two_var_taylor.ipynb

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      "text/latex": [
       "$\\displaystyle - 2 x + \\frac{9 y^{2} \\left(x + 5\\right)}{100} + \\frac{9 y^{2}}{20}$"
      ],
      "text/plain": [
       "-2*x + 9*y**2*(x + 5)/100 + 9*y**2/20"
      ]
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     "execution_count": 10,
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   "source": [
    "def Taylor_polynomial_sympy(function_expression, variable_list, evaluation_point, degree):\n",
    "    \"\"\"\n",
    "    Mathematical formulation reference:\n",
    "    https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Multivariable_Calculus/3%3A_Topics_in_Partial_Derivatives/Taylor__Polynomials_of_Functions_of_Two_Variables\n",
    "    :param function_expression: Sympy expression of the function\n",
    "    :param variable_list: list. All variables to be approximated (to be \"Taylorized\")\n",
    "    :param evaluation_point: list. Coordinates, where the function will be expressed\n",
    "    :param degree: int. Total degree of the Taylor polynomial\n",
    "    :return: Returns a Sympy expression of the Taylor series up to a given degree, of a given multivariate expression, approximated as a multivariate polynomial evaluated at the evaluation_point\n",
    "    \"\"\"\n",
    "    from sympy import factorial, Matrix, prod, fraction, poly\n",
    "    import itertools\n",
    "\n",
    "    n_var = len(variable_list)\n",
    "    point_coordinates = [(i, j) for i, j in (zip(variable_list, evaluation_point))]  # list of tuples with variables and their evaluation_point coordinates, to later perform substitution\n",
    "\n",
    "    deriv_orders = list(itertools.product(range(degree + 1), repeat=n_var))  # list with exponentials of the partial derivatives\n",
    "    deriv_orders = [deriv_orders[i] for i in range(len(deriv_orders)) if sum(deriv_orders[i]) <= degree]  # Discarding some higher-order terms\n",
    "    n_terms = len(deriv_orders)\n",
    "    deriv_orders_as_input = [list(sum(list(zip(variable_list, deriv_orders[i])), ())) for i in range(n_terms)]  # Individual degree of each partial derivative, of each term\n",
    "\n",
    "    polynomial = 0\n",
    "    for i in range(n_terms):\n",
    "        partial_derivatives_at_point = function_expression.diff(*deriv_orders_as_input[i]).subs(point_coordinates)  # e.g. df/(dx*dy**2)\n",
    "        denominator = prod([factorial(j) for j in deriv_orders[i]])  # e.g. (1! * 2!)\n",
    "        distances_powered = prod([(Matrix(variable_list) - Matrix(evaluation_point))[j] ** deriv_orders[i][j] for j in range(n_var)])  # e.g. (x-x0)*(y-y0)**2\n",
    "        polynomial += partial_derivatives_at_point / denominator * distances_powered\n",
    "    return polynomial\n"
   ]
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   "execution_count": 11,
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 2 x + \\frac{9 y^{2} \\left(x + 5\\right)}{100} + \\frac{9 y^{2}}{20}$"
      ],
      "text/plain": [
       "-2*x + 9*y**2*(x + 5)/100 + 9*y**2/20"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import Symbol, exp, sin, cos, sqrt\n",
    "\n",
    "x = Symbol('x')\n",
    "y = Symbol('y')\n",
    "\n",
    "#input function here\n",
    "f = sqrt(4*x**2+ 9*y**2)\n",
    "#instead of e^x use exp(x), for sin and cos simply use sin(x)/cos(x)\n",
    "#hope this helps ^-^\n",
    "\n",
    "Taylor_polynomial_sympy(f,[x,y],(-5,0),3)"
   ]
  }
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