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    "# The Order of Partial Differentiation\n",
    "\n",
    "For a sufficiently smooth function, mixed partial derivatives\n",
    "commute.  We illustrate Clairaut's theorem with\n",
    "\n",
    "$$\n",
    "f(x,y,z)=\\frac{x^5y^3}{z},\n",
    "$$\n",
    "\n",
    "differentiating once with respect to each variable in several\n",
    "orders.\n"
   ]
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   "source": [
    "## Define the function\n",
    "\n",
    "The explicit type annotation makes the symbolic domain clear to\n",
    "the Egison kernel.\n"
   ]
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   "source": [
    "declare symbol x, y, z : MathValue\n",
    "\n",
    "def f (x : MathValue) (y : MathValue) (z : MathValue) : MathValue :=\n",
    "  x^5 * y^3 / z\n"
   ]
  },
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   "execution_count": 2,
   "id": "3f7c13e8",
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     "data": {
      "text/html": [
       "$x^{5} y^{3} z^{-1}$"
      ]
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   "source": [
    "f x y z\n"
   ]
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   "id": "d03c8ce4",
   "metadata": {},
   "source": [
    "## A first mixed derivative\n",
    "\n",
    "Differentiating in the order $x$, then $y$, then $z$ should give\n",
    "\n",
    "$$\n",
    "\\partial_z\\partial_y\\partial_x f\n",
    "  =-\\frac{15x^4y^2}{z^2}.\n",
    "$$\n"
   ]
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   "execution_count": 3,
   "id": "752fe2ba",
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       "$-15 x^{4} y^{2} z^{-2}$"
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   "source": [
    "∂/∂ (∂/∂ (∂/∂ (f x y z) x) y) z\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed4e02d6",
   "metadata": {},
   "source": [
    "## Permuting the order\n",
    "\n",
    "We repeat the calculation with $z,y,x$ and $y,z,x$.  Equality of\n",
    "all three outputs is the computational form of Clairaut's theorem\n",
    "on the region $z\\ne0$.\n"
   ]
  },
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   "execution_count": 4,
   "id": "ac6a3511",
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       "$-15 x^{4} y^{2} z^{-2}$"
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   "source": [
    "∂/∂ (∂/∂ (∂/∂ (f x y z) z) y) x\n"
   ]
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   "execution_count": 5,
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       "$-15 x^{4} y^{2} z^{-2}$"
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   "source": [
    "∂/∂ (∂/∂ (∂/∂ (f x y z) y) z) x\n"
   ]
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   "id": "a49bb78e",
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   "source": [
    "Every order produces the same rational expression.  The example\n",
    "also shows that Egison keeps the variables of differentiation\n",
    "explicit instead of encoding the order in auxiliary function names.\n"
   ]
  }
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