09aManifold_Tensors_onModules.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 9a. Tensors on modules\n",
    "\n",
    "This notebook is part of the [Introduction to manifolds in SageMath](https://sagemanifolds.obspm.fr/intro_to_manifolds.html) by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'SageMath version 9.6, Release Date: 2022-05-15'"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "version()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reminder.  Vectors and linear forms on modules\n",
    "\n",
    "<br> \n",
    "\n",
    "Recall that for free modules of finite rank (in [notebook 6](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/06Manifold_VectSpaces_Modules.ipynb))  we have introduced  **vectors** and **linear forms**.\n",
    "\n",
    "<br>\n",
    "\n",
    "**Example 9.1a**\n",
    "\n",
    "Define a free module of finite rank and some vector and linear form."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "N=3                                     # dimension of the module\n",
    "%display latex\n",
    "M=FiniteRankFreeModule(SR,3,name='M')   # 3-dim module over SR\n",
    "e = M.basis('e')                        # basis of M\n",
    "v=var('v',n=3)                          # components of vector v\n",
    "v=M(list(v))                            # define vector v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Element of the 3-dimensional vector space M over the Symbolic Ring\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[v_{0}, v_{1}, v_{2}\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[v_{0}, v_{1}, v_{2}\\right]$"
      ],
      "text/plain": [
       "[v0, v1, v2]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(v)                                # information on v\n",
    "v.comp()[:]                             # components of v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "General linear forms  on a  3-dimensional module can be defined as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "a=M.linear_form()                       # linear form a on M\n",
    "a[:]=var('a',n=4)                       # components of a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle a_{0} e^{0} + a_{1} e^{1} + a_{2} e^{2}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle a_{0} e^{0} + a_{1} e^{1} + a_{2} e^{2}$"
      ],
      "text/plain": [
       "a0 e^0 + a1 e^1 + a2 e^2"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.disp()                                # show a as combination\n",
    "                                        # of dual basis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The linear form applied to a  vector gives a scalar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle a_{0} v_{0} + a_{1} v_{1} + a_{2} v_{2}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle a_{0} v_{0} + a_{1} v_{1} + a_{2} v_{2}$"
      ],
      "text/plain": [
       "a0*v0 + a1*v1 + a2*v2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a(v)                                   # value a(v)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The value of a  vector $v\\in M$ on a linear form $a\\in M^*$ is by definition equal to $a(v)$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle a_{0} v_{0} + a_{1} v_{1} + a_{2} v_{2}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle a_{0} v_{0} + a_{1} v_{1} + a_{2} v_{2}$"
      ],
      "text/plain": [
       "a0*v0 + a1*v1 + a2*v2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v(a)                                   # value v(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "Now we are ready to define more general objects."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "\n",
    "## Tensors on modules\n",
    "\n",
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If $M$ is a vector space or a module, by a **multilinear** or more precisely  $k$-**linear form** we mean a function $t:\n",
    "M^k\\to  R$ which is linear in each of its arguments, i.e., for $i=1,\\ldots,k$\n",
    "\n",
    "$$t(v_1\\ldots,\\alpha v_i+\\beta w_i,\\ldots v_k)=\n",
    "\\alpha t(v_1\\ldots,v_i,\\ldots v_k)+\\beta t(v_1\\ldots,w_i,\\ldots v_k),\\quad \\alpha,\\beta\\in R,\\quad v_i,w_i\\in M.$$\n",
    "\n",
    "<br>\n",
    "\n",
    "Assume that $M$ is a module."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "### Tensor module $T^{(k,l)}M$\n",
    "\n",
    "<br>\n",
    "\n",
    "\n",
    "**Tensors** of type (k,l) on $ M$ \n",
    "are multilinear maps:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$    T^{(k,l)}M = \\{t:\\underbrace{M^*\\times\\cdots\\times M^*}_{k\\ \\; \\mbox{times}}\n",
    "    \\times \\underbrace{M\\times\\cdots\\times M}_{l\\ \\; \\mbox{times}}\\to R\\},$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\ M^*\\ $ denotes the dual module, i.e. the module of linear forms on $M$ (cf. [notebook 6](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/06Manifold_VectSpaces_Modules.ipynb)).\n",
    "\n",
    "Since $a\\to v(a)$ defines a linear form  $M^*\\to R$, the elements $v\\in M$ can be considered as elements of $T^{(1,0)}M$. On the other hand $T^{(0,1)}M$ as the space of linear forms on $M$ is equal to $M^*$. Thus,  **tensors generalize vectors and linear forms**. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 9.2a**\n",
    "\n",
    "Let us show how to use `tensor_module` method of SageMath Manifolds."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle M\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle M$"
      ],
      "text/plain": [
       "3-dimensional vector space M over the Symbolic Ring"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# continuation\n",
    "M.tensor_module(1,0)    # elements of T^(1,0) are vectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3-dimensional vector space M over the Symbolic Ring\n"
     ]
    }
   ],
   "source": [
    "print(M.tensor_module(1,0))         # T^(1,0) = M\n",
    "# since the ring SR is a field we obtain vector space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle M^*\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle M^*$"
      ],
      "text/plain": [
       "Free module of type-(0,1) tensors on the 3-dimensional vector space M over the Symbolic Ring"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.tensor_module(0,1)               # T^(0,1) = M*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Free module of type-(0,1) tensors on the 3-dimensional vector space M over the Symbolic Ring\n"
     ]
    }
   ],
   "source": [
    "print(M.tensor_module(0,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Module $T^{(0,k)}M$ - of covariant tensors of rank $k$\n",
    "\n",
    "<br>\n",
    "\n",
    "is the module of multilinear maps"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$    T^{(0,k)}M = \\{t:\\underbrace{M\\times\\cdots\\times M}_{k\\ \\; \\mbox{times}}\\to R\\}.$$\n",
    "For k=1 we obtain the module of linear forms or **covectors**   on $M$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In $T^{(0,k)}M$ we introduce the **algebraic operations** by\n",
    "\n",
    "$$ (at+bs)(v_1\\ldots,v_k)=at(v_1\\ldots,v_k)+bs(v_1\\ldots,v_k),$$\n",
    "\n",
    "where $t,s\\in T^{(0,k)}M$, $v_1,\\ldots,v_k\\in M$ and $a,b\\in R$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For $t\\in T^{(0,k)},\\ s\\in T^{(0,l)}M$ we define the **tensor product** $\\ t\\otimes s\\in T^{(0,k+l)}M\\ $ by\n",
    "\n",
    "\\begin{equation}\n",
    "(t ⊗ s )(v_1 ,\\ldots , v_{k+l} ) = t (v_1 ,\\ldots, v_k ) s (v_{k+1} , . . . , v_{k+l} ),\n",
    "\\label{eq:tensor_product}\\tag{9.1a}\n",
    "\\end{equation}\n",
    "\n",
    "for $v_1,\\ldots,v_{k+l}\\in M$.<br><br>\n",
    "In SageMath Manifolds the symbol of tensor product is simply $*$.<br>\n",
    "\n",
    "<br>\n",
    "\n",
    "**Example 9.3a**\n",
    "\n",
    "Let us check the last formula in the case of two general tensors from $T^{(0,2)}$ and 2-dimensional module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "N=2;                              # dimension of module\n",
    "%display latex\n",
    "Mo=FiniteRankFreeModule(SR,2,name='Mo')   # module Mo\n",
    "e = Mo.basis('e')                 # basis of Mo\n",
    "t=Mo.tensor((0,2))                # (0,2) type tensor t\n",
    "s=Mo.tensor((0,2))                # (0,2) type tensor s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\left[{v^0_0}, {v^1_0}\\right], \\left[{v^0_1}, {v^1_1}\\right], \\left[{v^0_2}, {v^1_2}\\right], \\left[{v^0_3}, {v^1_3}\\right]\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\left[{v^0_0}, {v^1_0}\\right], \\left[{v^0_1}, {v^1_1}\\right], \\left[{v^0_2}, {v^1_2}\\right], \\left[{v^0_3}, {v^1_3}\\right]\\right]$"
      ],
      "text/plain": [
       "[[v00, v10], [v01, v11], [v02, v12], [v03, v13]]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M=4                               # number of vectors \n",
    "vv=[[var('v'+str(i)+str(j),       # components of four vectors\n",
    "    latex_name='v'+'^'+str(i)+'_'+str(j))\n",
    "        for i in range(N)] for j in range(M)]\n",
    "vv "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "v=[Mo(vv[k]) for k in range(M)]   # list of vectors\n",
    "tt=[[var('t'+str(i)+str(j))       # components of tensor t\n",
    "    for j in range(M)] for i in range(N)]\n",
    "ss=[[var('t'+str(i)+str(j))       # components of tensor s\n",
    "    for j in range(M)] for i in range(N)]\n",
    "\n",
    "\n",
    "t[:]=tt                 # define tensor t components t_{ij}\n",
    "s[:]=ss                 # define tensor s components s_{ij}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\mathrm{True}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\mathrm{True}$"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# check the equality (9.1a) defining tensor product\n",
    "bool((t*s)(v[0],v[1],v[2],v[3])==t(v[0],v[1])*s(v[2],v[3]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "Tensor product has the following properties (proofs  for modules $T_pM,$ where $M$ denotes a manifold  are given in the [next notebook](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09Manifold_LinearFormsTensors_onTp.ipynb))\n",
    "\n",
    "\\begin{equation}\n",
    "\\begin{matrix}\n",
    "(ar + bs ) ⊗ t = ar ⊗ t + bs ⊗ t ,\\\\\n",
    "r ⊗ (as + bt ) = ar ⊗ s + br ⊗ t ,\\\\\n",
    "(r ⊗ s ) ⊗ t = r ⊗ (s ⊗ t ),\n",
    "\\end{matrix}\n",
    "\\label{eq:tensor_product_prop}\\tag{9.2a}\n",
    "\\end{equation}\n",
    "\n",
    "for $a,b\\in R\\ $ and for arbitrary covariant tensors $r,s,t$ (the addition is defined only for tensors of the same rank $(0,k)$). "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Covariant tensors in  components\n",
    "\n",
    "<br>\n",
    "\n",
    "\n",
    "One can check that if $e_1,\\ldots,e_n$ is a basis of the module $M$\n",
    "and $e^1,\\ldots,e^n\\ $ its dual basis (defined in [notebook 6](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/06Manifold_VectSpaces_Modules.ipynb)), then the elements\n",
    "\n",
    "\\begin{equation}\n",
    "e^{j_1}\\otimes\\ldots \\otimes e^{j_m}, \\quad j_q\\in\\{1,\\ldots,n\\}\n",
    "\\tag{9.2a}\n",
    "\\end{equation}\n",
    "\n",
    "form a basis for $T^{(0,m)}M$ and if we put\n",
    "\n",
    "$$t_{j_1\\ldots j_m}\n",
    "=t(e_{j_1},\\ldots,e_{j_m}),\n",
    "$$\n",
    "\n",
    "then \n",
    "\n",
    "\\begin{equation}\n",
    "t=t_{j_1\\ldots j_m}\n",
    "e^{j_1}\\otimes\\ldots\\otimes e^{j_m}, \\quad \\text{for }\\  t\\in   T^{(0,m)}M.\n",
    "\\tag{9.3a}\n",
    "\\end{equation}\n",
    "\n",
    "In fact, since $\\ v_1=e^{j_1}(v_1)e_{j_1},\\ldots, v_m=e^{j_m}(v_m)e_{j_m},\\ $ then\n",
    "\n",
    "$$t(v_1,\\ldots,v_m)=t(e^{j_1}(v_1)e_{j_1},\\ldots,e^{j_m}(v_m)e_{j_m})\\\\\n",
    "=e^{j_1}(v_1)\\ldots e^{j_m}(v_m)t(e_{j_1},\\ldots,e_{j_m})\\\\\n",
    "=t_{j_1\\ldots j_m}e^{j_1}\\otimes \\ldots \\otimes e^{j_m}(v_1,\\ldots v_m).\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 9.4a**\n",
    "\n",
    "Let us show the representation of a tensor $\\ t\\in T^{(0,3)}M\\ $ in   components."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\left[\\left[t_{000}, t_{001}\\right], \\left[t_{010}, t_{011}\\right]\\right], \\left[\\left[t_{100}, t_{101}\\right], \\left[t_{110}, t_{111}\\right]\\right]\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\left[\\left[t_{000}, t_{001}\\right], \\left[t_{010}, t_{011}\\right]\\right], \\left[\\left[t_{100}, t_{101}\\right], \\left[t_{110}, t_{111}\\right]\\right]\\right]$"
      ],
      "text/plain": [
       "[[[t000, t001], [t010, t011]], [[t100, t101], [t110, t111]]]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N=2                                     # dimension of module\n",
    "st=[[[var('t'+str(i0)+str(i1)+str(i2))  # components of t\n",
    "    for i2 in range(N)] \n",
    "        for i1 in range(N)] \n",
    "            for i0 in range(N)]\n",
    "st                                      # show components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "Mo=FiniteRankFreeModule(SR,2,name='Mo') # module Mo\n",
    "e = Mo.basis('e')                       # basis of Mo\n",
    "%display latex\n",
    "t = Mo.tensor((0,3), name='t')          # tensor of (0,3) type\n",
    "t[:]=st                                 # define all components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "General tensor of type (0,3) on a 2-dimensional module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle t = t_{000} e^{0}\\otimes e^{0}\\otimes e^{0} + t_{001} e^{0}\\otimes e^{0}\\otimes e^{1} + t_{010} e^{0}\\otimes e^{1}\\otimes e^{0} + t_{011} e^{0}\\otimes e^{1}\\otimes e^{1} + t_{100} e^{1}\\otimes e^{0}\\otimes e^{0} + t_{101} e^{1}\\otimes e^{0}\\otimes e^{1} + t_{110} e^{1}\\otimes e^{1}\\otimes e^{0} + t_{111} e^{1}\\otimes e^{1}\\otimes e^{1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle t = t_{000} e^{0}\\otimes e^{0}\\otimes e^{0} + t_{001} e^{0}\\otimes e^{0}\\otimes e^{1} + t_{010} e^{0}\\otimes e^{1}\\otimes e^{0} + t_{011} e^{0}\\otimes e^{1}\\otimes e^{1} + t_{100} e^{1}\\otimes e^{0}\\otimes e^{0} + t_{101} e^{1}\\otimes e^{0}\\otimes e^{1} + t_{110} e^{1}\\otimes e^{1}\\otimes e^{0} + t_{111} e^{1}\\otimes e^{1}\\otimes e^{1}$"
      ],
      "text/plain": [
       "t = t000 e^0⊗e^0⊗e^0 + t001 e^0⊗e^0⊗e^1 + t010 e^0⊗e^1⊗e^0 + t011 e^0⊗e^1⊗e^1 + t100 e^1⊗e^0⊗e^0 + t101 e^1⊗e^0⊗e^1 + t110 e^1⊗e^1⊗e^0 + t111 e^1⊗e^1⊗e^1"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.disp()                                # show t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tensor of type (0,3) on a 2-dimensional module with concrete components:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle t = e^{0}\\otimes e^{0}\\otimes e^{0} + 2 e^{0}\\otimes e^{0}\\otimes e^{1} + 3 e^{0}\\otimes e^{1}\\otimes e^{0} + 4 e^{0}\\otimes e^{1}\\otimes e^{1} + 5 e^{1}\\otimes e^{0}\\otimes e^{0} + 6 e^{1}\\otimes e^{0}\\otimes e^{1} + 7 e^{1}\\otimes e^{1}\\otimes e^{0} + 8 e^{1}\\otimes e^{1}\\otimes e^{1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle t = e^{0}\\otimes e^{0}\\otimes e^{0} + 2 e^{0}\\otimes e^{0}\\otimes e^{1} + 3 e^{0}\\otimes e^{1}\\otimes e^{0} + 4 e^{0}\\otimes e^{1}\\otimes e^{1} + 5 e^{1}\\otimes e^{0}\\otimes e^{0} + 6 e^{1}\\otimes e^{0}\\otimes e^{1} + 7 e^{1}\\otimes e^{1}\\otimes e^{0} + 8 e^{1}\\otimes e^{1}\\otimes e^{1}$"
      ],
      "text/plain": [
       "t = e^0⊗e^0⊗e^0 + 2 e^0⊗e^0⊗e^1 + 3 e^0⊗e^1⊗e^0 + 4 e^0⊗e^1⊗e^1 + 5 e^1⊗e^0⊗e^0 + 6 e^1⊗e^0⊗e^1 + 7 e^1⊗e^1⊗e^0 + 8 e^1⊗e^1⊗e^1"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# continuation\n",
    "ct=range(1,9)      #  [1,...,8] consecutive components\n",
    "for i0 in range(N):\n",
    "    for i1 in range(N):\n",
    "        for i2 in range(N):\n",
    "            t[i0,i1,i2]=ct[i0*N*N+i1*N+i2]\n",
    "t.disp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Basis (e_0,e_1) on the 2-dimensional vector space Mo over the Symbolic Ring\n"
     ]
    }
   ],
   "source": [
    "print(e)                              # basis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check that the coefficient $t_{011}$ is equal to $t(e_0,e_1,e_1)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 4\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 4$"
      ],
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t(e[0],e[1],e[1])                    # t(e_0,e_1,e_1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Module $T^{(k,0)}M $ of contravariant tensors of rank $k$\n",
    "\n",
    "<br>\n",
    "\n",
    "is the module of $k$-linear forms:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$    T^{(k,0)}M = \\{t:\\underbrace{M^*\\times\\cdots\\times M^*}_{k\\ \\;\\mbox{times}}\n",
    "    \\to R\\}.$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In $T^{(k,0)}M$ we introduce the module structure by\n",
    "\n",
    "$$ (at+bs)(\\alpha_1\\ldots,\\alpha_k)=at(\\alpha_1\\ldots,\\alpha_k)+bs(\\alpha_1\\ldots,\\alpha_k),$$\n",
    "\n",
    "where $t,s\\in T^{(k,0)}M$, $\\alpha_1,\\ldots,\\alpha_k\\in M^*$ and $a,b\\in R$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For $t,s\\in T^{(k,0)}M$ we define the **tensor product** $t\\otimes s\\ $ by\n",
    "\n",
    "\\begin{equation}\n",
    "(t ⊗ s )(\\alpha_1 ,\\ldots , \\alpha_{k+l} ) ≡ t (\\alpha_1 ,\\ldots, \\alpha_k ) s(\\alpha_{k+1} , \\ldots , \\alpha_{k+l} ),\n",
    "\\tag{9.4a}\n",
    "\\end{equation}\n",
    "\n",
    "for $\\alpha_1,\\ldots,\\alpha_{k+l}\\in M^*$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### Contravariant tensors in  components\n",
    "\n",
    "<br>\n",
    "\n",
    "One can check that if $e_1,\\ldots,e_n$ is a basis of the module $M$\n",
    "and $e^1,\\ldots,e^n$ its dual basis then the elements\n",
    "\n",
    "$$e_{i_1}\\otimes\\ldots\\otimes e_{i_k}, \\quad i_p\\in\\{1,\\ldots,n\\}\n",
    "$$\n",
    "\n",
    "form a basis for $T^{(k,0)}M$ and if we put\n",
    "\n",
    "$$t^{i_1\\ldots i_k}\n",
    "=t(e^{i_1},\\ldots,e^{i_k}),\n",
    "$$\n",
    "\n",
    "then \n",
    "\n",
    "\\begin{equation}\n",
    "t=t^{i_1\\ldots i_k}\n",
    "e_{i_1}\\otimes\\ldots\\otimes e_{i_k}, \\quad \\text{for }\\  t\\in   T^{(k,0)}M.\n",
    "\\tag{9.5a}\n",
    "\\end{equation}\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This follows from  $\\alpha(v)=\\alpha(e^i(v)e_\n",
    "i)=\\alpha(e_i)e^i(v),\\ $ and  $\\ v(\\alpha)=\\alpha(v)$ for linear form $\\ \\alpha\\in M^*\\ $ and $\\ v\\in M,\\ $since we have\n",
    "\n",
    "$$t(\\alpha_1,\\ldots,\\alpha_k)=\n",
    "t(\\alpha_1(e_{i_1})e^{i_1},\\ldots,\\alpha_k(e_{i_k})e^{i_k})\\\\\n",
    "=\\alpha_1(e_{i_1})\\ldots \\alpha_k(e_{i_k})t(e^{i_1},\\ldots,e^{i_k})\\\\\n",
    "=e_{i_1}(\\alpha_1)\\ldots e_{i_k}(\\alpha_k)t(e^{i_1},\\ldots, e^{i_k})\\\\\n",
    "=t^{i_1\\ldots i_k} e_{i_1}\\otimes \\ldots \\otimes e_{i_k}(\\alpha_1,\\ldots,\\alpha_k).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 9.5a**\n",
    "\n",
    "Let us show the  representation of a tensor of type (3,0) in components."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\left[\\left[{t^{000}}, {t^{001}}\\right], \\left[{t^{010}}, {t^{011}}\\right]\\right], \\left[\\left[{t^{100}}, {t^{101}}\\right], \\left[{t^{110}}, {t^{111}}\\right]\\right]\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\left[\\left[{t^{000}}, {t^{001}}\\right], \\left[{t^{010}}, {t^{011}}\\right]\\right], \\left[\\left[{t^{100}}, {t^{101}}\\right], \\left[{t^{110}}, {t^{111}}\\right]\\right]\\right]$"
      ],
      "text/plain": [
       "[[[t000, t001], [t010, t011]], [[t100, t101], [t110, t111]]]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "N=2                                      # dimension of module\n",
    "st=[[[var('t'+str(i0)+str(i1)+str(i2),\n",
    "     latex_name='t'+'^'+'{'+str(i0)+str(i1)+str(i2)+'}') \n",
    "        for i2 in range(N)] \n",
    "           for i1 in range(N)]           # components of t\n",
    "               for i0 in range(N)]       # with superscripts\n",
    "st                                       # show components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "Mo=FiniteRankFreeModule(SR,2,name='Mo')  # module Mo\n",
    "e = Mo.basis('e')                        # basis of Mo\n",
    "t = Mo.tensor((3,0), name='t')           # tensor of (3,0) type\n",
    "t[:]=st                                  # define all components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "General tensor of type (3,0) on a 2-dimensional module:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle t = {t^{000}} e_{0}\\otimes e_{0}\\otimes e_{0} + {t^{001}} e_{0}\\otimes e_{0}\\otimes e_{1} + {t^{010}} e_{0}\\otimes e_{1}\\otimes e_{0} + {t^{011}} e_{0}\\otimes e_{1}\\otimes e_{1} + {t^{100}} e_{1}\\otimes e_{0}\\otimes e_{0} + {t^{101}} e_{1}\\otimes e_{0}\\otimes e_{1} + {t^{110}} e_{1}\\otimes e_{1}\\otimes e_{0} + {t^{111}} e_{1}\\otimes e_{1}\\otimes e_{1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle t = {t^{000}} e_{0}\\otimes e_{0}\\otimes e_{0} + {t^{001}} e_{0}\\otimes e_{0}\\otimes e_{1} + {t^{010}} e_{0}\\otimes e_{1}\\otimes e_{0} + {t^{011}} e_{0}\\otimes e_{1}\\otimes e_{1} + {t^{100}} e_{1}\\otimes e_{0}\\otimes e_{0} + {t^{101}} e_{1}\\otimes e_{0}\\otimes e_{1} + {t^{110}} e_{1}\\otimes e_{1}\\otimes e_{0} + {t^{111}} e_{1}\\otimes e_{1}\\otimes e_{1}$"
      ],
      "text/plain": [
       "t = t000 e_0⊗e_0⊗e_0 + t001 e_0⊗e_0⊗e_1 + t010 e_0⊗e_1⊗e_0 + t011 e_0⊗e_1⊗e_1 + t100 e_1⊗e_0⊗e_0 + t101 e_1⊗e_0⊗e_1 + t110 e_1⊗e_1⊗e_0 + t111 e_1⊗e_1⊗e_1"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.disp()                                 # show t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tensor of type (3,0) on a 2-dimensional module with concrete components:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle t = e_{0}\\otimes e_{0}\\otimes e_{0} + 2 e_{0}\\otimes e_{0}\\otimes e_{1} + 3 e_{0}\\otimes e_{1}\\otimes e_{0} + 4 e_{0}\\otimes e_{1}\\otimes e_{1} + 5 e_{1}\\otimes e_{0}\\otimes e_{0} + 6 e_{1}\\otimes e_{0}\\otimes e_{1} + 7 e_{1}\\otimes e_{1}\\otimes e_{0} + 8 e_{1}\\otimes e_{1}\\otimes e_{1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle t = e_{0}\\otimes e_{0}\\otimes e_{0} + 2 e_{0}\\otimes e_{0}\\otimes e_{1} + 3 e_{0}\\otimes e_{1}\\otimes e_{0} + 4 e_{0}\\otimes e_{1}\\otimes e_{1} + 5 e_{1}\\otimes e_{0}\\otimes e_{0} + 6 e_{1}\\otimes e_{0}\\otimes e_{1} + 7 e_{1}\\otimes e_{1}\\otimes e_{0} + 8 e_{1}\\otimes e_{1}\\otimes e_{1}$"
      ],
      "text/plain": [
       "t = e_0⊗e_0⊗e_0 + 2 e_0⊗e_0⊗e_1 + 3 e_0⊗e_1⊗e_0 + 4 e_0⊗e_1⊗e_1 + 5 e_1⊗e_0⊗e_0 + 6 e_1⊗e_0⊗e_1 + 7 e_1⊗e_1⊗e_0 + 8 e_1⊗e_1⊗e_1"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# continuation\n",
    "ct=range(1,9)            # [1,...,8]- consecutive components\n",
    "for i0 in range(N):\n",
    "    for i1 in range(N):\n",
    "        for i2 in range(N):\n",
    "            t[i0,i1,i2]=ct[i0*N*N+i1*N+i2]\n",
    "t.disp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Dual basis (e^0,e^1) on the 2-dimensional vector space Mo over the Symbolic Ring\n"
     ]
    }
   ],
   "source": [
    "d=e.dual_basis()                        # dual basis\n",
    "print(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check that the coefficient $t^{011}$ is equal to $t( e^0,e^1, e^1)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 4\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 4$"
      ],
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t(d[0],d[1],d[1])                       # t(e^0,e^1,e^1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "### General tensors from $T^{(k,l)}M$ in  components\n",
    "\n",
    "<br>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For $t\\in T^{(k_1,l_1)}M,\\ \\ s\\in T^{(k_2,l_2)}M$ we define the **tensor product** $t\\otimes s\\in T^{(k_1+k_2,l_1+l_2)}$ by\n",
    "\n",
    "\\begin{equation}\n",
    "(t ⊗ s )(\\alpha_1,\\ldots\\alpha_{k_1+k_2},v_1 ,\\ldots , v_{l_1+l_2} )\\\\\n",
    "=t(\\alpha_1,\\ldots,\\alpha_{k_1},v_1 ,\\ldots, v_{l_1} ) s (\\alpha_{k_1+1},\\ldots,\\alpha_{k_1+k_2},v_{l_1+1},\\ldots, v_{l_1+l_2} ),\n",
    "\\tag{9.6a}\n",
    "\\end{equation}\n",
    "\n",
    "for $\\alpha_1,\\ldots,\\alpha_{k_1+k_2}\\in M^*$ and $v_1,\\ldots,v_{l_1+l_2}\\in M$.<br><br>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generalizing the formulas (9.3a) and (9.5a) we obtain the following expression for the general tensor $t\\in T^{(k,l)}M$ in components\n",
    "\n",
    "\\begin{equation}\n",
    "t=\n",
    "t\\big(e^{i_1},..,e^{i_k},e_{j_1},..,e_{j_l})\n",
    " e_{i_1}\\otimes\\dots\\otimes e_{i_k}\\otimes e^{j_1}\\otimes\\ldots\\otimes e^{j_l}.\n",
    " \\label{}\\tag{9.7a}\n",
    "\\end{equation}\n",
    "\n",
    "Very often the notation\n",
    "$$t^{i_1\\ldots i_k}_{j_1\\ldots j_l}=t_p\\big(e^{i_1},..,e^{i_k},e_{j_1},..,e_{j_l})\n",
    "$$\n",
    "is used and then \n",
    "$$t=t^{i_1\\ldots i_k}_{j_1\\ldots j_l}e_{i_1}\\otimes\\dots\\otimes e_{i_k}\\otimes e^{j_1}\\otimes\\ldots\\otimes e^{j_l}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To check that the elements\n",
    "\\begin{equation}\n",
    "e_{i_1}\\otimes\\dots\\otimes e_{i_k}\\otimes e^{j_1}\\otimes\\ldots\\otimes e^{j_l}\n",
    "\\label{} \\tag{9.8a}\n",
    "\\end{equation}\n",
    "\n",
    "are linearly independent, assume that the linear combination\n",
    "\n",
    "$$a^{i_1\\ldots i_k}_{j_1\\ldots j_l}e_{i_1}\\otimes\\dots\\otimes e_{i_k}\\otimes e^{j_1}\\otimes\\ldots\\otimes e^{j_l}\n",
    "$$\n",
    "\n",
    "vanishes. If we apply this combination to $\\quad(e^{m_1},\\ldots,e^{m_k}, e_{n_1},..,e_{n_l}\\big)$ we get $\\displaystyle a^{m_1\\ldots m_k}_{n_1\\ldots n_l}=0$.<br>\n",
    "Since as in previous cases we can check that elements of the type (9.8a) span $T^{(k,l)}M$, we have proved that these elements form a **basis** for   $T^{(k,l)}M$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "\n",
    "**Example 9.6a**\n",
    "\n",
    "Now we show the  representation of a tensor of (2,2) type in components."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\left[\\left[\\left[\\left[{t^{00}_{00}}, {t^{00}_{01}}\\right], \\left[{t^{00}_{10}}, {t^{00}_{11}}\\right]\\right], \\left[\\left[{t^{01}_{00}}, {t^{01}_{01}}\\right], \\left[{t^{01}_{10}}, {t^{01}_{11}}\\right]\\right]\\right], \\left[\\left[\\left[{t^{10}_{00}}, {t^{10}_{01}}\\right], \\left[{t^{10}_{10}}, {t^{10}_{11}}\\right]\\right], \\left[\\left[{t^{11}_{00}}, {t^{11}_{01}}\\right], \\left[{t^{11}_{10}}, {t^{11}_{11}}\\right]\\right]\\right]\\right]\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\left[\\left[\\left[\\left[{t^{00}_{00}}, {t^{00}_{01}}\\right], \\left[{t^{00}_{10}}, {t^{00}_{11}}\\right]\\right], \\left[\\left[{t^{01}_{00}}, {t^{01}_{01}}\\right], \\left[{t^{01}_{10}}, {t^{01}_{11}}\\right]\\right]\\right], \\left[\\left[\\left[{t^{10}_{00}}, {t^{10}_{01}}\\right], \\left[{t^{10}_{10}}, {t^{10}_{11}}\\right]\\right], \\left[\\left[{t^{11}_{00}}, {t^{11}_{01}}\\right], \\left[{t^{11}_{10}}, {t^{11}_{11}}\\right]\\right]\\right]\\right]$"
      ],
      "text/plain": [
       "[[[[t0000, t0001], [t0010, t0011]], [[t0100, t0101], [t0110, t0111]]],\n",
       " [[[t1000, t1001], [t1010, t1011]], [[t1100, t1101], [t1110, t1111]]]]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%display latex\n",
    "N=2                                            # dimension of module \n",
    "st=[[[[var('t'+str(i0)+str(i1)+str(i2)+str(i3),\n",
    "  latex_name='t'+'^'+'{'+str(i0)+str(i1)+'}'+'_'+\n",
    "           '{'+str(i2)+str(i3)+'}') \n",
    "    for i3 in range(N)] for i2 in range(N)]    # comp. with upper\n",
    "      for i1 in range(N)] for i0 in range(N)]  # and lower indices\n",
    "st                                             # show components"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "Mo=FiniteRankFreeModule(SR,2,name='Mo')   # module Mo \n",
    "e = Mo.basis('e')                         # basis of Mo\n",
    "t = Mo.tensor((2,2), name='t')            # tensor of (2,2) type\n",
    "t[:]=st                                   # define all components"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "General tensor from $T^{(2,2)}M$ on a 2-dimensional module ($2^4$ components):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle t = {t^{00}_{00}} e_{0}\\otimes e_{0}\\otimes e^{0}\\otimes e^{0} + {t^{00}_{01}} e_{0}\\otimes e_{0}\\otimes e^{0}\\otimes e^{1} + {t^{00}_{10}} e_{0}\\otimes e_{0}\\otimes e^{1}\\otimes e^{0} + {t^{00}_{11}} e_{0}\\otimes e_{0}\\otimes e^{1}\\otimes e^{1} + {t^{01}_{00}} e_{0}\\otimes e_{1}\\otimes e^{0}\\otimes e^{0} + {t^{01}_{01}} e_{0}\\otimes e_{1}\\otimes e^{0}\\otimes e^{1} + {t^{01}_{10}} e_{0}\\otimes e_{1}\\otimes e^{1}\\otimes e^{0} + {t^{01}_{11}} e_{0}\\otimes e_{1}\\otimes e^{1}\\otimes e^{1} + {t^{10}_{00}} e_{1}\\otimes e_{0}\\otimes e^{0}\\otimes e^{0} + {t^{10}_{01}} e_{1}\\otimes e_{0}\\otimes e^{0}\\otimes e^{1} + {t^{10}_{10}} e_{1}\\otimes e_{0}\\otimes e^{1}\\otimes e^{0} + {t^{10}_{11}} e_{1}\\otimes e_{0}\\otimes e^{1}\\otimes e^{1} + {t^{11}_{00}} e_{1}\\otimes e_{1}\\otimes e^{0}\\otimes e^{0} + {t^{11}_{01}} e_{1}\\otimes e_{1}\\otimes e^{0}\\otimes e^{1} + {t^{11}_{10}} e_{1}\\otimes e_{1}\\otimes e^{1}\\otimes e^{0} + {t^{11}_{11}} e_{1}\\otimes e_{1}\\otimes e^{1}\\otimes e^{1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle t = {t^{00}_{00}} e_{0}\\otimes e_{0}\\otimes e^{0}\\otimes e^{0} + {t^{00}_{01}} e_{0}\\otimes e_{0}\\otimes e^{0}\\otimes e^{1} + {t^{00}_{10}} e_{0}\\otimes e_{0}\\otimes e^{1}\\otimes e^{0} + {t^{00}_{11}} e_{0}\\otimes e_{0}\\otimes e^{1}\\otimes e^{1} + {t^{01}_{00}} e_{0}\\otimes e_{1}\\otimes e^{0}\\otimes e^{0} + {t^{01}_{01}} e_{0}\\otimes e_{1}\\otimes e^{0}\\otimes e^{1} + {t^{01}_{10}} e_{0}\\otimes e_{1}\\otimes e^{1}\\otimes e^{0} + {t^{01}_{11}} e_{0}\\otimes e_{1}\\otimes e^{1}\\otimes e^{1} + {t^{10}_{00}} e_{1}\\otimes e_{0}\\otimes e^{0}\\otimes e^{0} + {t^{10}_{01}} e_{1}\\otimes e_{0}\\otimes e^{0}\\otimes e^{1} + {t^{10}_{10}} e_{1}\\otimes e_{0}\\otimes e^{1}\\otimes e^{0} + {t^{10}_{11}} e_{1}\\otimes e_{0}\\otimes e^{1}\\otimes e^{1} + {t^{11}_{00}} e_{1}\\otimes e_{1}\\otimes e^{0}\\otimes e^{0} + {t^{11}_{01}} e_{1}\\otimes e_{1}\\otimes e^{0}\\otimes e^{1} + {t^{11}_{10}} e_{1}\\otimes e_{1}\\otimes e^{1}\\otimes e^{0} + {t^{11}_{11}} e_{1}\\otimes e_{1}\\otimes e^{1}\\otimes e^{1}$"
      ],
      "text/plain": [
       "t = t0000 e_0⊗e_0⊗e^0⊗e^0 + t0001 e_0⊗e_0⊗e^0⊗e^1 + t0010 e_0⊗e_0⊗e^1⊗e^0 + t0011 e_0⊗e_0⊗e^1⊗e^1 + t0100 e_0⊗e_1⊗e^0⊗e^0 + t0101 e_0⊗e_1⊗e^0⊗e^1 + t0110 e_0⊗e_1⊗e^1⊗e^0 + t0111 e_0⊗e_1⊗e^1⊗e^1 + t1000 e_1⊗e_0⊗e^0⊗e^0 + t1001 e_1⊗e_0⊗e^0⊗e^1 + t1010 e_1⊗e_0⊗e^1⊗e^0 + t1011 e_1⊗e_0⊗e^1⊗e^1 + t1100 e_1⊗e_1⊗e^0⊗e^0 + t1101 e_1⊗e_1⊗e^0⊗e^1 + t1110 e_1⊗e_1⊗e^1⊗e^0 + t1111 e_1⊗e_1⊗e^1⊗e^1"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.disp()                                  # show t"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a version with concrete components:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "ct=range(1,17)            # [1,..,16]-consecutive components\n",
    "for i0 in range(N):               \n",
    "    for i1 in range(N):\n",
    "        for i2 in range(N):\n",
    "            for i3 in range(N):\n",
    "                t[i0,i1,i2,i3]=ct[i0*N*N*N+i1*N*N+i2*N+i3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle t = e_{0}\\otimes e_{0}\\otimes e^{0}\\otimes e^{0} + 2 e_{0}\\otimes e_{0}\\otimes e^{0}\\otimes e^{1} + 3 e_{0}\\otimes e_{0}\\otimes e^{1}\\otimes e^{0} + 4 e_{0}\\otimes e_{0}\\otimes e^{1}\\otimes e^{1} + 5 e_{0}\\otimes e_{1}\\otimes e^{0}\\otimes e^{0} + 6 e_{0}\\otimes e_{1}\\otimes e^{0}\\otimes e^{1} + 7 e_{0}\\otimes e_{1}\\otimes e^{1}\\otimes e^{0} + 8 e_{0}\\otimes e_{1}\\otimes e^{1}\\otimes e^{1} + 9 e_{1}\\otimes e_{0}\\otimes e^{0}\\otimes e^{0} + 10 e_{1}\\otimes e_{0}\\otimes e^{0}\\otimes e^{1} + 11 e_{1}\\otimes e_{0}\\otimes e^{1}\\otimes e^{0} + 12 e_{1}\\otimes e_{0}\\otimes e^{1}\\otimes e^{1} + 13 e_{1}\\otimes e_{1}\\otimes e^{0}\\otimes e^{0} + 14 e_{1}\\otimes e_{1}\\otimes e^{0}\\otimes e^{1} + 15 e_{1}\\otimes e_{1}\\otimes e^{1}\\otimes e^{0} + 16 e_{1}\\otimes e_{1}\\otimes e^{1}\\otimes e^{1}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle t = e_{0}\\otimes e_{0}\\otimes e^{0}\\otimes e^{0} + 2 e_{0}\\otimes e_{0}\\otimes e^{0}\\otimes e^{1} + 3 e_{0}\\otimes e_{0}\\otimes e^{1}\\otimes e^{0} + 4 e_{0}\\otimes e_{0}\\otimes e^{1}\\otimes e^{1} + 5 e_{0}\\otimes e_{1}\\otimes e^{0}\\otimes e^{0} + 6 e_{0}\\otimes e_{1}\\otimes e^{0}\\otimes e^{1} + 7 e_{0}\\otimes e_{1}\\otimes e^{1}\\otimes e^{0} + 8 e_{0}\\otimes e_{1}\\otimes e^{1}\\otimes e^{1} + 9 e_{1}\\otimes e_{0}\\otimes e^{0}\\otimes e^{0} + 10 e_{1}\\otimes e_{0}\\otimes e^{0}\\otimes e^{1} + 11 e_{1}\\otimes e_{0}\\otimes e^{1}\\otimes e^{0} + 12 e_{1}\\otimes e_{0}\\otimes e^{1}\\otimes e^{1} + 13 e_{1}\\otimes e_{1}\\otimes e^{0}\\otimes e^{0} + 14 e_{1}\\otimes e_{1}\\otimes e^{0}\\otimes e^{1} + 15 e_{1}\\otimes e_{1}\\otimes e^{1}\\otimes e^{0} + 16 e_{1}\\otimes e_{1}\\otimes e^{1}\\otimes e^{1}$"
      ],
      "text/plain": [
       "t = e_0⊗e_0⊗e^0⊗e^0 + 2 e_0⊗e_0⊗e^0⊗e^1 + 3 e_0⊗e_0⊗e^1⊗e^0 + 4 e_0⊗e_0⊗e^1⊗e^1 + 5 e_0⊗e_1⊗e^0⊗e^0 + 6 e_0⊗e_1⊗e^0⊗e^1 + 7 e_0⊗e_1⊗e^1⊗e^0 + 8 e_0⊗e_1⊗e^1⊗e^1 + 9 e_1⊗e_0⊗e^0⊗e^0 + 10 e_1⊗e_0⊗e^0⊗e^1 + 11 e_1⊗e_0⊗e^1⊗e^0 + 12 e_1⊗e_0⊗e^1⊗e^1 + 13 e_1⊗e_1⊗e^0⊗e^0 + 14 e_1⊗e_1⊗e^0⊗e^1 + 15 e_1⊗e_1⊗e^1⊗e^0 + 16 e_1⊗e_1⊗e^1⊗e^1"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t.disp()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "d=e.dual_basis()                      # dual basis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Check that the coefficient $t^{10}_{01}$ is equal to $t(e^1,e^0,e_0,e_1)$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle 10\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle 10$"
      ],
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t(d[1],d[0],e[0],e[1])                # t(e^1,e^0,e_0,e_1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What's next?\n",
    "\n",
    "Take a look at the notebook [Tensors on $T_pM$](https://nbviewer.org/github/sagemanifolds/IntroToManifolds/blob/main/09Manifold_LinearFormsTensors_onTp.ipynb)."
   ]
  }
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