$\n",
    "\\ip{\\frac{1}{\\sqrt{1 + f(t,x)^2}}}{\\frac{f(t,x)}{\\sqrt{1 + f(t,x)^2}}}.\n",
    "$$\n",
    "\n",
    "Next we need to get python to plot the vector field $\\ip{U}{V}$ ad the points $\\ip{T}{X}$. The ```matplotlib``` library \n",
    "is a library of plotting functions. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(T, X, U, V) # draws the arrows at (X,Y) with slope dYdX"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expression ```plt.figure()``` creates a figure on which to draw. The expression ```plt.title('Direction \n",
    "Field')``` sets the title of the plot. Finally, the expression ```Q = plt.quiver(T,X,U,V)``` is the one that draws \n",
    "the vectors. \n",
    "\n",
    "At this point, we should try to determine whether we have chosen a good range of $t$ and $v$ values (or using \n",
    "the variable names in the code, ```t``` and ```x``` values). The problem domain suggests that all velocities \n",
    "should be negative *for this model* if we are dropping something from a height (and we have a coordinate system \n",
    "in which downward velocities are negative.) Also, there are certain features that we want to see in the direction \n",
    "field and these features can be deduced from the model itself (more on this below.) \n",
    "\n",
    "Sol, let's change the $v$ range to be from $-40$ to $-60$ (or, using the variables in the code, we will change the\n",
    "```x``` vector to be ```x = np.arange(-60,-40,5)```).  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(0,10,1)\n",
    "x = np.arange(-60,-40,5)\n",
    "T, X = np.meshgrid(t, x)\n",
    "dXdT = f(T,X)\n",
    "U = (1 / (1 + dXdT**2)**.5)*np.ones(T.shape)\n",
    "V = (1 / (1 + dXdT**2)**.5)*dXdT\n",
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(T, X, U, V) # draws the arrows at (X,Y) with slope dYdX"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will analyze this below, but first I want to describe ways to change the apperance. Some of the changes in \n",
    "appearance is purely esthetic \n",
    "I prefer for the vectors to be red (this is inline with what is in the textbook as well). To do this, we can \n",
    "add color in the following way: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(T, X, U, V, color='red') # draws the arrows at (X,Y) with slope dYdX"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In addition, the arrowheads aren't really needed. We really only need short line segments that represent the \n",
    "tangent lines to the solution curves:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXkAAAEICAYAAAC6fYRZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+j8jraAAAgAElEQVR4nO3deZxU1Zn/8c8DKAhKIAECKJsOanABpVETNbgn0Sg6o9EYjZNkxJgxGKOTxC2jjsbEmKjz0zES1wkuwxiNRHFDE8Zdu0HZRA2yi7KjoEDT/fz+OFW51W110921nOL29/161aur6lbd++26Vc8999xTt8zdERGRdOoQO4CIiJSOiryISIqpyIuIpJiKvIhIiqnIi4ikmIq8iEiKqcjLNsHMfmdml5dpWbPN7LByLKs1yzOzwWbmZtapielXmNmEogeUbVreN4tIOZnZAuDzwBagDpgD/Dcw3t3rAdz9+yVa9t3AEne/LHufu+9VomX9FTiI8H9mHV2q5YmAWvJSOY53952AQcAvgZ8Cd7TkiU21bCvUee6+Y87lpdiBJN1U5KWiuPs6d58EnAqcZWZ7Q2hxm9nVmeuHmdkSM/upmb0P3GVmHczsZ2Y2z8xWmdlEM/tsdr5mdoiZvWhma81ssZn9s5mNBb4F/MTM1pvZnzOPXWBmR2WudzazG83svczlRjPr3CjHhWa23MyWmdl3Wvs/N1pes/9Ho+cNMbOpZvaRmT0N9GrtsiX9VOSlIrn7q8AS4NAmHtIX+Cyh5T8WGAecCIwG+gNrgFsAzGwg8Djw/4DewAjgdXcfD9wLXJdpVR+fZzmXErpYRgDDgQOAy3Km9wU+A+wMfA+4xcx6tu2/hub+jzzuA2oIxf0/gLMKWK6klIq8VLL3CIU8n3rg3919k7t/ApwDXOruS9x9E3AFcHKmK+dbwBR3v9/da919lbu/3sIM3wKucvfl7r4CuBI4M2d6bWZ6rbtPBtYDezQzv//M7E2sNbNpeaY393/8XWbDNQq4PPMa/B/w5xb+T9KObEt9mdL+7AysbmLaCnffmHN7EPCwmdXn3FdHOKA7AJjXxgz9gYU5txdm7sta5e65B1I/BnZsZn7j3P32ZqY39380zrXG3Tc0yjagmXlLO6SWvFQkMxtFKPLPN/GQxqdPXQx8zd175Fy6uPvSzLTdWjifxt4jFN6sgZn7SqW5/yPXMqCnmXVrlE2kARV5qShm1t3Mvg48AExw95ktfOrvgGvMbFBmPr3NbExm2r3AUWb2DTPrZGafM7MRmWkfALs2M9/7gcsy8+sF/Bwo5Vj05v6Pv3P3hUA1cKWZbW9mhwD5jilIO6ciL5Xiz2b2EaEleynwW6A1I1VuAiYBT2Xm8zJwIIC7LwKOBS4kdP+8TjiICmGY5rBMH/mf8sz3akIxnQHMBKZl7iuVJv+PPE7PTFsN/DvhuwUiDZh+NEREJL3UkhcRSTEVeRGRFFORFxFJMRV5EZEUq6gvQ/Xq1csHDx4cO4aIyDalpqZmpbv3zjetoor84MGDqa6ujh1DRGSbYmYLm5qm7hoRkRRTkRcRSTEVeRGRFFORFxFJMRV5EZEUU5EXEUkxFXkRkRRLT5Ffvx6uugo2bYqbY8ECuOkmiH12z6lT4eGH42YA+MMfoKYmdgq4/npYvDhuhtpauPLK8F6NacUKuPZaqKuLm+ONN+Cuu+JmAHjkEfjLX2KngFtvhbfeKv583b1iLiNHjvQ2qa93P/lkd3A/6CD3pUvbNp9CbdjgPmJEyHHaae7r18fJsXChe69eIcdll7nX1cXJ8dxz7p06uXfu7H7PPXEyuLuPHx9eiz593KdOjZfjvPNCjr32cn/nnTgZamvdDz885PjKV9xXrYqTY+VK98GDQ44f/MB906Y4OWbOdO/Wzb1jR/cbbgi1JIaHHw6vRffu7pMmtfrpQLU3UVfT0ZL/8ENYmPnC18svw8iR8OKL5c+xZAmsWhWuP/AAfOlLMH9++XO8/TZszPz86dVXwwknwNq15c8xezbU14e9q7POgvPPD63ZcpsxI/xdvhyOPBJuuaX8e1qbN8PcueH67NkwahQ88UR5M0B4DZYtC9effDLkmNnSH98qovnzkz2a//qvsF4++KD8OebODe/Jujq44AL49rfhk0/KnyO7Dj78MHxer7oqfHaKoanqH+PS5pa8u/snn7j/8z+HrSG4b7ed+223tX1+bbV8ufthhyU5PvtZ96eeKn+O2bPdhw5NcgwdGu4rt6eeCq9BNsfo0e4ffFD+HLfdFt4T2Rzf+U54z5RTba37RRclGczcf/GL8rce161zHzMmydGtm/vEieXN4O6+YIH7fvslOXbe2f2VV8qf45VXwrKzOfbbL2Qrt4kT3bt2TXKceGJYVy1AMy356IU991JQkXcPH5abbw5dBNkX6uyz3TduLGy+rbV5s/v55ycZOnRwv+668n+Y16xxP+64JMeOO4bdwnJ79133ffdNcgwY4F5dXf4cL7zg3rdvkmPUKPfFi8uf47773HfYIclx8snuH31U3gx1de5XXplkAPef/cx9y5by5vj4Y/czzkgybL+9+x13lDeDu/uyZe6HHJLk6NXL/dlny59jxgz3XXdNcuy5p/vcuVt9Wvsp8llTp4b+1+wL9cUvxumn/+//Dv3R2Rwx+unr6twvv7zhh/nyy8vfT79+vfuppyYZYvXTL10ajttkc/Tp4/5//1f+HNOnJ33S4L733nH66SdNCv3A2Rwx+unr60N/eMeOSY4Y/fSbNoXlZjPE6qdftSqsh2yO7t3d//znZp/S/oq8e2ihjRqVvFB9+4aWXLlVV4eWazbHvvuGlm25PfRQaMlncxx3XGjpl1N9fdij6dAhyTFuXNjzKaeNG8MeXjZDp05hD7DcH+YVK9yPPDLJ0aOH++OPlzeDe2gp7rlnkmPXXUOLstyefTYZMAChZf3+++XPcfvtYY8im+OMM8IeRzlt2RL2rHIbZ1de2WTjrH0WeffK6af/4IPQF53bT//00+XPUUn99D17JjlGjw7HMsrtd7+rjH76Cy9MMsTspz/hhCRHe++nf/ll9/79kxwV3k/ffou8e2X1048bl2To0MH9179uv/308+ZVRj/9889XRj/9vfeqnz6rkvrpDz44yVHB/fTtu8hnVUo//T33NOyn/+Y3w/j6cqrUfvouXcJxjHKrpH76QYOSHOqnr4x++nPPTTJUaD+9inzWokWV2U8/fLj66XP76c8/P04//b/8S5JB/fTqp89V4f30JS/ywEWAA71y7tsXeAmYDcwEumxtPiUv8u75++lvv730y20sXz99jF3BfP30b71V/hyN++kPOyzOtzHz9dPX1pY3Q75++l/9qrwZ3PP30z/0UPlz5Ounj9G1l6+ffsmS8ufI00/fXJEv+BuvZjYAOBpYlHNfJ2AC8H133ws4DIjwVcc8unSBO++Em2+GTp3Ct906dy5/jj594OmnYdy4cHvDBthxx/LnGDYMXn0Vjjsu3F6/Hnbaqfw5jj4aqqth333D7Y8/hq5dy5/jnHPCeUz69g236+qgY8fyZujUKZxr5957YYcdwke53BkAuncP5z+68spwO9Y6GTQIXngBzjgj3F6/PmQrtwMPDOdhOvjgcDvWZ/aUU8I3+3fdNdzOfru9KU1V/5ZegAeB4cACMi154FhgQmvnVZaWfK6pU8PuTmz33BP33C7uoT/+5z93f+mluDnWrw99oDFaSLmWLnUfO7b8u+SNTZ8eviUb65wqWZMmud94Y9wM9fUhw1bGjJfcpk3uF1zg/uabcXOsWhXeo6tXN9uSNw9FuU3M7ATgSHc/38wWAFXuvtLMfgSMBPoAvYEH3P26JuYxFhgLMHDgwJELFzb5o+MiIpKHmdW4e1W+aZ1a8OQpQN88ky4FLgGOaWK+hwCjgI+BZzIhnmn8QHcfD4wHqKqqavsWR0REPmWrRd7dj8p3v5ntAwwB3jAzgF2AaWZ2ALAEmOruKzOPnQzsD3yqyIuISOm0+cCru8909z7uPtjdBxMK+/7u/j7wJLCvmXXNHIQdDcwpSmIREWmxrbbk28Ld15jZb4HXCEMrJ7v7Y6VYloiINK1oRT7Tms+9PYEwjFJERCJJxy9DiYhIXiryIiIppiIvIpJiKvIiIimmIi8ikmIq8iIiKaYiLyKSYiryIiIppiIvIpJiKvIiIimmIi8ikmIq8iIiKaYiLyKSYiryIiIppiIvIpJiKvIiIimmIi8ikmIq8iIiKVaUIm9mF5mZm1mvzO3tzOweM5tpZm+a2cXFWI6IiLROwb/xamYDgKOBRTl3nwJ0dvd9zKwrMMfM7nf3BYUuT0REWq4YLfkbgJ8AnnOfA93MrBOwA7AZ+LAIyxIRkVYoqMib2QnAUnd/o9GkB4ENwDJCC/96d1/dxDzGmlm1mVWvWLGikDgiItLIVrtrzGwK0DfPpEuBS4Bj8kw7AKgD+gM9gefMbIq7v9v4ge4+HhgPUFVV5Y2ni4hI2221yLv7UfnuN7N9gCHAG2YGsAswzcwOAE4HnnD3WmC5mb0AVAGfKvIiIlI6be6ucfeZ7t7H3Qe7+2BgCbC/u79P6KI5woJuwEHA3KIkFhGRFivVOPlbgB2BWcBrwF3uPqNEyxIRkSYUPIQyK9Oaz15fTxhGKSIiEekbryIiKaYiLyKSYiryIiIppiIvIpJiKvIiIimmIi8ikmIq8iIiKaYiLyKSYiryIiIppiIvIpJiKvIiIimmIi8ikmIq8iIiKaYiLyKSYiryIiIpVrTzyW+z3GHePKiuhpoaGDAAxo0rf44tW2DOnJChpgaOPhrGjCl/jk8+gRkzkhzjxsHw4eXP8eGHMG1ayFBdDePHw047lT/HBx8kr8WSJXDbbeXP4A4LFyY5unaFyy4rf466Onj77STHiBFw1lnlz7F5M8yalXxmTz8dRo8uf44NG2D69OT1uPpqGDiw/DlWr252cvsq8tmCnl0p1dWhkKxblzzmgANKX+S3bIE330zepDU18MYbocBmbdpU+iK/cWNYbjZDTU348NTVJY/Zb7/SF/ncgp5dL++80/Ax554LX/5yaXMsX54sP7ew5/rVr6BHj9JlcIdFixpmqKmBVauSxwwaVPoi37ig19SEdbRhQ/KY448vfZHPFvTc98bMmeH+rD59Sl/kN2yA119vuF7mzoX6+uQxxx9f+iK/enVYD7k55s9v9ikFFXkzuwI4G1iRuesSd5+cmXYx8D2gDhjn7k8WsqxWa1zQs5fcgt5Yhw7hee4Qfpy8cNmCnls8Ghf0xjp3bvgmLoZ8BX327JCvKTvtBOvXFzdH44JeUxOKSXN694aVK4ubI1vQc9dL44Le2MCBsHhx8Yp8Swp6Y2awww7w8cehRV8M+Qr69OnNr/tOnZp/77RF44JeUxP2Kpv7LHTt2vxnqS2yBT33/dG4oDfWowesXVvcHKtXf7p+baWg51OMlvwN7n597h1mNgw4DdgL6A9MMbPd3b0u3wyKYuFCePnl5APTuIXeWIcOMGwYjBwJVVXh7/DhhX9wZs+GV19tuoXeWOfOYbkjRyaXvfaC7bZre4b6enjtteZb6I3ttBPsv3/DHEOHhteprTZuDOskt5A2bqE31rt3svzsetlll8I2umvXwksvNd9Cb2zgwIYZ9t8/ZCvEsmXw4outK+h77NFwney3X+HdVn/7W7JeWlrQ99mn4XrZZ5/w3i3E9OlhnWTXS+MWemNdu4Yuotz1suee0LFj2zPU1obPa3Mt9MZ69my4TkaOhCFDCnuPrl8f3qOtKej9+yfLv+KKJh9Wqu6aMcAD7r4JmG9mfwMOAF4q0fLCbvStt+afVqqCns/YseGDnE8pCnpTjj4aPvoo/7RSFPR83nsPDj+86emlKOj5/PWvcNJJTU8vRUHPZ/z4pj+MZrD77kmGYhX0fH76U3joofzTSlXQ8znllLC3nU8pCno+GzfCoYeGvap8SlHQ85k5E445punp/folr0P2b9++yfQSF/nzzOzbQDVwobuvAXYGXs55zJLMfZ9iZmOBsQADC+nPGjky/C1nQc+nqioU+c6dYd99G35oS1XQG+vQIRSqqVPLV9DzGTIkfEjWrClfQc8n+96A8hX05nKUqoXemhwPPVTegt5UjnnzylfQ89lpp7Au5s4NXS65GUpV0PMZPjx8JuvrG7bQs5d+/do8a/OmtmDZB5hNAfrmmXQpoZCvBBz4D6Cfu3/XzG4BXnL3CZl53AFMdvc/Nresqqoqr66ubv1/AWEUxLvvlreg5/POO6FPr1wFvSnTp4fXoVwFvSnPPx8OFJaroOfjDlOmhEJSroKez7p1ofuunAU9n0WLwudln32gS5d4OWbPDn/LVdCb8sor4X1RroLelL/8JbwWbSjoZlbj7lV5p22tyLdiIYOBR91978xBV9z92sy0J4Er3L3Z7pqCiryISDvVXJEvqIlnZrmbnJOAWZnrk4DTzKyzmQ0BhgKvFrIsERFpvUL75K8zsxGE7poFwDkA7j7bzCYCc4AtwL+WdGSNiIjkVVCRd/czm5l2DXBNIfMXEZHC6Nw1IiIppiIvIpJiKvIiIimmIi8ikmIq8iIiKaYiLyKSYiryIiIppiIvIpJiKvIiIimmIi8ikmIq8iIiKaYiLyKSYiryIiIppiIvIpJiKvIiIimmIi8ikmIq8iIiKVbob7xeYWZLzez1zOXYzP1Hm1mNmc3M/D2iOHFFRKQ1Cv2NV4Ab3P36RvetBI539/fMbG/gSWDnIixLRERaoRhF/lPcfXrOzdlAFzPr7O6bSrE8ERHJrxh98ueZ2Qwzu9PMeuaZ/k/A9KYKvJmNNbNqM6tesWJFEeKIiEiWuXvzDzCbAvTNM+lS4GVC14wD/wH0c/fv5jx3L2AScIy7z9tamKqqKq+urm55ehERwcxq3L0q37Stdte4+1EtXMjvgUdzbu8CPAx8uyUFXkREiq/Q0TX9cm6eBMzK3N8DeAy42N1fKGQZIiLSdoX2yV+XGSY5AzgcuCBz/3nAPwCX5wyv7FPgskREpJUKGl3j7mc2cf/VwNWFzFtERAqnb7yKiKSYiryISIqpyIuIpJiKvIhIiqnIi4ikmIq8iEiKqciLiKSYiryISIqpyIuIpJiKvIhIiqnIi4ikmIq8iEiKqciLiKSYiryISIqpyIuIpJiKvIhIiqnIi4ikmIq8iEiKFfpD3leY2dKc33E9ttH0gWa23swuKixmkdXXx04QKEfDDO6xU1TGawHK0Vgl5HCvjBytzFCMlvwN7j4ic5nceBrweBGWUTzvvQf77w9Tp8bN8eqrMHIkvPNO3BwTJsAxx8Dq1XFzXHwxnHMObN4cL0NdHYwZA7/5TdwNztq1cOCB8Mgj8TIAvP02jBgB06fHzfHkk/ClL8HSpXFz3HADfOMbsGFDvAzu8N3vwiWXtLzYu3ubL8AVwEVNTDsR+HVzj2l8GTlypJfUxx+7jxrlDu6dOrnPmFHa5TVlyRL3fv1Cjl693NeujZPjpZfcO3cOOQ44wL2+Pk6Ou+8OGcD9e9+Lk8Hd/cc/TnL853/GyVBb6/6Vr4QMZu7PPhsnx+rV7rvvHnJ07eq+eHGcHG++6f6Zz4Qcu+3mvmlTnByPPebeoUPI8fWvx8ng7n7ddcl79NJL/343UO1N1emmJrTkkingC4AZwJ1Az8z93YCXgB23VuSBsUA1UD1w4MDSvTj19e6nnZa8QN/9bpyitmGD+/77JzluvLH8GdzdFy50//znQ4aOHd2ffjpOjuefd99++5CjZ0/3t9+Ok+P225N1MmyY+7p1cXKcf36S4+tfd9+ypfwZNm92P/LIJMfFF5c/g7v7ypWhsGdz3H9/nByzZrnvtFPIsMMO7tXVcXJMmhQ2/OA+YID7smV/n1RQkQemALPyXMYAnwc6Erp9rgHuzDzneuAbvpXWfuNLSVvyV12VvFkOPTROi6Cuzv3kk5McZ58dZ0Pz0Ufuw4cnOW6+ufwZ3N3nz3fv3TvZs3rmmTg5pk513267kONzn3OfNy9OjttuS9bJ3nu7f/hhnBw/+EGS48QTw/u23DZvdj/ssCTH5ZeXP4O7+4oV7kOGJDkmToyTY8YM9x13TPaspk9vMLlkLfkGM4LBwKzM9ecyLfwFwFpgNXDe1uZRsiI/cWKykoYMcV++vDTL2Zqf/zzJcdhh8TY0J56Y5PjBD8qfwT0UsH32SXL87ndxcsybFwo7hEI/dWqcHM8+GzZ0EDZ88+fHyXHzzck6GT48NAjKrb7efezYJMc//VOcDc2mTaFBmM1x5ZXlz+Du/sEH7oMGJTn++MdPPaSU3TX9cq5fADyQ5zFxW/LV1WEXC8Iu16xZxV9GS9x/f7KSdtst7IrGcMklSY4jjwwtpnLbssX9+OOTHD/8YfkzuIcumWHDkhy33x4nxzvvhK6q7Ibmuefi5HjqqdB1B6Erb+HCODluuilZJ/vt575+ffkz1NeHLt1sjlNPjbPXvXGj+8EHJzmuvjrvw0pZ5P8AzMz0yU/KLfo5j4lX5N97z33nnf3vB7EefbS482+pV15x79Il5Oje3X3OnDg5JkxI3ixDh4aDazH85CdJjmOOCQcby23LFvdjj01yXHBB+TO4u69Z477nnkmOu+6Kk+Ott9x79AgZtt/e/cUX4+R44onkAGffvvEO+P7mN8k6qaoKgzbKrb7e/ayzkhynn97khqYs3TXFuBS1yOeOpAH3668v3rxbY/HiZCRNhw7ujz8eJ0fuSJoePdznzo2TI3ckzR57hCIXQ+5Imq99Lc4BztyRNOB+0UXlz+AeNvZDhyY5/vCHODnmzElG0nTpEhpHMTz6aLKh6d/ffenSODlyR9IceKD7J580+dD2V+Q1kqahRYsajqR56qk4OSp1JE2sIazjxiU5NJKm4Uia++6Lk6PxSJrXXouTI3ckzS67NBhJk0/7K/IaSZNYv959xIgkh0bSaCRNrkocSXPZZeXP4F65I2mmTdvqU9pXkf/f/01W0uDBlTGSZvToeBuak05Kcpx7bvkzuH96JM2tt8bJkTuSplMn97/+NU6O3JE0vXq5v/tunBy33JKsk0oZSfOP/xhvJM2Xv5zkuOKK8mdw//RImgcfbNHT2k+RbzySZubMwubXVhpJk9iyxf2EE5IcGklTGSNpnn46GUnTp49G0mxDI2nyaR9FfulSjaTJpZE0CY2kaahSRtI8/rhG0mTV17t/+9tJjmZG0uST/iJfKSNpcs9Jo5E0GkmTq7Y2bOSyOf7t38qfwb3hOWlij6Tp3t2jj6TJPSdNzJE0v/pVsk62MpImn+aKfDrOJ794cTi7JIQztP34x3FyvPkmrFsXrt9wA3z1q3FyTJsWzubYsSNMnAh77BEnxyuvhL89e8Kjj0KPHuXPsHlzchbFYcPg/vvD61Juy5fD/Pnh+vHHw7XXlj8DwLx5sGJFuH7xxXDGGXFyzJwJn3wSrt91FxxwQJwcr70Wzua4ww7hrJ/9+5c/g3s4Ky3AgAHwpz9Bly7FnH/8Fnz2UnB3zXnnxTtLXdb06WF0QKwzOmY98kgYwRFTfX3YFY51JsWszZvdL7ww3kiarNWrw2iWWCNpst55J3RZxTjAmWvq1DAWPLa77443kiarri4M1mjBSJp8aKYlb2F6ZaiqqvLq6urYMUREtilmVuPuVfmmpaO7RkRE8lKRFxFJMRV5EZEUU5EXEUkxFXkRkRRTkRcRSTEVeRGRFFORFxFJsYKKvJldYWZLzez1zOXYnGn7mtlLZjbbzGaaWRG/pysiIi3RqQjzuMHdr8+9w8w6AROAM939DTP7HFBbhGWJiEgrFKPI53MMMMPd3wBw91UlWo6IiDSjGH3y55nZDDO708x6Zu7bHXAze9LMppnZT5p6spmNNbNqM6tekT07noiIFMVWi7yZTTGzWXkuY4Bbgd2AEcAy4DeZp3UCDgG+lfl7kpkdmW/+7j7e3avcvap3797F+J9ERCRjq9017n5US2ZkZr8HHs3cXAJMdfeVmWmTgf2BZ9qYU0RE2qDQ0TX9cm6eBMzKXH8S2NfMumYOwo4G5hSyLBERab1CD7xeZ2YjAAcWAOcAuPsaM/st8Fpm2mR3f6zAZYmISCsVVOTd/cxmpk0gDKMUEZFI9I1XEZEUU5EXEUkxFXkRkRRTkRcRSTEVeRGRFFORFxFJMRV5EZEUU5EXEUkxFXkRkRRTkRcRSTEVeRGRFFORFxFJMRV5EZEUU5EXEUkxFXkRkRRTkRcRSTEVeRGRFCv0N16vMLOlZvZ65nJs5v7tzOweM5tpZm+a2cXFiSsiIq1R6G+8Atzg7tc3uu8UoLO772NmXYE5Zna/uy8owvJERKSFStVd40A3M+sE7ABsBj4s0bJERKQJxSjy55nZDDO708x6Zu57ENgALAMWAde7++oiLEtERFphq0XezKaY2aw8lzHArcBuwAhCQf9N5mkHAHVAf2AIcKGZ7drE/MeaWbWZVa9YsaIY/5OIiGRstU/e3Y9qyYzM7PfAo5mbpwNPuHstsNzMXgCqgHfzzH88MB6gqqrKW5hbRERaoNDRNf1ybp4EzMpcXwQcYUE34CBgbiHLEhGR1it0dM11ZjaCcKB1AXBO5v5bgLsIRd+Au9x9RoHLEhGRViqoyLv7mU3cv54wjFJERCLSN15FRFJMRV5EJMVU5EVEUiz9RX7TJvif/4Ha2rg5Fi+Gp54CjzxKdNo0mD49bgaAZ56BBQtip4CHH4bVkb+n5w733QcbN8bNsXYt/OlPUF8fN8fcufDCC3EzQMgwtwIGBU6eDO+/3/bnu3vFXEaOHOlFd9dd7uA+YID7rFnFn39L/ehHIcfw4e7r1sXLccQRIceYMe719XEybNzo3q+fe8eO7pdcEieDu/uCBSFD167ud98dL8fkyWGd9Onj/vzz8XL88pchx9Ch7osXx8vxzW+GHIcc4r55c5wM9fXuw4aFHGefHSeDu/vq1e7durlvv737b3/b5MOAam+irhbjBGWVyx2uz5w7rbYW/uEf4uRYswZ+//twvWdP6N49To5p0+DZZ8P1QYPALE6O++6DZcvC9d12i5MB4MYboa4OPv4Y9torXo5f/zr83XWuSQQAAAeTSURBVLABhg2Lk2HTJrjppnC9Uyfo3z9OjoULYeLEcH3nnWG77eLkePxxmDMnXB88OE4GgNtuC+8LgN13b9s8mqr+MS5Fb8lnW0jg/otfFHferXHttUmOxx6LlyPbQurQwX3+/DgZcltIffuGVn0M2RYSuB92WJwM7u7V1cl74/zz4+XI7vGC+x13xMuR3eMF99dei5fj8MNDhm7d3FetipMhu8cL4TNTV9fkQ2mmJR+9sOdeil7ks10T28iKKqls1wS4n3pqnAzulbPhzXZNgPujj8bLUSkb3r32Cjk+/3lteHM3vOPGxcvRig1v+yzyNTWVsaLuvFMtpFy5LaTVq+NkyN3wfuEL2vDmbnivuSZejtw93pgb3tNOSza8774bJ0PuhrcFe7zts8hXSgtJXRMJdU00VCkbXu3xJubPr7wNbwv2eNtfka/EFpK6JpINb8eOldE1oQ1v5ezxasPbUO6GtwV7vO2vyFfKiqq0rolKOSZw2mlxMrhXzoZXB+MT2vA2lLvhbeEeb/sq8pWyotQ10VClbHjVNZHQHm9DlXJMoA0b3vZV5CtlRWUP3qhronI2vJXSNaGD8Q1V2h7vNngwvv0U+UpZUZV48EajJiqna6LSDsaPHh0ng3vlDFfM3fDefnu8HG3c8LafIq+uiYbUNZFQ10RDlXYwvlI2vNvo9wTaR5FX10RDldI1oQ1vQ60cNVESlbLHW4kb3ph7vAVseNtHka+UFpJGTSS04W2oDaMmSkIb3oa2sVMY5FPSIg/8EHgLmA1cl3P/xcDfMtO+0pJ5FVTkU7CiiqYSW0iVsuFV14Q2vLkq5ZhAgRvekhV54HBgCtA5c7tP5u8w4A2gMzAEmAd03Nr82lzkU7KiiqZSWkg6JpDQhrchHRNIFOHcQc0V+UJ/NORc4JfuvilzRsvlmfvHAA+4+yZ3n59p0R9Q4LKatnw5DBgAHTrAj35UssVs1YYN8JnPQN++8K1vxcvhDttvD6NHQ1VVnAwbN8KOO4br3/sefPazcXK8/344rTLAhReG90gMS5fCrruG6xddFCcDhNNe9+kD3brB978fL0dtLXTtCl/4Anzta3Ey1NdDx47hcsop8U4pvGYN9OsXro8bB507F3X2FjYCbXyy2evAI8BXgY3ARe7+mpndDLzs7hMyj7sDeNzdH8wzj7HAWICBAweOXLhwYdvC1NbCyy/DoYe27fnF8tFH4ddkRo2Km+P998MvHsU6P3nW3Lnh/Pmxzk+e9eKLMHJk0T9ArVJfD889Fza+MW3cGH4d7ItfjJtj9WpYtAhGjIibY8GCsG6yG+FYpk8PG5qePVv9VDOrcfe8LbqtFnkzmwL0zTPpUuAa4FngfGAU8D/ArsDNwEuNivxkd/9jc8uqqqry6urq5v8bERFpoLkiv9VfhnL3o5qZ8bnAQ5k+oVfNrB7oBSwBBuQ8dBfgvValFhGRghXaQfkn4AgAM9sd2B5YCUwCTjOzzmY2BBgKvFrgskREpJUK/Y3XO4E7zWwWsBk4K9Oqn21mE4E5wBbgX929rsBliYhIKxVU5N19M3BGE9OuIfTZi4hIJJHGk4mISDmoyIuIpJiKvIhIiqnIi4ikWEHfeC02M1sBtPErr0AYo7+ySHG2dXotGtLrkdBr0VAaXo9B7t4734SKKvKFMrPqpr711d7otWhIr0dCr0VDaX891F0jIpJiKvIiIimWtiI/PnaACqLXoiG9Hgm9Fg2l+vVIVZ+8iIg0lLaWvIiI5FCRFxFJsVQUeTP7qpm9ZWZ/M7Ofxc4Tk5kNMLO/mNmbZjbbzM6PnSk2M+toZtPN7NHYWWIzsx5m9qCZzc28RyL/PFRcZnZB5nMyy8zuN7MusTMV2zZf5M2sI3AL8DXCD4h/08wi/+ZdVFuAC939C8BBwL+289cDwi+XvRk7RIW4CXjC3fcEhtOOXxcz2xkYB1S5+95AR+C0uKmKb5sv8oQfCP+bu7+bOfXxA4QfEm+X3H2Zu0/LXP+I8CHeOW6qeMxsF+A44PbYWWIzs+7Al4E7IJwq3N3Xxk0VXSdgBzPrBHQlhb9gl4YivzOwOOf2EtpxUctlZoOB/YBX4iaJ6kbgJ0B97CAVYFdgBXBXpvvqdjPrFjtULO6+FLgeWAQsA9a5+1NxUxVfGoq85bmv3Y8LNbMdgT8CP3L3D2PnicHMvg4sd/ea2FkqRCdgf+BWd98P2AC022NYZtaTsNc/BOgPdDOzvD+CtC1LQ5HXj4Y3YmbbEQr8ve7+UOw8ER0MnGBmCwjdeEeY2YS4kaJaAixx9+ye3YOEot9eHQXMd/cV7l4LPAR8KXKmoktDkX8NGGpmQ8xse8KBk0mRM0VjZkboc33T3X8bO09M7n6xu+/i7oMJ74tn3T11LbWWcvf3gcVmtkfmriMJv8PcXi0CDjKzrpnPzZGk8EB0oT/kHZ27bzGz84AnCUfH73T32ZFjxXQwcCYw08xez9x3ibtPjphJKscPgXszDaJ3ge9EzhONu79iZg8C0wij0qaTwlMc6LQGIiIplobuGhERaYKKvIhIiqnIi4ikmIq8iEiKqciLiKSYiryISIqpyIuIpNj/B5bn3CZtaLqmAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(T, X, U, V, headlength=0, headwidth=1, color='red') # draws the arrows at (X,Y) with slope dYdX"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ```plt.quiver``` function doesn't know what scale to draw the lines/arrows at and it basically makes an \n",
    "educated guess. However, we want to tell it to use the scale in the units of the plot. We do this by \n",
    "adding ```angles='xy', scale_units='xy', scale=1``` to the ```plt.quiver``` call: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(T, X, U, V, headlength=0, headwidth=1, color='red', angles='xy', scale_units='xy', scale=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ```angles='xy'``` tells ```np.quiver``` to draw the vectors with respect to the scale in the \n",
    "plot. (They use ```x``` and ```y``` instead of ```t``` and ```x``` respectivley; this is kind of confusing \n",
    "and annoying, but the conventions in ODE don't match with the conventions used in python; it's just something \n",
    "one has to get used to.) \n",
    "\n",
    "(For those more interested, the default choice ```np.quiver``` makes is bascially independent of the scale of the \n",
    "figure. For example, in the code below, I make a figure that goes from $0$ to $100$ in the vertical axis and \n",
    "$0$ to $10$ in the horizontal axis. The vector field is the constant field $\\ip{1}{1}$. Observe that these are drawn at actual $45$ degree angles, i.e. the $\\tan \\theta = 1$ . However, if using the scale of figure, they \n",
    "should be drawn so that $\\tan \\theta = 10$):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "## This code goes along with the explanation of axis='xy' directly above and is not part of the ODE direction \n",
    "## field!\n",
    "a = np.arange(0,10,1)\n",
    "b = np.arange(0,100, 10)\n",
    "A, B = np.meshgrid(a, b)\n",
    "C = D = np.ones(A.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(A, B, C, D, headlength=0, headwidth=1, color='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we tell ```np.quiver``` to determine the angle at which it draws the vector from the figure's scale, we \n",
    "get: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXAAAAEICAYAAABGaK+TAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjEsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+j8jraAAAgAElEQVR4nO3deZhcdZ3v8fc3ey/pJQRIAoSAYtgUFxDmwgVGXFh0uIogLtyo3CeKsriNoIgsombu4yCoPCKiqAOiXkRBBlcQnVEGMYCyREUghJCEbL0m6bV+949fnanqprq66vc73adP8nk9T56uruo69alT53zPqUonH3POISIi+TMt6wAiIhJGA1xEJKc0wEVEckoDXEQkpzTARURySgNcRCSnNMAlc2Z2nZldMkmP9ZiZHT8Zj1XP45nZEjNzZjZjjNsvM7ObUg8ouVZxYxFJi5mtBvYEhoBh4HHgO8D1zrkCgHPu/RP02N8C1jrnPpVc55w7ZIIe617gKPzzTLxuoh5PBHQGLpPjTc65ucC+wArgQuAbtdxxrDPSKepc51xz2Z/7sg4kOzcNcJk0zrku59wdwNuAZWZ2KPgzZTO7snj5eDNba2YXmtkG4EYzm2ZmF5nZk2a2xcx+YGbzkuWa2TFm9nsz6zSzZ83s3Wa2HHgn8HEz6zWznxR/drWZvbZ4ebaZXW1m64p/rjaz2aNyfNTMNprZejN7T73PedTjVX0eo+63n5n9xsx6zOyXwPx6H1t2fhrgMumcc38A1gL/c4wfWQDMw5+xLwfOB/4XcBywCOgArgUws8XAT4EvA7sDLwceds5dD9wM/N/i2fCbKjzOxfiPPV4OHAa8GvhU2e0LgFZgL+Bs4Fozaw971lDteVTwXWAlfnB/BlgW8biyk9IAl6ysww/pSgrApc65fufcDuB9wMXOubXOuX7gMuCtxY9X3gn8yjl3i3Nu0Dm3xTn3cI0Z3glc4Zzb6JzbBFwOnFV2+2Dx9kHn3F1AL7C0yvK+VHwX0GlmD1a4vdrz+G/Fg9IRwCXFdfBb4Cc1PifZheTp80XZuewFbB3jtk3Oub6y7/cFfmRmhbLrhvF/OboP8GRghkXAM2XfP1O8LrHFOVf+l5LbgeYqyzvfOXdDldurPY/RuTqcc9tGZdunyrJlF6QzcJl0ZnYEfoD/5xg/Mvq/yHwWOMk511b2Z45z7rnibS+qcTmjrcMP1cTi4nUTpdrzKLceaDezplHZREbQAJdJY2YtZvZG4HvATc65R2q863XAZ81s3+JydjezU4u33Qy81szOMLMZZrabmb28eNvzwP5VlnsL8Kni8uYDnwYm8netqz2P/+acewb4I3C5mc0ys2OASp/hyy5OA1wmw0/MrAd/BnoxcBVQz290XAPcAfyiuJz/Ao4EcM6tAU4GPor/SOZh/F9Igv9VxYOLn0n/uMJyr8QPyj8DjwAPFq+bKGM+jwreUbxtK3Ap/nfnRUYwFTqIiOSTzsBFRHJKA1xEJKc0wEVEckoDXEQkpyb1H/LMnz/fLVmyZDIfUkQk91auXLnZObf76OsndYAvWbKEP/7xj5P5kCIiuWdmz1S6Xh+hiIjklAa4iEhOaYCLiOSUBriISE5pgIuI5JQGuIhITuWz0ME52LYNOjqgs9N/reVyZyccfTR8//vp5CgUoKen/hwdHfDBD8Ill6STY3gYuroqP9/x1sd118Fb3pJOjsHB2h6z0u333w8HHJBOjr6++h47udzXBxs2pJPBOdi+PWx9HHAA3HNPejl6esJynHYafPnL6eQoFPw2GpLj8svh/e9PJ8fQkF9uSI4f/xiOOSadHAMD4z92+dcx5GeAn3oqrFpVekJDQ+Pfp5J1Ef9f/7ZtcNxxpRXe1eU3zBDPPx+e409/gmXLSi9wd3f4sjZvDr/vD38Il11WyrFt27h3GdPWscp5avD5z8O//dvIQRyqrw/mzAm773vfC7/7XSnH4GDYchoawu6XOOYYfyBKBtXwcNhyYg5mzz7r99lkf+3q8geTEJs2hef49a/hQx8q5ejpCV9WzDZ6/fXwpS+VcmzfHr6sMvkZ4E8/DU88Ufp+9mxob/d/2tpGfh3v+lANDfDggyM3xMbGsBzzxqqDrNGf/jTy+7lzw3Lstlt4hu3b4dFHS9+bQWvr5OfYssUf3BPTp49c9ng5yi/Pnh2eY+1a+NvfSt/PnDn+4411OcYTT8DGjaXvGxrGXg/VLse8JrNmwUMPjbyuuXnsx5yoHIOD8Oc/j7wu2UYnM0dXFzz2WOn7adOqP+boryeeWHGxk/r/gR9++OEu+F9iPvqof9LJEw09S4r1wAPQ0lJaubNmTX6GHTv8xpC8uK2tMCODY/Hmzf5MK8nR0uJfo8m2Zo0/q0lyNDf7g8lk+8tf/NlukqOhIZscDz00cmjHHJRCDQ35k51kf21t9Qe0ydbVBX//eylHS4s/wE+29ev9u+4kR3NzXfuKma10zh3+gutzM8BFRHZRYw1w/RaKiEhOaYCLiOSUBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE5pgIuI5FRNA9zMPmxmj5nZo2Z2i5nNMbN5ZvZLM3ui+DWyB0pEROoxbg+Xme0FnA8c7JzbYWY/AM4EDgbuds6tMLOLgIuACyc0bWJwsHqLc7W252OPhR/9KJ0cfX1hOTo64IIL4Ior4jMk7eeh6+Ob34TTT08nR0/P+I8/VqaHH4alS+NzDA/7kueQ9dHX5yu40jA46JcVsj6WLvXlyGno769tPVS67owz4Gtfi8/gnK8BDM3xuc/Bueemk6O3NzzHT3/q50esQsFvo/Vsn2OotUhxBtBgZoNAI7AO+ARwfPH2bwP3MpED/A1v8D2QnZ1x7ecxLezbtsERR5RWbH9/+LJiGq4fegjOPLP04oa2n4NfRqjvfx8uvriUo1AIX1aVjXRcV1wBN97oc3R3h7efQ1wr/TvfCf/xHz5Hb294hpgWdoBXvco3ynd0+MEZKmYbXbMGXv/60iAaGAhfVsw2+qtfwTnn+Bydnb6rM1TMNnrttfCv/+qX0dUVt6+UGXeAO+eeM7MvAGuAHcAvnHO/MLM9nXPriz+z3sz2qHR/M1sOLAdYvHhxeNING+C555KFhreft7WFZ2ho8MW1yYCot/28/PrW1vAc06fX1n5eS46WlvAcg4Pw5JOl7xsawtdHc3N4jp4eWL269H1zczat9EnJc6K1tb4m+uRyzDYKfj95/nl/OWk/n+wcc+bAX/9a+n7GjOqPWS1HzL7inC81TsyeHZ5j7tzwHDt2wNNPl75vbKwtR/L1uOMqLnbcUuPiZ9s/BN4GdAL/D7gV+Ipzrq3s5zqcc1U/B48qNX7iidJGkFX7OcAjj5QOHlm1n/f1+cGZdft5R4cfFGkMvxjr1/sz3mToZNF+Dn4Hda40dLJoPwdYtao0IOpsP0/N8DA8/nhpMDU2ZrON9vb6dwNJjtB3V7E2bfLvaJKhPGtWXXcPbqU3s9OBE51zZxe//9/AUcAJwPHFs++FwL3OuaofYqqVXkSkfjGt9GuAo8ys0cwMP7hXAXcAy4o/swy4Pa2wIiIyvlo+A7/fzG4FHgSGgIeA64Fm4AdmdjZ+yKfwqwwiIlKrmn4LxTl3KXDpqKv78WfjIiKSAf1LTBGRnNIAFxHJKQ1wEZGc0gAXEckpDXARkZzSABcRySkNcBGRnNIAFxHJKQ1wEZGc0gAXEckpDXARkZzSABcRySkNcBGRnNr1BvjwcNYJfGuLcpQUCnE9lmmZCusClKPcVNlGnUutxzJNtZYaTx3V2s9raWE/4QT4yU/ic8S0n3d2wj//M1x5ZXyOau3nteT47nd9+3ismPbzri5fAxbbSj9e+/l462NgwNfVxYptPz/kEPjDH+Jz1NN+Xum6d70LbrghPsfwcKlUOCTHVVfBeefF5xgcDN9POjvhnnvG7KasS19f/TnGkJ8Bftxx8Oij8e3nXV3h9+3thZe+NJ3285gcK1fCm98c334em+Pmm+HjH49vP4e4xu9LLoGvfz2+/Rz882hoCLvvW98Kv/lNfPt5zGsCcPDBvgQ8tv28uzv8vs88A8ce69dFzHIgbn38/Odw9tk+x7ZtcTlinsc118CKFX4b7e+Py1EmPwO8q8uXgoLfwcZq0Z7I9vPGRr9hJoO7uTk8R+iQAF+IOrr9PCRHW1tcyatzsG6dv5y0n9eTo/z2mCLi/v5SC3u19vPxcsQWEff0+GZ68Ou1lrbxtNvPwe8nHR3+clNTeI6mpvAMDQ2+TDjR0hKeI2ZfmTYNnnvOXzZ74XZQT446i4hHGBryB1Xw29h4bfSjLx9xRMXFjltqnKaoUuNnn/UrsK0tu/Zz8G3wra0+x4yMjn8DA36jbGvzO0ZW7efJ2/Nk6GTROg5+aPb3+xxZtZ+DP5glB7Ks2s8BVq/2w7e1NW7oxCgU4Omn/fBpacluX9m+HTZu9DnmzvWvTxaSd8vt7f61qXMbDW6lT5Na6UVE6hfTSi8iIlOQBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE5pgIuI5JQGuIhITtU0wM2szcxuNbO/mNkqM/sHM5tnZr80syeKX9snOmy0QsE3dGRtaCid8txY/f3xHZKxnPOvSdbN487FdyamYapso8PD8T2naRgcTLVDMlhfX1zP6QSptefoGuBnzrm3mtksoBH4JHC3c26FmV0EXARcOEE5S8rbz2tplC6/3N0NJ58Md94Zl2F0+3m9OXp74ZOfhM9+Nj5Hb299Ddvll3fsgO99D972trgcSft5rQ3fldrgV62CAw+My5G0n9fz2OWt42Z++4qtZEvaz0PWR1cXvPKV8MADcRmg1H4ekqO7G977XvjGN+IyJAfokP0kKSK+5ho4//z4HD094Tn6+uDXv4bjj4/LMTxc2lfGey3Kv45h3AFuZi3AscC7/XpwA8CAmZ0KJM/m28C9TOQAP/JIeOSR+LOCmKN5by+8+MXptZ+HeuABOOkk/8LGnLlOmxZ3tved78AFF/ihE1PNN2OG37lCXXghfPWrccsA32PZ1xdeovumN8E998SfQceece6/v+/njF1OzP1Xr4ZXv9pvo4OD4csxi9tX7roLzjrL5ygUwpczfXrcO7QvfAGuvNIP7xRrLGs5A98f2ATcaGaHASuBC4A9nXPrAZxz681sj0p3NrPlwHKAxYsXhyft7/cv5PTp4zdLV2ucjilXbWyETZv8hjBzZm3N0pWujy0injMHtmwpXa43R3n7eczZ5vTppbODpqbwHLFFxMnZFfh1G5ojtoh4cNAP76T9fLyW80qX0ygi3rHD7y9J+3lIjtgi4qYmv6+Afz7V1nu1HLFFxLNmwdat/nJj4/jPfazbA4qIRzDzJzrgn1OtOZKvhx5aebHjlRqb2eHAfwFHO+fuN7NrgG7gPOdcW9nPdTjnqn4OHlVq/Pzz/swoy/Zz8Gc2ra3Ztp8PDvoBnnX7+bZtfmBl2X4OpbOrLNvPATZv9gf2LNvPATZsgObm+KETo1DwOdrbw9/RpKGvzw/OtjaYPTu7HD09/qAaeBI5VqlxLUtaC6x1zt1f/P5W/Ofdz5vZwuLZ90JgY92p6rHnnhO6+JotWpR1Aj8kFizIOoUfEE1NWafwO8VUMH9+1gm8qbBtTJs2NfaVOXOyPclJzJ3r/6Rs3NME59wG4FkzW1q86gTgceAOYFnxumXA7amnExGRMdV6Ln8ecHPxN1CeAt6DH/4/MLOzgTXA6RMTUUREKqlpgDvnHgZe8PkL/mxcREQyoH+JKSKSUxrgIiI5pQEuIpJTGuAiIjmlAS4iklMa4CIiOaUBLiKSUxrgIiI5pQEuIpJTGuAiIjmlAS4iklMa4CIiOaUBLiKSU7vOAE+KiJNaoyxz9PT4fs0sFQqlstYsDQ/7dqGsG78HB32jTtaSIuIsOefblrq7s8/R3R3fMRqrUEinB3cCZNg/FSBpPw9tgx8YgFNOiW+lr9R+XmuOzk4/rNJopa/Ufl5rjq4uvz7TaKWPbT+H+Fb6etvPR1+3bZtvOoptpa+3/Xz0dX19cPjh8a30tbSfV8sxOJhOK/3goN/WQtvgC4V0Wun7+8PnRlJEHNtKn5xEhuQYQ34G+Mtf7lvpY5qlZ8yIa4Tu7fU1UWm0n8eccd5/P7zmNfFnJk1Ncc3j3/wmfOAD8e3nLS1xjd8f+Qh85Svx7efNzXGt9K9/Pdx9d3z7eayFC32HbMy2PmtW3P2ffhpe9rL4d5oNDXGv6513whlnxDXbQ2nbCLViBVx6aepn8fkZ4GZ+xwhpP0++xhYRNzaWBk1I+3nyNbajr6mptvbzajna2uKLiGfPrq39vFqO1tb4IuKZM/1OXt5+Xm8rfRpFxNOm+W20oSG8DT6tImLnRraf15tjzpy4HM3Nfnib+dc4pA0+jSLiOXP88J42rbbnPtZ1sdvorFl+eM+cWV+O5OuLXlRxseO20qcpqpW+s9MP0CzbzwG2bs2+/XxoyB9Ism4/7+vzWbJsPwe/LqZNix86sbq7/cDJsv0c/L7S3JztNloo+I9OWlrSeVcRamDAb6dz52a7bWzf7g+qgSeRMa30U8NUaR6fNy/rBH7HbG3NOsXUaPsGfwCZClpask7gTYV9JTnjzdqsWdmf9IEf3BNg1/ktFBGRnYwGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhO1TzAzWy6mT1kZncWv59nZr80syeKX6fA/95exfCwb9PZuDHbHIODPkPWzeN9fbB+fVwXZayk/Xzt2rjewzRydHf7HFlK2s83bMg2x9AQbN4MW7Zkm2NgwHd7JsXXWUiKiNeti+vEnCD1NPJcAKwCktqRi4C7nXMrzOyi4vcXppxvpB07Kjdp19LInmwEJ58M//7v4RmSoROaIxmYn/gEfO5z4TkKhVL7eUiOZGO85RY488zwHMPDpdbxWh6/Uvs5wOOPw0EHhecYHBx/PVRbH4WCbzoaGIir3urvH/s5j7c+kvbzV70KQqsHoTR0QnMkRcTveY8vro7J0dtbX47yy0kR8dVXwwUXhOcoFPy6Dc2RFBHfcw/84z+G5xga8ssM2T7HUNMAN7O9gVOAzwIfKV59KnB88fK3gXuZyAG+dCn87W9xy2hpiesr7O31NVExjfJJEXFMl+V998Exx8S3n7e3+wEc6vrr4X3vC78/lIqIY1rDzz0Xrr02LkdSRBzTSn/ccfDb38blaG4Of/xEe7s/qMZobfUFvKGeegpe8pK47WvatPh6uNtvhze/2R9MQs2Y4ddpzLvEz3wGPv3p8PuPodYz8KuBjwNzy67b0zm3HsA5t97M9qh0RzNbDiwHWLx4cXjSxsbSihyrQXqi28+bmvzQnD07PEdLS3wR8dy5PkdjY1iOtNrPm5tLeaq1a1fLkUYRcfJcWlvDX5c0iogbG/2BcbyG84lsP09ybN9eX+t5+eU0iohbWvzwbmgIXx9pFBE3N/vh3dwcniOwiHiEpLe1tbW2HKMvL1xYcbHjttKb2RuBk51zHzCz44GPOefeaGadzrm2sp/rcM5V/Rw8qpW+r8/vZFk2Syc5si7zLRT8u4Csy1oHB/3rkWX7Ofi3uDNmxB8YY/X3+9dkKmyjWe8rzvnXJY0DY4yhIZ8l5t1EGgYG/EEx8MAY00p/NPBPZnYyMAdoMbObgOfNbGHx7HshMLF/O5j10ExMhRzTpmU/vCH7nSIxFdYFZD+sElNhGzWbGusj65OLxARto+OesjjnPuGc29s5twQ4E7jHOfcu4A5gWfHHlgG3T0hCERGpKOY95wrgdWb2BPC64vciIjJJ6np/4Zy7F//bJjjntgAnpB9JRERqoX+JKSKSUxrgIiI5pQEuIpJTGuAiIjmlAS4iklMa4CIiOaUBLiKSUxrgIiI5pQEuIpJTGuAiIjmlAS4iklMa4CIiOaUBLiKSU1PkfzufYEkRcVIOuvfe2eQoLyJubIQ9KrbQTbzyIuLdd/f1V1koLyJesiS7YoakiLi72/c4ZqG8iHhw0K+PrHIkRcQzZ45Z5TXhyouIW1th3rxscgwNlfaVRYv8fjuF5GeAV2o/r+dyUkR80klw113hOSq1n9d6uaurVER80UXw+c+H5+jrq631vdLl7u7Scr77XXj728MyOOe7F+tZD+XXbdtWWtZjj8HBB4fn6OkJz9HX55cT20pfKPjXODRHUpr7ylfCypVhGeCF7ef15OjsLBURx7bSDwyE769dXaUi4i9+ET70ofAcyYExJEdPT2k5Ma30yUlkva9FGq30U8IBB8DTT4ffP2k/b20NX0Zvry9ajZG0n8fUXv3ud76VPkZzs88R0yN53XXwgQ/E5UiKiAcGwpdxzjnwta+F3z9pP49tpT/mGLjvvvAcSWl3e9Vq2fG1tvoDa6g5c/z6SIp4Qzz5JLz4xeH3B//4bW1x78xuuw1OOy0uR0uLz5GcfIW47DK44oq4HBXkZ4C3t8PmzbU1bVe6Lq3285kz/deQHGm1n7e1jWw/D8mRRldgW5tfH2M1zo+XI4328yRHeft5vTnSaD9PcpS3n1drGa/Uhp5G+zn45SWvy3ht55VypNGp2dY28sBYz+MnJ1ppfKSWbOshr0d7u99G09pXZs8Oz9HWVnGx47bSpymqlb5QyL51fKrkSF6zrNvPCwWfYSrkyPo1UY6RnPN/ss6xk2yjMa30U0PWG0JiKuTIemNMTIV1Acox2lTIMRWGJkyNdQETlmOKPDsREamXBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE5pgIuI5JQGuIhITo1b6GBm+wDfARYABeB659w1ZjYP+D6wBFgNnOGc65i4qBGSIuLOTl/htf/+2eRI2s+Tpu1FiyY/Q3n7eUcH7LVXfAdjaI6k/byjAw48MJ26uXqVt5/39MDLXjb5GWBk+/nwMCxdmk2OgYFSmW5DAyxenE2OHTtKOebPhz32mPwMSRFxkmO//Xxl3mSr0sVZSyPPEPBR59yDZjYXWGlmvwTeDdztnFthZhcBFwEXphC3su3b62+UrtR+HtNKP3ro1JsjaT+HuFb6au3nteRI2s8hrpU+rfZziGulD2k/T76Wt5/HttKHtJ8nX8vbz2Nb6ZNtNCRHeRnyu98NN94YlsG50oExpA2+v7+0rJhW+uHh0oExJMfQUGlZd98Nr3lNWI7kJDIkR1fXmIsdd4A759YD64uXe8xsFbAXcCpwfPHHvg3cy0QO8EMPDW+lNyu1n8ccybdt8yWnocqLiGOWc999ca305UXEMcWxX/96XCv9nDmlHOU7Sr3OPz+ulb68iLi/P7zU94QT4lrpW1t9jth3ZnvuGd5KnxQRt7XBbruFZ3jqqbhW+qSIOCl7DnX77XGt9EkRcVLSHOrKK7NvpTezJcArgPuBPYvDHefcejOrOBnNbDmwHGBxzNuxBQv8AK21dXx0s3QanXRNTb7BvLk5LEdzczo9gfPm1dZ+PlaOhob0ciRDp94cabWfg3+LPW9eeI6ZM9PJsfvu/k89rfTJ5dZWf4BPw4IFfoCH5Jg7N519JRm89Tx2+eWmpnS20fZ2/5zqbYNPLjc0xGcAfzCsZz2Uf62yr9TcSm9mzcBvgM86524zs07nXFvZ7R3Ouaofpka10ouI7KLGaqWv6VBrZjOBHwI3O+duK179vJktLN6+ENiYVlgRERnfuAPczAz4BrDKOXdV2U13AMuKl5cBt6cfT0RExlLLZ+BHA2cBj5jZw8XrPgmsAH5gZmcDa4DTJyaiiIhUUstvofwnMNbfJpyQbhwREamV/iWmiEhOaYCLiOSUBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE5pgIuI5JQGuIhITmmAi4jklAa4iEhO1VWplkvlRcSdnb4+66CDJj9Heft5R4evWNp338nPUd5+3tkJS5b4SrLJVt5+3tEBhx2WXn1VPcrbz3t64MgjJz/D6PZzgJe+dPJzFAp+HSQ5mpvjei1DJUXESY5Fi2DhwsnPkRQRJzkOPDCuyzZUecHzKPkZ4Fu2wNat4zedV7qtvP38xBPhpz8Ny1AowKZNtTWuj76tvP0c4MILYcWKsByDg6Uc1R6/Uo7y9nOAm2+Gd7wjLEdfX205xms/B3j0UTjkkLAcvb2weXNYjvKdY/p0v25Duxg7Ovw2GpKjvNT5Fa+ABx8MywCwcWPYftLZ6bfxxLJl8K1vhWUYHg7L0dHhT3TKXXUVfPjDYTkGBsLXR2/vyGX96le+uDrE9u1+XwlZH319Yy42PwP8iCPCW+nL289jipW3b/eFsaGamko5Ys56//CHuFb68iLimMbvG28Mb6WfNs3nSNZHjd2sFX3sY+Gt9OXt57Gt9KecEt5KX95+vv/+YctI7LdfeCt9eRHxokXhGVavjjt7T4qIk8LnUHfeGd5Kb1baRmPLr//lX8Jb6adPH3kSWiY/A3zJEr+TV2uQnuj286Ym2HPPUit9vTnSaj/fbTfYY4/KDdbj5Uiz/XyPPfxb25AcabWfgx80e+8dliOt9nOAffaBdevGfvzJaD8HfwAob6WvJ8esWelkSE5Sqj3WWLe1tvoDaxrmz/f7bEiOlpb0ttEFC/x2WmuO8svNzWPmqLmVPg1qpRcRqV9UK72IiEw9GuAiIjmlAS4iklMa4CIiOaUBLiKSUxrgIiI5pQEuIpJTGuAiIjmlAS4iklMa4CIiOaUBLiKSUxrgIiI5pQEuIpJTGuAiIjmlAS4iklMa4CIiOaUBLiKSU1G9RWZ2InANMB24wTkX2NI7gfr7R5aGNjT4BvTJ5NzI9vOODl9F9qIXTX6O3t6R62PpUl85NZkKBV9cW74+jnj72EIAAAWASURBVDoqrp8zxPDwyNbx3l44/vjJzQC+SLm8yHbaNN8BO9n6+kbmaGuDgw+e3AzO+Tq48hxLlvi6usnO0dMzMsdhh/mKs8mU7CtjCB7gZjYduBZ4HbAWeMDM7nDOPR66zKqee84301drb650eceOkcuJbaVfs6b2xy6/PDAwclkxrfT9/fDss9Ubritd7uwc2X4Oca30vb3+dalnPXR2QlfXyPZziGul37oVNmyoP8foHSO2lX7DhrGbx6vl2LZt5HJiW+mfeab+7bOz84Xt5zGt9ENDpX2l3hyDgyOXFdNKv2MHrF0blmP0NhrTSt/VBevX1/7Y5ftKldrLmDPwVwN/d849BWBm3wNOBSZmgB97LDz1VP33MxtZFHrAAeEZduzwjd8hZswYmWPhwvAcK1fC0UeH3XfWrJHFvi0t4TluugnOOSfsvuXt5+3tccXCF18M110Xdt/y9vPYVvrTToPf/77++41uP4896z3kkBceFGoxffrIct0lS8IzrFkT/g5z5syRxb7z54fn+NnP4C1vCbvvnDkjc8SUo3/xi3D55WH3bW72J0sVxAzwvYBny75fCxw5+ofMbDmwHGDx4sXhj/aSl8Ds2dm2nzc2+o06pJU+zfbz+fNh333DWtjTbD9fsMA3oIfkSKv9HGDxYn9grjdHmu3n4NfF1q3Ztp8DHHSQP9moN0dzc3rbaHu7/9gjpA2+oSG9HLvv7vfZenPEDuzR9trLH9DqbaVva/MHtDHWR3ArvZmdDrzBOfd/it+fBbzaOXfeWPdRK72ISP0mopV+LVD+Nwt7A+siliciInWIGeAPAAeY2X5mNgs4E7gjnVgiIjKe4A8AnXNDZnYu8HP8rxF+0zn3WGrJRESkqqi/wXHO3QXclVIWERGpg/4lpohITmmAi4jklAa4iEhOaYCLiORU8D/kCXows03AM4F3nw9sTjFO3ml9lGhdjKT1MdLOsD72dc7tPvrKSR3gMczsj5X+JdKuSuujROtiJK2PkXbm9aGPUEREckoDXEQkp/I0wK/POsAUo/VRonUxktbHSDvt+sjNZ+AiIjJSns7ARUSkjAa4iEhO5WKAm9mJZvZXM/u7mV2UdZ6smNk+ZvZrM1tlZo+Z2QVZZ5oKzGy6mT1kZndmnSVrZtZmZrea2V+K28k/ZJ0pK2b24eJ+8qiZ3WJmKVbsTA1TfoCXlSefBBwMvN3MJrkqe8oYAj7qnDsIOAr44C68LspdAKzKOsQUcQ3wM+fcgcBh7KLrxcz2As4HDnfOHYr/L6/PzDZV+qb8AKesPNk5NwAk5cm7HOfceufcg8XLPfidc69sU2XLzPYGTgFuyDpL1sysBTgW+AaAc27AOdeZbapMzQAazGwG0MhO2BiWhwFeqTx5lx5aAGa2BHgFcH+2STJ3NfBxoJB1kClgf2ATcGPxI6UbzKwp61BZcM49B3wBWAOsB7qcc7/INlX68jDAK9Ux79K/+2hmzcAPgQ8557qzzpMVM3sjsNE5tzLrLFPEDOCVwFedc68AtgG75N8ZmVk7/p36fsAioMnM3pVtqvTlYYCrPLmMmc3ED++bnXO3ZZ0nY0cD/2Rmq/Efrb3GzG7KNlKm1gJrnXPJu7Jb8QN9V/Ra4Gnn3Cbn3CBwG/A/Ms6UujwMcJUnF5mZ4T/fXOWcuyrrPFlzzn3CObe3c24Jfru4xzm3051l1co5twF41syWFq86AXg8w0hZWgMcZWaNxf3mBHbCv9CN6sScDCpPHuFo4CzgETN7uHjdJ4vdpCIA5wE3F092ngLek3GeTDjn7jezW4EH8b+99RA74T+p1z+lFxHJqTx8hCIiIhVogIuI5JQGuIhITmmAi4jklAa4iEhOaYCLiOSUBriISE79f7495nd2dQDkAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(A, B, C, D, headlength=0, headwidth=1, color='red', angles='xy')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ```scale_units='xy', scale=1``` tells ```np.quiver``` by how much to scale the line segments (that's the \n",
    "```scale=1``` part) and which units to use (that's the ```scale_units='xy'```part). If you don't give it \n",
    "```scale_units``` then ```np.quiver``` first chooses to make all arrows unit length, and then picks a \n",
    "```scale``` factor based on the data you give it (i.e. the vector field) to make it look the best. By giving \n",
    "it ```scale_units``` and a ```scale```, you are telling ```np.quiver``` to scale the vectors using the \n",
    "units of the figure. So, in our little side example above this gives: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(A, B, C, D, headlength=0, headwidth=1, color='red', angles='xy', scale_units='xy', scale=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The bigger ```scale``` is, the shorter the line segments, and the smaller ```scale``` is, the shorter: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Scale = .5\n",
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(A, B, C, D, headlength=0, headwidth=1, color='red', angles='xy', scale_units='xy', scale=.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# scale = 2\n",
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(A, B, C, D, headlength=0, headwidth=1, color='red', angles='xy', scale_units='xy', scale=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we are (mainly) intersted in the *direction* of the line segments and not their lengths we pick the \n",
    "```scale``` parameter to give us the \"best\" picture. The fact that our primary concern is with direction and \n",
    "not lengths is manifested in the fact that we normalize the vectors prior to drawing them. Then the \n",
    "```angles='xy', scale_units='xy', scale=1``` part let's us scale all the vectors simulteanously. \n",
    "\n",
    "Now let's get back to the direction field; all the code is copied again below for convinience: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(0,10,1)\n",
    "x = np.arange(-60,-40,5)\n",
    "T, X = np.meshgrid(t, x)\n",
    "dXdT = f(T,X)\n",
    "U = (1 / (1 + dXdT**2)**.5)*np.ones(T.shape)\n",
    "V = (1 / (1 + dXdT**2)**.5)*dXdT\n",
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(T, X, U, V, headlength=0, headwidth=1, color='red', angles='xy', scale_units='xy', scale=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Observe the qualitative difference between the line segments along the $v=-50$ line and the ones along the \n",
    "$v= -46$ line: their slopes have opposite sign. Given a continuous vector field, there should be some $c$ for \n",
    "which the line segments on $v=c$ have zero slope. We know from class this happens when $f(t,v)=0$, that is \n",
    "$v=-49$. We want to add this to our plot. The following code does that:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(0,10,1)\n",
    "x = np.arange(-60,-40,5)\n",
    "x = np.append(x,-49)\n",
    "x.sort()\n",
    "T, X = np.meshgrid(t, x)\n",
    "dXdT = f(T,X)\n",
    "U = (1 / (1 + dXdT**2)**.5)*np.ones(T.shape)\n",
    "V = (1 / (1 + dXdT**2)**.5)*dXdT\n",
    "plt.figure()\n",
    "plt.title('Direction Field')\n",
    "Q = plt.quiver(T, X, U, V, headlength=0, headwidth=1, color='red', angles='xy', scale_units='xy', scale=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ```np.append(x,49)``` expression creates a new array that is just ```x``` with ```-49``` appeneded to it. It \n",
    "doesn't change ```x``` so we have to assign this back to ```x``` which is why I wrote \n",
    "```x = np.arange(-60,-40,5)```. This just adds ```-49``` to the end of ```x```. It isn't necessary to get \n",
    "the plot we want, but the ```x.sort()``` line sorts the ```x``` array. This is good thing to do because we are \n",
    "going to be thinking of ```x``` as being sorted, and it's possible that it will cause problems later (though, \n",
    "not actually for today's example) if it isn't. In general, it's best that your variables have the properties that \n",
    "you think they have. \n",
    "\n",
    "Observe that our scale makes the line segments at $v=-49$ look like a line. Also, an optical illusion makes it \n",
    "look slanted. We will change the scale below. We will also add some to the ```x``` array to sample a few more \n",
    "$v$ values. We also recenter the vertical axis so that the $v=-49$ line is (more or less) in the middle. We also \n",
    "add some labels to the axes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.arange(0,10,1)\n",
    "x = np.arange(-60,-39,2)\n",
    "x = np.append(x,-49)\n",
    "x.sort()\n",
    "T, X = np.meshgrid(t, x)\n",
    "dXdT = f(T,X)\n",
    "U = (1 / (1 + dXdT**2)**.5)*np.ones(T.shape)\n",
    "V = (1 / (1 + dXdT**2)**.5)*dXdT\n",
    "plt.figure()\n",
    "plt.title('Direction Field for dv/dt = -9.8 - v/5')\n",
    "plt.xlabel('Time In Seconds')\n",
    "plt.ylabel('Velocity')\n",
    "Q = plt.quiver(T, X, U, V,  angles='xy', headlength=0, headwidth=1, scale_units='xy', scale=1.5, color='red')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is all we will do with the plot. I made several \"style\" decisions that you might like to do differently \n",
    "(e.g., maybe you like blue vector fields or maybe you wanted to keep the arrowheads.) The \"correct\" choices to \n",
    "make are the choices that convey the information in the best way."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Field Mice \n",
    "\n",
    "Let's do a direction field using the field mouse example. First, let's define the $f(t,p)$ for the ODE. Recall \n",
    "that the equation is: \n",
    "$$\n",
    "\\frac{dp}{dt} \n",
    "= \\frac{p}{2} - 450.\n",
    "$$\n",
    "\n",
    "So we define our ```f(t,x)``` by: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(t,x):\n",
    "    return x/2 - 450"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To set the range of $t$ and $p$ values (in python this will be ```t``` and ```x``` values) note that we want to \n",
    "model things going forward in time, so we want ```t``` to be positive. Also, since the units of time is months, \n",
    "we probably want ```t``` to go through a year or so. We know that $p\\quiv 900$ is the equilibrium solution, so \n",
    "we want to include that in the ```x``` range."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Field Mouse Population')"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "## t is time measured in months \n",
    "## x is field mouse population \n",
    "t = np.arange(0,13,1)\n",
    "x = np.arange(800,1000,20)\n",
    "\n",
    "\n",
    "T,X = np.meshgrid(t,x)\n",
    "dXdT = f(T,X)\n",
    "#U = 1/(1+dXdT**2)**0.5*np.ones(T.shape) # Normalizes the arrows to see near-zero slopes.\n",
    "#V = 1/(1+dXdT**2)**0.5*dXdT\n",
    "U = np.ones(T.shape) # Normalizes the arrows to see near-zero slopes.\n",
    "V = dXdT\n",
    "plt.figure()\n",
    "plt.title('Direction Field for dp/dt = p/2 - 450')\n",
    "Q = plt.quiver(T, X, U, V, angles='xy', scale_units='xy', scale=3, headlength=0, headwidth=1, color='red') \n",
    "plt.xlabel('Time In Months')\n",
    "plt.ylabel('Field Mouse Population')"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}