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  {
   "cell_type": "markdown",
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   "source": [
    "# Vector Fields\n",
    "\n",
    "The standard way to represent $N$-dimensional vector fields in *tntorch* is via a list of $N$ tensors, each of which has $N$ dimensions. Functions that accept or return vector fields do so in that form. For example, `gradient()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3D TT tensor:\n",
      "\n",
      " 64  64  64\n",
      "  |   |   |\n",
      " (0) (1) (2)\n",
      " / \\ / \\ / \\\n",
      "1   10  10  1\n",
      ", 3D TT tensor:\n",
      "\n",
      " 64  64  64\n",
      "  |   |   |\n",
      " (0) (1) (2)\n",
      " / \\ / \\ / \\\n",
      "1   10  10  1\n",
      ", 3D TT tensor:\n",
      "\n",
      " 64  64  64\n",
      "  |   |   |\n",
      " (0) (1) (2)\n",
      " / \\ / \\ / \\\n",
      "1   10  10  1\n",
      "]\n"
     ]
    }
   ],
   "source": [
    "import torch\n",
    "torch.set_default_dtype(torch.float64)\n",
    "import tntorch as tn\n",
    "\n",
    "t = tn.rand([64]*3, ranks_tt=10)\n",
    "grad = tn.gradient(t)\n",
    "print(grad)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can check that the curl of any gradient is 0:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor(1.0344e-06)\n",
      "tensor(1.4569e-06)\n",
      "tensor(2.5995e-06)\n"
     ]
    }
   ],
   "source": [
    "curl = tn.curl(tn.gradient(t))  # List of 3 3D tensors\n",
    "print(tn.norm(curl[0]))\n",
    "print(tn.norm(curl[1]))\n",
    "print(tn.norm(curl[2]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's also check that the divergence of any curl is zero (we'll use a random, non-gradient vector field here):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(4.8832e-08)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "vf = [tn.rand([64]*3, ranks_tt=1) for n in range(3)]\n",
    "tn.norm(tn.divergence(tn.curl(vf)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "... and that the Laplacian of a scalar field `t` equals the divergence of `t`'s gradient:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "tensor(0.)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tn.norm(tn.laplacian(t) - tn.divergence(tn.gradient(t)))"
   ]
  }
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