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   "source": [
    "# PDE python exercise 2"
   ]
  },
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   "metadata": {},
   "source": [
    "In this exercise we will focus on work the manipulation of arrays and 3D plotting. We will use a simple mathematical expression as a background for the development of our python code."
   ]
  },
  {
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   "source": [
    "First we import our mandatory modules"
   ]
  },
  {
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import axes3d"
   ]
  },
  {
   "attachments": {},
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   "source": [
    "In this exercise we will use a function $F$ which is dependent upon two variables $x$ and $y$. This equation reads:\n",
    "$$ F = x^2*y^2+\\sin(x)+\\cos(y)$$"
   ]
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   "source": [
    "In this problem, you need to:\n",
    "1. $x$ and $y$ must be defined using a `numpy meshgrid` in the range 0 to 1\n",
    "2. Define a function which computes $F$ with both $x$ and $y$ as input\n",
    "3. Define a function which computes the partial derivative of $F$ with respect to $x$\n",
    "4. Define a function which computes the partial derivative of $F$ with respect to $y$\n",
    "5. Plot the resulting function $F$ and its derivatives $\\frac{\\partial F}{\\partial x}$ and $\\frac{\\partial F}{\\partial y}$ using a 3D plot\n",
    "6. Plot the resulting function $F$ and its derivatives $\\frac{\\partial F}{\\partial x}$ and $\\frac{\\partial F}{\\partial y}$ using a 2D contour plot\n",
    "\n",
    "In this python exercise, $F$ and its derivatives have to be computed for a given range of $x$ and $y$.\n",
    "The function must be evaluated as:\n",
    "´F = func(X,Y)´\n",
    "\n",
    "Without using a `for` loop:\n",
    "\n",
    "7. Compute the partial derivates of $F$ with respect to both $x$ and $y$ (1st derivative) using a backward difference.\n",
    "8. Compute the partial derivates of $F$ with respect to both $x$ and $y$ (1st derivative) using a forward difference."
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