RiemannCurvatureT2-Detail.ipynb

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   "source": [
    "def x := [| θ, φ |]"
   ]
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  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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   ],
   "source": [
    "def X := [| `(a * cos θ + b) * cos φ, `(a * cos θ + b) * sin φ,  a * sin θ |]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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   ],
   "source": [
    "def e_i := ∂/∂ X x_i"
   ]
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     "data": {
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       "$$\\begin{pmatrix} -a sin(θ) cos(φ) & -a sin(θ) sin(φ) & a cos(θ) \\\\ -(a cos(θ) + b) sin(φ) & (a cos(θ) + b) cos(φ) & 0 \\\\ \\end{pmatrix}_{\\#}^{\\;}$$"
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   "source": [
    "e_#"
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   "metadata": {},
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   ],
   "source": [
    "def g_i_j := generateTensor (\\[a, b] -> e_a~i . e_b_i) [2, 2]"
   ]
  },
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   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
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   ],
   "source": [
    "def g~i~j := M.inverse g_#_#"
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
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     "data": {
      "text/html": [
       "$$(\\begin{pmatrix} a^2 & 0 \\\\ 0 & (a cos(θ) + b)^2 \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}, \\begin{pmatrix} \\frac{1}{a^2} & 0 \\\\ 0 & \\frac{1}{(a cos(θ) + b)^2} \\\\ \\end{pmatrix}_{\\;\\;}^{\\#\\#})$$"
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   ],
   "source": [
    "(g_#_#, g~#~#)"
   ]
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   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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   ],
   "source": [
    "def Γ_i_j_k := (1 / 2) * (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i)"
   ]
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   "execution_count": 9,
   "metadata": {},
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   ],
   "source": [
    "def Γ~i_j_k := withSymbols [m] g~i~m . Γ_m_j_k"
   ]
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   "execution_count": 10,
   "metadata": {},
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     "data": {
      "text/html": [
       "$$(\\begin{pmatrix} 0 & 0 \\\\ 0 & (a cos(θ) + b) a sin(θ) \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}, \\begin{pmatrix} 0 & -(a cos(θ) + b) a sin(θ) \\\\ -(a cos(θ) + b) a sin(θ) & 0 \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;})$$"
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   ],
   "source": [
    "(Γ_1_#_#, Γ_2_#_#)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": []
    }
   ],
   "source": [
    "def R~i_j_k_l := withSymbols [m]\n",
    " ∂/∂ Γ~i_j_l x~k - ∂/∂ Γ~i_j_k x~l + Γ~m_j_l . Γ~i_m_k - Γ~m_j_k . Γ~i_m_l"
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   "execution_count": 12,
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     "data": {
      "text/html": [
       "$$(\\begin{pmatrix} 0 & \\frac{(a cos(θ) + b) cos(θ)}{a} \\\\ \\frac{-(a cos(θ) + b) cos(θ)}{a} & 0 \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}, \\begin{pmatrix} 0 & \\frac{-a cos(θ)}{(a cos(θ) + b)} \\\\ \\frac{a cos(θ)}{(a cos(θ) + b)} & 0 \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;})$$"
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   "source": [
    "(R~1_2_#_#, R~2_1_#_#)"
   ]
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   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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   ],
   "source": [
    "def Ric_i_j := withSymbols [m] sum (contract R~m_i_m_j)"
   ]
  },
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   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
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     "data": {
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       "$$\\begin{pmatrix} \\frac{a cos(θ)}{(a cos(θ) + b)} & 0 \\\\ 0 & \\frac{(a cos(θ) + b) cos(θ)}{a} \\\\ \\end{pmatrix}_{\\#\\#}^{\\;\\;}$$"
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   "source": [
    "Ric_#_#"
   ]
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   "execution_count": 15,
   "metadata": {},
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   ],
   "source": [
    "def scalarCurvature := withSymbols [i, j] g~i~j . Ric_i_j"
   ]
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   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
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       "$$\\frac{2 cos(θ)}{(a cos(θ) + b) a}$$"
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   "source": [
    "scalarCurvature"
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   "execution_count": null,
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   "source": []
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