\n",
"Let us first consider an open subset of $\\mathcal{M}$, for instance the complement $U$ of the region defined by $\\{y=0, x\\geq 0\\}$ (note that `(y!=0, x<0)` stands for $y\\not=0$ OR $x<0$; the condition $y\\not=0$ AND $x<0$ would have been written `[y!=0, x<0]` instead):\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [],
"source": [
"U = M.open_subset('U', coord_def={X: (y!=0, x<0)})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"開部分集合$U$に制限された座標系 `X` を `X_U`と呼ぶことにします。\n",
"\n",
"Let us call `X_U` the restriction of the chart `X` to the open subset $U$:\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\left(U,(x, y, z)\\right)\\)"
],
"text/latex": [
"$\\displaystyle \\left(U,(x, y, z)\\right)$"
],
"text/plain": [
"Chart (U, (x, y, z))"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"X_U = X.restrict(U)\n",
"X_U"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$U$上の別の座標系 `Y`を球面座標 $(r,\\theta,\\phi)$ を使って導入します。\n",
" = U.chart(r'r:(0,+oo) th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n",
"Y"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"ここでの関数 `chart()` は引数を持っています。それはraw文字列でLaTexのシンボルを含んでいます。また、この文字列には座標の(既定値の $(-\\infty, +\\infty)$ とは異なる）範囲指定が含まれています。フィールドは`:`で区切られており、座標は空白文字で区切られています。 座標 $r$ については、LaTexシンボルを指定していないので、下限と上限の二つだけが指定されています。LaTexシンボルは出力の際に使われます。\n",
" \n",
"
The function chart() has now some argument; it is a string, which contains specific LaTeX symbols, hence the prefix 'r' to it (for raw string). It also contains the coordinate ranges, since they are different from the default value, which is $(-\\infty, +\\infty)$. For a given coordinate, the various fields are separated by the character ':' and a space character separates the coordinates. Note that for the coordinate $r$, there are only two fields, since the LaTeX symbol has not to be specified. The LaTeX symbols are used for the outputs:
\n",
"
\n",
" var('r1 ph1 th1')\n",
"Y1.= U.chart(\n",
" coord_restrictions=[r1 > 0, th1 >= 0, th1 < pi, ph1 > 0, ph1 < 2*pi]\n",
" )\n",
"Y1\n",
"# 接続／曲率など具体的な計算には、default_chartとつながる変換関数の定義が必要です。\n",
"transit_Y1_to_X = Y1.transition_map(\n",
" X_U,\n",
" [r1*sin(th1)*cos(ph1), r1*sin(th1)*sin(ph1), r1*cos(th1)]\n",
")\n",
"var('r2 ph2 th2')\n",
"Y2= U.chart(\n",
" names=(\"r2\",\"th2:\\\\theta_2\",\"ph2:\\\\phi_2\"),\n",
" coord_restrictions={((r2 > 0) , (0<= th2 < pi) , (0 <= ph2 < 2*pi))}\n",
" )\n",
"for c in ((r2 > 0) , (0<= th2 < pi) , (0 <= ph2 < 2*pi)):\n",
" assume(c)\n",
"Y2\n",
"
"
]
},
{
"cell_type": "raw",
"metadata": {},
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Sageが使っている座標の条件は、`assumptions()`メソッドを使って確認することができます。\n",
"\n",
"
The declared coordinate ranges are now known to Sage, as we may check by means of the command assumptions():
\n",
"
\n",
"
They are used in simplifications:
\n",
"
\n",
"
After having been declared, the chart Y can be fully specified by its relation to the chart X_U, via a transition map:
\n",
"
\n",
"
The inverse of the transition map can be specified by means of the method set_inverse():
\n",
"
\n",
"A check of the provided inverse is performed by composing \n",
"it with the original transition map, on the left and\n",
"on the right respectively. As indicated, the reported failure for th and ph is actually due to a lack of\n",
"simplification of expressions involving arctan2 .\n",
"We have then\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\left\\{\\begin{array}{lcl} r & = & \\sqrt{x^{2} + y^{2} + z^{2}} \\\\ {\\theta} & = & \\arctan\\left(\\sqrt{x^{2} + y^{2}}, z\\right) \\\\ {\\phi} & = & \\arctan\\left(y, x\\right) \\end{array}\\right.\\)"
],
"text/latex": [
"$\\displaystyle \\left\\{\\begin{array}{lcl} r & = & \\sqrt{x^{2} + y^{2} + z^{2}} \\\\ {\\theta} & = & \\arctan\\left(\\sqrt{x^{2} + y^{2}}, z\\right) \\\\ {\\phi} & = & \\arctan\\left(y, x\\right) \\end{array}\\right.$"
],
"text/plain": [
"r = sqrt(x^2 + y^2 + z^2)\n",
"th = arctan2(sqrt(x^2 + y^2), z)\n",
"ph = arctan2(y, x)"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"transit_Y_to_X.inverse().display()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"この段階で、多様体のアトラス( `user atlas`であって最大アトラスではありません）は三つの座標系を含んでいます。\n",
"\n",
"
At this stage, the manifold's atlas (the \"user atlas\", not the maximal atlas!) contains three charts:
\n",
"
\n",
"
The first chart defined on the manifold is considered as the manifold's default chart (it can be changed by the method set_default_chart()):
\n",
"
\n",
"
Each open subset has its own atlas (since an open subset of a manifold is a manifold by itself):
\n",
"
\n",
"We can draw the chart $Y$ in terms of the chart $X$ via the command `Y.plot(X)`, which shows the lines of constant coordinates from the $Y$ chart in a \"Cartesian frame\" based on the $X$ coordinates:\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Y.plot(X, color={r:'blue', th:'green', ph:'red'}, aspect_ratio=1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"コマンド`plot()`は多くのオプションを持っています。オプションを使い、描画される座標線の数、スタイル、色、そして座標の範囲などを指定します。\n",
"(参照:[list of all options](http://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/chart.html#sage.manifolds.chart.RealChart.plot)):\n",
"\n",
"The command plot() allows for many options, to control the number of coordinate lines to be drawn, their style and color, as well as the coordinate ranges (cf. the [list of all options](http://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/chart.html#sage.manifolds.chart.RealChart.plot)):\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n"
],
"text/plain": [
"Graphics3d Object"
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Y.plot(X, ranges={r:(1,2), th:(0,pi/2)}, number_values=4, \n",
" color={r:'blue', th:'green', ph:'red'}, aspect_ratio=1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"逆に、（制限された）座標系 $X|_{U}$ を座標系$Y$を使って表示することも可能です。この場合、座標系 $X$ の全てを表示することはできません、なぜなら $X$ の定義域(ドメイン)は $Y$ のそれより広いからです。\n",
"\n",
"
Conversly, the chart $X|_{U}$ can be plotted in terms of the chart $Y$ (this is not possible for the whole chart $X$ since its domain is larger than that of chart $Y$):
\n",
"
\n",
"
A point on $\\mathcal{M}$ is defined by its coordinates in a given chart:
\n",
"
\n",
"
Since $X=(\\mathcal{M}, (x,y,z))$ is the manifold's default chart, its name can be omitted:
\n",
"
\n",
"
Of course, $p$ belongs to $\\mathcal{M}$:
\n",
"
\n",
"
Indeed the coordinates of $p$ have $y\\not=0$:
\n",
"
\n",
"
Note in passing that since $X$ is the default chart on $\\mathcal{M}$, its name can be omitted in the arguments of coord():
\n",
"
\n",
"
The coordinates of $p$ can also be obtained by letting the chart acting of the point (from the very definition of a chart!):
\n",
"
\n",
"
Let $q$ be a point with $y = 0$ and $x \\geq 0$:
\n",
"
\n",
"
This time, the point does not belong to $U$:
\n",
"
\n",
"
Accordingly, we cannot ask for the coordinates of $q$ in the chart $Y=(U, (r,\\theta,\\phi))$:
\n",
"
\n",
"
but we can for point $p$:
\n",
"
\n",
"
Points can be compared:
\n",
"
\n",
"
In SageMath's terminology, points are elements, whose parents are the manifold on which they have been defined:
\n",
"
\n",
"A **scalar field** is a differentiable map $U \\to \\mathbb{R}$, where $U$ is an open subset of $\\mathcal{M}$.\n",
"\n",
"A scalar field is defined by its expressions in terms of charts covering its domain (in general more than one chart is necessary to cover all the domain):\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 70,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Scalar field f on the Open subset U of the 3-dimensional differentiable manifold M\n"
]
}
],
"source": [
"f = U.scalar_field({X_U: x+y^2+z^3}, name='f') ; print(f)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"座標で書かれた表式は、座標系をキーとしたPythonの辞書型データ`{X_U: x+y^2+z^3}`として渡されます。\n",
"この例では、辞書には一つの座標系しかありませんので、次のような表現も可能です。\n",
"\n",
"
The coordinate expressions of the scalar field are passed as a Python dictionary, with the charts as keys, hence the writing {X_U: x+y^2+z^3}.
\n",
"
Since in the present case, there is only one chart in the dictionary, an alternative writing is
\n",
"
\n",
"
Since X_U is the domain's default chart, it can be omitted in the above declaration:
\n",
"
\n",
"As a mapping $U\\subset\\mathcal{M}\\longrightarrow\\mathbb{R}$, a scalar field acts on points, not on coordinates:\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle 4\\)"
],
"text/latex": [
"$\\displaystyle 4$"
],
"text/plain": [
"4"
]
},
"execution_count": 73,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f(p)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"スカラー場の`display()` メソッドは与えられた座標系でのスカラー場の表現を表示します。\n",
"\n",
"The method `display()` provides the expression of the scalar field in terms of a given chart:\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 74,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & z^{3} + y^{2} + x \\end{array}\\)"
],
"text/latex": [
"$\\displaystyle \\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & z^{3} + y^{2} + x \\end{array}$"
],
"text/plain": [
"f: U → ℝ\n",
" (x, y, z) ↦ z^3 + y^2 + x"
]
},
"execution_count": 74,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f.display(X_U)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"引数なしで呼び出された場合、`display()`メソッドは 定義域 $U$ の上で定義された全ての座標系について、座標表現を表示します。\n",
"（但し*副座標系*,即ち制限された副定義域のみで定義された座標系は除きます。）\n",
"\n",
"If no argument is provided, the method `display()` shows the coordinate expression of the scalar field in all the charts defined on the domain (except for *subcharts*, i.e. the restrictions of some chart to a subdomain):\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 75,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & z^{3} + y^{2} + x \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r^{3} \\cos\\left({\\theta}\\right)^{3} + r^{2} \\sin\\left({\\phi}\\right)^{2} \\sin\\left({\\theta}\\right)^{2} + r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\end{array}\\)"
],
"text/latex": [
"$\\displaystyle \\begin{array}{llcl} f:& U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & z^{3} + y^{2} + x \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r^{3} \\cos\\left({\\theta}\\right)^{3} + r^{2} \\sin\\left({\\phi}\\right)^{2} \\sin\\left({\\theta}\\right)^{2} + r \\cos\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) \\end{array}$"
],
"text/plain": [
"f: U → ℝ\n",
" (x, y, z) ↦ z^3 + y^2 + x\n",
" (r, th, ph) ↦ r^3*cos(th)^3 + r^2*sin(ph)^2*sin(th)^2 + r*cos(ph)*sin(th)"
]
},
"execution_count": 75,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f.display()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"スカラー場 $f$の 座標$(r,\\theta,\\phi)$ での表現はユーザによって与えられたものではなく、座標系の遷移写像で定義された座標変換のルールを使って自動的に導出されていることに注意してください。\n",
"\n",
"
Note that the expression of $f$ in terms of the coordinates $(r,\\theta,\\phi)$ has not been provided by the user but has been automatically computed by means of the change-of-coordinate formula declared above in the transition map.
\n",
"
In each chart, the scalar field is represented by a function of the chart coordinates (an object of the type CoordFunctionSymb described above), which is accessible via the method coord_function():
\n",
"
\n",
"
The \"raw\" symbolic expression is returned by the method expr():
\n",
"
A scalar field can also be defined by some unspecified function of the coordinates:
\n",
"
The parent of $f$ is the set $C^\\infty(U)$ of all smooth scalar fields on $U$, which is a commutative algebra over $\\mathbb{R}$:
\n",
"The base ring of the algebra is the field $\\mathbb{R}$, which is represented here by SageMath's Symbolic Ring (`SR`):
"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\text{SR}\\)"
],
"text/latex": [
"$\\displaystyle \\text{SR}$"
],
"text/plain": [
"Symbolic Ring"
]
},
"execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"CU.base_ring()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"スカラー場の演算は代数構造を通じて定義されます。\n",
"\n",
"
Arithmetic operations on scalar fields are defined through the algebra structure:
\n",
"
\n",
"
The tangent vector space to the manifold at point $p$ is obtained as follows:
\n",
"
$T_p\\, \\mathcal{M}$ is a 3-dimensional vector space over $\\mathbb{R}$ (represented here by SageMath's Symbolic Ring (SR)) :
\n",
"
$T_p\\, \\mathcal{M}$ is automatically endowed with vector bases deduced from the vector frames defined around the point:
\n",
"
For the tangent space at the point $q$, on the contrary, there is only one pre-defined basis, since $q$ is not in the domain $U$ of the frame associated with coordinates $(r,\\theta,\\phi)$:
\n",
"
\n",
"
Note that, despite what the above simplified writing may suggest (the mention of the point $p$ or $q$ is omitted in the basis vectors), $u$ and $v$ are different vectors, for they belong to different vector spaces:
\n",
"
In particular, it is not possible to add $u$ and $v$:
\n",
"
\n",
" 一般に二つの点のベクトルの差分として、共変微分の積分を考えても、それは積分の経路に依存することになるので、一意に定義することはできない。\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 106,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Error: unsupported operand parent(s) for +: 'Tangent space at Point q on the 3-dimensional differentiable manifold M' and 'Tangent space at Point p on the 3-dimensional differentiable manifold M'\n"
]
}
],
"source": [
"try:\n",
" s = u + v\n",
"except TypeError as exc:\n",
" print(\"Error: \", str(exc))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## べクトル場：Vector Fields\n",
"全ての座標系は座標系のドメインの上のベクトルフレームを持っています。これは、`coordinate basis`と呼ばれます、\n",
"\n",
"
Each chart defines a vector frame on the chart domain: the so-called coordinate basis:
# this frame is defined only on
"
]
},
{
"cell_type": "code",
"execution_count": 110,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle U\\)"
],
"text/latex": [
"$\\displaystyle U$"
],
"text/plain": [
"Open subset U of the 3-dimensional differentiable manifold M"
]
},
"execution_count": 110,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Y.frame().domain() # このフレームは部分多様体 $U$でのみ定義されています。"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`frames()`メソッドによって、与えられた開空間で定義されたフレームのリストが入手可能です。\n",
"\n",
"
The list of frames defined on a given open subset is returned by the method frames():
\n",
"
Unless otherwise specified (via the command set_default_frame()), the default frame is that associated with the default chart:
\n",
"
\n",
"
Individual elements of a frame can be accessed by means of their indices:
\n",
"
We may define a new vector field as follows:
\n",
"
A vector field can be defined by its components with respect to a given vector frame. When the latter is not specified, the open set's default frame is of course assumed:
\n",
"Since version 8.8 of SageMath, it is possible to initialize the components of the vector field while declaring it, so that the above is equivalent to\n",
"
"
]
},
{
"cell_type": "code",
"execution_count": 121,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle v = \\left( y + 1 \\right) \\frac{\\partial}{\\partial x } -x \\frac{\\partial}{\\partial y } + x y z \\frac{\\partial}{\\partial z }\\)"
],
"text/latex": [
"$\\displaystyle v = \\left( y + 1 \\right) \\frac{\\partial}{\\partial x } -x \\frac{\\partial}{\\partial y } + x y z \\frac{\\partial}{\\partial z }$"
],
"text/plain": [
"v = (y + 1) ∂/∂x - x ∂/∂y + x*y*z ∂/∂z"
]
},
"execution_count": 121,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v = U.vector_field(1+y, -x, x*y*z, name='v') # valid only in SageMath 8.8 and higher\n",
"v.display() "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$U$上のベクトル場はSageの **要素(element)** オブジェクトです。その **親(parent)** は $U$上で定義されたベクトル場の集合$\\mathfrak{X}(U)$ に設定されています。\n",
"\n",
"
Vector fields on $U$ are Sage element objects, whose parent is the set $\\mathfrak{X}(U)$ of vector fields defined on $U$:
\n",
"
The set $\\mathfrak{X}(U)$ is a module over the commutative algebra $C^\\infty(U)$ of scalar fields on $U$:
\n",
"
A vector field acts on scalar fields:
\n",
"
Unset components are assumed to be zero:
\n",
"
A vector field on $U$ can be expanded in the vector frame associated with the chart $(r,\\theta,\\phi)$:
\n",
"
By default, the components are expressed in terms of the default coordinates $(x,y,z)$. To express them in terms of the coordinates $(r,\\theta,\\phi)$, one should add the corresponding chart as the second argument of the method display():
\n",
"
The components of a tensor field w.r.t. the default frame can also be obtained as a list, via the command [:]:
\n",
"
An alternative is to use the method display_comp():
\n",
"
To obtain the components w.r.t. to another frame, one may go through the method comp() and specify the frame:
\n",
"
However a shortcut is to provide the frame as the first argument of the square brackets:
\n",
"
Components are shown expressed in terms of the default's coordinates; to get them in terms of the coordinates $(r,\\theta,\\phi)$ instead, add the chart name as the last argument in the square brackets:
\n",
"
or specify the chart in display_comp():
\n",
"To get some vector component as a scalar field instead of a coordinate expression, use double square brackets:
"
]
},
{
"cell_type": "code",
"execution_count": 143,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Scalar field on the Open subset U of the 3-dimensional differentiable manifold M\n"
]
}
],
"source": [
"print(v[[1]])"
]
},
{
"cell_type": "code",
"execution_count": 144,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\begin{array}{llcl} & U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & y + 1 \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) + 1 \\end{array}\\)"
],
"text/latex": [
"$\\displaystyle \\begin{array}{llcl} & U & \\longrightarrow & \\mathbb{R} \\\\ & \\left(x, y, z\\right) & \\longmapsto & y + 1 \\\\ & \\left(r, {\\theta}, {\\phi}\\right) & \\longmapsto & r \\sin\\left({\\phi}\\right) \\sin\\left({\\theta}\\right) + 1 \\end{array}$"
],
"text/plain": [
"U → ℝ\n",
"(x, y, z) ↦ y + 1\n",
"(r, th, ph) ↦ r*sin(ph)*sin(th) + 1"
]
},
"execution_count": 144,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v[[1]].display()"
]
},
{
"cell_type": "code",
"execution_count": 145,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\left[y + 1, -x, x y z\\right]\\)"
],
"text/latex": [
"$\\displaystyle \\left[y + 1, -x, x y z\\right]$"
],
"text/plain": [
"[y + 1, -x, x*y*z]"
]
},
"execution_count": 145,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"[v[[i]].expr(X_U) for i in range(1,4)]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"ベクトル場は関数オブジェクトを要素として定義することも可能です。\n",
"\n",
"
A vector field can be defined with components being unspecified functions of the coordinates:
\n",
"
The value of a vector field at some point of the manifold is obtained via the method at():
\n",
"
Indeed, recall that, w.r.t. chart X_U=$(x,y,z)$, the coordinates of the point $p$ and the components of the vector field $v$ are
\n",
"
Note that to simplify the writing, the symbol used to denote the value of the vector field at point $p$ is the same as that of the vector field itself (namely $v$); this can be changed by the method set_name():
\n",
"
Of course, $v|_p$ belongs to the tangent space at $p$:
\n",
"
A 1-form on $\\mathcal{M}$ is a field of linear forms. For instance, it can be the differential of a scalar field:
\n",
"
In the above writing, the 1-form is expanded over the basis $(\\mathrm{d}x, \\mathrm{d}y, \\mathrm{d}z)$ associated with the chart $(x,y,z)$. This basis can be accessed via the method coframe():
\n",
"
The list of all coframes defined on a given manifold open subset is returned by the method coframes():
\n",
"
As for a vector field, the value of the differential form at some point on the manifold is obtained by the method at():
\n",
"
The linear form $\\mathrm{d}f|_p$ belongs to the dual of the tangent vector space at $p$:
\n",
"
As such, it is acting on vectors at $p$, yielding a real number:
\n",
"
The differential 1-form of the unspecified scalar field $h$:
\n",
"
A 1-form can also be defined from scratch:
\n",
"
It can be specified by providing its components in a given coframe:
\n",
"Since version 8.8 of SageMath, it is possible to initialize the components of the 1-form while declaring it, so that the above is equivalent to
"
]
},
{
"cell_type": "code",
"execution_count": 173,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\omega = \\left( x^{2} + y^{2} \\right) \\mathrm{d} x + z \\mathrm{d} y + \\left( x - z \\right) \\mathrm{d} z\\)"
],
"text/latex": [
"$\\displaystyle \\omega = \\left( x^{2} + y^{2} \\right) \\mathrm{d} x + z \\mathrm{d} y + \\left( x - z \\right) \\mathrm{d} z$"
],
"text/plain": [
"omega = (x^2 + y^2) dx + z dy + (x - z) dz"
]
},
"execution_count": 173,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"om = U.one_form(\n",
" x^2+y^2, z, x-z, # valid only in SageMath 8.8 and higher\n",
" name='omega', \n",
" latex_name=r'\\omega'\n",
")\n",
"om.display()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"もちろん、既定のフレームとは異なるフレームでの要素を指定することもできます。\n",
"\n",
"
Of course, one may set the components in a frame different from the default one:
\n",
"
The components in the coframe $(\\mathrm{d}x,\\mathrm{d}y,\\mathrm{d}z)$ are updated automatically:
\n",
"
Let us revert to the values set previously:
\n",
"
This time, the components in the coframe $(\\mathrm{d}r, \\mathrm{d}\\theta,\\mathrm{d}\\phi)$ are those that are updated:
\n",
"
A 1-form acts on vector fields, resulting in a scalar field:
\n",
"
In the case of a differential 1-form, the following identity holds:
\n",
"1-forms are Sage *element* objects, whose *parent* is the $C^\\infty(U)$-module $\\Omega^{1}(U)$ of all 1-forms defined on $U$:
"
]
},
{
"cell_type": "code",
"execution_count": 182,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\Omega^{1}\\left(U\\right)\\)"
],
"text/latex": [
"$\\displaystyle \\Omega^{1}\\left(U\\right)$"
],
"text/plain": [
"Free module Omega^1(U) of 1-forms on the Open subset U of the 3-dimensional differentiable manifold M"
]
},
"execution_count": 182,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df.parent()"
]
},
{
"cell_type": "code",
"execution_count": 183,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Free module Omega^1(U) of 1-forms on the Open subset U of the 3-dimensional differentiable manifold M\n"
]
}
],
"source": [
"print(df.parent())"
]
},
{
"cell_type": "code",
"execution_count": 184,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Free module Omega^1(U) of 1-forms on the Open subset U of the 3-dimensional differentiable manifold M\n"
]
}
],
"source": [
"print(om.parent())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$\\Omega^{1}(U)$は自由モジュール $\\mathfrak{X}(U)$の双対です。\n",
"\n",
"$\\Omega^{1}(U)$ is actually the dual of the free module $\\mathfrak{X}(U)$:
"
]
},
{
"cell_type": "code",
"execution_count": 185,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\mathrm{True}\\)"
],
"text/latex": [
"$\\displaystyle \\mathrm{True}$"
],
"text/plain": [
"True"
]
},
"execution_count": 185,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df.parent() is v.parent().dual()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 微分形式と外積, 外微分 :Differential forms and exterior calculus\n",
"二つの1ー形式の **外積** は `wedge()` メソッドを使って求められます。その結果は 2-形式となります。\n",
"\n",
"
Differential forms and exterior calculus
\n",
"
The exterior product of two 1-forms is taken via the method wedge() and results in a 2-form:
\n",
"
A matrix view of the components:
\n",
"
Displaying only the non-vanishing components, skipping the redundant ones (i.e. those that can be deduced by antisymmetry):
\n",
"
The 2-form $\\omega\\wedge\\mathrm{d}f$ can be expanded on the $(\\mathrm{d}r,\\mathrm{d}\\theta,\\mathrm{d}\\phi)$ coframe:
\n",
"
As a 2-form, $A:=\\omega\\wedge\\mathrm{d}f$ can be applied to a pair of vectors and is antisymmetric:
\n",
"
The exterior derivative of a differential form:
\n",
"Instead of invoking the method `exterior_derivative()`, one can use the function `diff()` (available in SageMath 9.2 or higher):
"
]
},
{
"cell_type": "code",
"execution_count": 194,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\mathrm{d}\\omega\\)"
],
"text/latex": [
"$\\displaystyle \\mathrm{d}\\omega$"
],
"text/plain": [
"2-form domega on the Open subset U of the 3-dimensional differentiable manifold M"
]
},
"execution_count": 194,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"dom = diff(om)\n",
"dom"
]
},
{
"cell_type": "code",
"execution_count": 195,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"3-form dA on the Open subset U of the 3-dimensional differentiable manifold M\n"
]
},
{
"data": {
"text/html": [
"\\(\\displaystyle \\mathrm{d}A = \\left( -6 \\, y z^{2} - 2 \\, y - 1 \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\wedge \\mathrm{d} z\\)"
],
"text/latex": [
"$\\displaystyle \\mathrm{d}A = \\left( -6 \\, y z^{2} - 2 \\, y - 1 \\right) \\mathrm{d} x\\wedge \\mathrm{d} y\\wedge \\mathrm{d} z$"
],
"text/plain": [
"dA = (-6*y*z^2 - 2*y - 1) dx∧dy∧dz"
]
},
"execution_count": 195,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"da = diff(a) ; print(da) ; da.display()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**外微分** を続けて作用させると、結果は0になります。\n",
"\n",
"
The exterior derivative is nilpotent:
\n",
"
The Lie derivative of any tensor field with respect to a vector field is computed by the method lie_derivative(), with the vector field as the argument:
\n",
"
Let us check Cartan identity on the 1-form $\\omega$:
\n",
"
$\\mathcal{L}_v \\omega = v\\cdot \\mathrm{d}\\omega + \\mathrm{d}\\langle \\omega, v\\rangle$
\n",
"
and on the 2-form $A$:
\n",
"
$\\mathcal{L}_v A = v\\cdot \\mathrm{d}A + \\mathrm{d}(v\\cdot A)$
\n",
"
The Lie derivative of a vector field along another one is the commutator of the two vectors fields:
\n",
"Up to now, we have encountered tensor fields\n",
"\n",
"- of type (0,0) (i.e. scalar fields),\n",
"- of type (1,0) (i.e. vector fields),\n",
"- of type (0,1) (i.e. 1-forms),\n",
"- of type (0,2) and antisymmetric (i.e. 2-forms).\n",
"\n",
"More generally, tensor fields of any type $(p,q)$ can be introduced in SageMath. For instance a tensor field of type (1,2) on the open subset $U$ is declared as follows:
"
]
},
{
"cell_type": "code",
"execution_count": 203,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Tensor field T of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M\n"
]
}
],
"source": [
"t = U.tensor_field(1, 2, name='T') ; print(t)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"ベクトルや1-形式と同じように、テンソル場の既定のフレームに関する要素を `[]`演算子を使って設定できます。\n",
"\n",
"
As for vectors or 1-forms, the tensor's components with respect to the domain's default frame are set by means of square brackets:
\n",
"
Unset components are zero:
\n",
"
Display of the nonzero components:
\n",
"
Double square brackets return the component (still w.r.t. the default frame) as a scalar field, while single square brackets return the expression of this scalar field in terms of the domain's default coordinates:
\n",
"
A tensor field of type (1,2) maps a 3-tuple (1-form, vector field, vector field) to a scalar field:
\n",
"
As for vectors and differential forms, the tensor components can be taken in any frame defined on the manifold:
\n",
"
The tensor product $\\otimes$ is denoted by `*`:
\n",
"
The tensor product preserves the (anti)symmetries: since $A$ is a 2-form, it is antisymmetric with respect to its two arguments (positions 0 and 1); as a result, b is antisymmetric with respect to its last two arguments (positions 1 and 2):
\n",
"
Standard tensor arithmetics is implemented:
\n",
"**Tensor contractions** are dealt with by the methods`trace()` and `contract()`: for instance, let us contract the tensor $T$ w.r.t. its first two arguments (positions 0 and 1), i.e. let us form the tensor $c$ of components $c_i = T^k_{\\ \\, k i}$:
"
]
},
{
"cell_type": "code",
"execution_count": 217,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1-form on the Open subset U of the 3-dimensional differentiable manifold M\n"
]
}
],
"source": [
"c = t.trace(0,1)\n",
"print(c)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"`trace(0,1)`と入力する代わりに、**指標表記** を使って縮約を指定できます。指標は文字列として`[]`演算子の中に記述されます。\n",
"文字列中で、\"^\" は反変指標の前、\"_\" は共変指標の前、に置かれます。\n",
"\n",
"An alternative to the writing `trace(0,1)` is to use the **index notation** to denote the contraction: the indices are given in a string inside the `[]` operator, with `'^'` in front of the contravariant indices and `'_'` in front of the covariant ones:
"
]
},
{
"cell_type": "code",
"execution_count": 218,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1-form on the Open subset U of the 3-dimensional differentiable manifold M\n"
]
},
{
"data": {
"text/html": [
"\\(\\displaystyle \\mathrm{True}\\)"
],
"text/latex": [
"$\\displaystyle \\mathrm{True}$"
],
"text/plain": [
"True"
]
},
"execution_count": 218,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c1 = t['^k_ki']\n",
"print(c1)\n",
"c1 == c"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"縮約は繰り返された指標(ここでは $k$ です)について実行されます。それ以外の文字(ここでは $i$ )は任意です。\n",
"\n",
"The contraction is performed on the repeated index (here `k`); the letter denoting the remaining index (here `i`) is arbitrary:
"
]
},
{
"cell_type": "code",
"execution_count": 219,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\mathrm{True}\\)"
],
"text/latex": [
"$\\displaystyle \\mathrm{True}$"
],
"text/plain": [
"True"
]
},
"execution_count": 219,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t['^k_kj'] == c"
]
},
{
"cell_type": "code",
"execution_count": 220,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\mathrm{True}\\)"
],
"text/latex": [
"$\\displaystyle \\mathrm{True}$"
],
"text/plain": [
"True"
]
},
"execution_count": 220,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t['^b_ba'] == c"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"縮約されない指標については \".\"（ドット）で置き換え可能です。\n",
"\n",
"
It can even be replaced by a dot:
\n",
"
LaTeX notations are allowed:
\n",
"as well as Greek letters (only for SageMath 9.2 or higher):
"
]
},
{
"cell_type": "code",
"execution_count": 223,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\mathrm{True}\\)"
],
"text/latex": [
"$\\displaystyle \\mathrm{True}$"
],
"text/plain": [
"True"
]
},
"execution_count": 223,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t['^μ_μα'] == c"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"テンソル場 $T$ とベクトル場 $v$ の縮約 $T^i_{\\ j k} v^k$ は次のようにして、実行されます。ここで、2は$T$の最後の指標の場所、0は$v$の唯一の指標の場所です。\n",
"\n",
"
The contraction $T^i_{\\ j k} v^k$ of the tensor fields $T$ and $v$ is taken as follows (2 refers to the last index position of $T$ and 0 to the only index position of v):
\n",
"
Since 2 corresponds to the last index position of $T$ and 0 to the first index position of $v$, a shortcut for the above is
\n",
"
Instead of contract(), the index notation, combined with the * operator, can be used to denote the contraction:
\n",
"
The non-repeated indices can be replaced by dots:
\n",
"
A Riemannian metric on the manifold $\\mathcal{M}$ is declared as follows:
\n",
"
It is a symmetric tensor field of type (0,2):
\n",
"
The metric is initialized by its components with respect to some vector frame. For instance, using the default frame of $\\mathcal{M}$:
\n",
"
The components w.r.t. another vector frame are obtained as for any tensor field:
\n",
"
Of course, the metric acts on vector pairs:
\n",
"
The Levi-Civita connection associated to the metric $g$:
\n",
"
The Christoffel symbols with respect to the manifold's default coordinates:
\n",
"
The Christoffel symbols with respect to the coordinates $(r,\\theta,\\phi)$:
\n",
"A nice view is obtained via the method `display()` (by default, only the nonzero connection coefficients are shown):
"
]
},
{
"cell_type": "code",
"execution_count": 240,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -r \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -r \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, r }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, r }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}\\)"
],
"text/latex": [
"$\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -r \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -r \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\theta} \\, r }^{ \\, {\\theta} \\phantom{\\, {\\theta}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, r }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, r} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\phi} \\, {\\theta} }^{ \\, {\\phi} \\phantom{\\, {\\phi}} \\phantom{\\, {\\theta}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}$"
],
"text/plain": [
"Gam^r_th,th = -r \n",
"Gam^r_ph,ph = -r*sin(th)^2 \n",
"Gam^th_r,th = 1/r \n",
"Gam^th_th,r = 1/r \n",
"Gam^th_ph,ph = -cos(th)*sin(th) \n",
"Gam^ph_r,ph = 1/r \n",
"Gam^ph_th,ph = cos(th)/sin(th) \n",
"Gam^ph_ph,r = 1/r \n",
"Gam^ph_ph,th = cos(th)/sin(th) "
]
},
"execution_count": 240,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"nabla.display(frame=Y.frame(), chart=Y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"計量の`christoffel_symbols_display()`メソッドを使うこともできます。この関数は（既定の設定では）独立なChristoffel記号だけを表示します。\n",
"\n",
"One may also use the method `christoffel_symbols_display()` of the metric, which (by default) displays only the non-redundant Christoffel symbols:
"
]
},
{
"cell_type": "code",
"execution_count": 241,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\\(\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -r \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -r \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}\\)"
],
"text/latex": [
"$\\displaystyle \\begin{array}{lcl} \\Gamma_{ \\phantom{\\, r} \\, {\\theta} \\, {\\theta} }^{ \\, r \\phantom{\\, {\\theta}} \\phantom{\\, {\\theta}} } & = & -r \\\\ \\Gamma_{ \\phantom{\\, r} \\, {\\phi} \\, {\\phi} }^{ \\, r \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -r \\sin\\left({\\theta}\\right)^{2} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, r \\, {\\theta} }^{ \\, {\\theta} \\phantom{\\, r} \\phantom{\\, {\\theta}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\theta}} \\, {\\phi} \\, {\\phi} }^{ \\, {\\theta} \\phantom{\\, {\\phi}} \\phantom{\\, {\\phi}} } & = & -\\cos\\left({\\theta}\\right) \\sin\\left({\\theta}\\right) \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, r \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, r} \\phantom{\\, {\\phi}} } & = & \\frac{1}{r} \\\\ \\Gamma_{ \\phantom{\\, {\\phi}} \\, {\\theta} \\, {\\phi} }^{ \\, {\\phi} \\phantom{\\, {\\theta}} \\phantom{\\, {\\phi}} } & = & \\frac{\\cos\\left({\\theta}\\right)}{\\sin\\left({\\theta}\\right)} \\end{array}$"
],
"text/plain": [
"Gam^r_th,th = -r \n",
"Gam^r_ph,ph = -r*sin(th)^2 \n",
"Gam^th_r,th = 1/r \n",
"Gam^th_ph,ph = -cos(th)*sin(th) \n",
"Gam^ph_r,ph = 1/r \n",
"Gam^ph_th,ph = cos(th)/sin(th) "
]
},
"execution_count": 241,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"g.christoffel_symbols_display(Y)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 接続の共変微分としての作用。\n",
"\n",
"
The connection acting as a covariant derivative:
\n",
"
Being a Levi-Civita connection, $\\nabla_g$ is torsion.free:
\n",
"Levi-Civita接続は定義から、対称な接続係数なので、捩れはない。\n",
"
\n",
"
In the present case, it is also flat:
\n",
"
Let us consider a non-flat metric, by changing $g_{rr}$ to $1/(1+r^2)$:
\n",
"
For convenience, we change the default chart on the domain $U$ to Y=$(U,(r,\\theta,\\phi))$:
\n",
"
In this way, we do not have to specify Y when asking for coordinate expressions in terms of $(r,\\theta,\\phi)$:
\n",
"
We recognize the metric of the hyperbolic space $\\mathbb{H}^3$. Its expression in terms of the chart $(U,(x,y,z))$ is
\n",
"
A matrix view of the components may be more appropriate:
\n",
"
We extend these components, a priori defined only on $U$, to the whole manifold $\\mathcal{M}$, by demanding the same coordinate expressions in the frame associated to the chart X=$(\\mathcal{M},(x,y,z))$:
\n",
"
The Levi-Civita connection is automatically recomputed, after the change in $g$:
\n",
"
In particular, the Christoffel symbols are different:
\n",
"
The Riemann tensor is now
\n",
"
Note that it can be accessed directely via the metric, without any explicit mention of the connection:
\n",
"
The Ricci tensor is
\n",
"
The Weyl tensor is:
\n",
"
Weylテンソルについては Weinberg Page 145, 内山 page. 322, eq. (52.25) を参照
\n",
"
\n",
"
The Weyl tensor vanishes identically because the dimension of $\\mathcal{M}$ is 3.
\n",
"
Finally, the Ricci scalar is
\n",
"
We recover the fact that $\\mathbb{H}^3$ is a Riemannian manifold of constant negative curvature.
\n",
"
\n",
"
The most important tensor transformation induced by the metric $g$ is the so-called musical isomorphism, or index raising and index lowering:
\n",
"Raising the last index (position 2) of $T$ with $g$:
"
]
},
{
"cell_type": "code",
"execution_count": 267,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Tensor field T of type (1,2) on the Open subset U of the 3-dimensional differentiable manifold M Tensor field of type (2,1) on the Open subset U of the 3-dimensional differentiable manifold M\n"
]
}
],
"source": [
"s = t.up(g, 2)\n",
"print(t,s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"上記の式で引き上げられた指標は反変な指数のうち最後のものになることに注意しましょう。つまり、メソッド`up`が返すテンソル $s$ は\n",
"$$ s^{ab}_{\\ \\ \\ \\, c} = g^{bi} T^a_{\\ \\ ic}$$\n",
"です。\n",
"\n",
"Note that the raised index becomes the *last* one among the contravariant indices, i.e. the tensor $s$ returned by the method `up` is \n",
"$$ s^{ab}_{\\ \\ \\ \\, c} = g^{bi} T^a_{\\ \\ ic}$$
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"これについての詳細は[up()の解説](https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/tensorfield.html#sage.manifolds.differentiable.tensorfield.TensorField.up) をご覧ください。\n",
"\n",
"See the [up() documentation](https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/tensorfield.html#sage.manifolds.differentiable.tensorfield.TensorField.up) for more details."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$T$の全ての共変な指標(この例では 位置 1と2）を引き上げます。\n",
"\n",
"
Raising all the covariant indices of $T$ (i.e. those at the positions 1 and 2):
\n",
"Note that the lowered index becomes the *first* one among the covariant indices, i.e. the tensor $s$ returned by the method `down` is \n",
"$$ s_{abc} = g_{ai} T^i_{\\ \\ bc}$$
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"これについての詳細は[down()の解説](https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/tensorfield.html#sage.manifolds.differentiable.tensorfield.TensorField.down) をご覧ください。\n",
"\n",
"See the [down() documentation](https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/tensorfield.html#sage.manifolds.differentiable.tensorfield.TensorField.down) for more details.
"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Hodge双対(星印演算子) ： Hodge duality\n",
"計量 $g$についての体積 3-形式(Levi-Civita テンソル ）は\n",
"\n",
"
The volume 3-form (Levi-Civita tensor) associated with the metric $g$ is
\n",
"
We check the classical formula $\\star f = f\\, \\epsilon_g$, or, more precisely, $\\star f = f\\, \\epsilon_g|_U$ (for $f$ is defined on $U$ only):
\n",
"
The Hodge dual of a 1-form is a 2-form:
\n",
"
The Hodge dual of a 2-form is a 1-form:
\n",
"
Finally, the Hodge dual of a 3-form is a 0-form:
\n",
"
In dimension 3 and for a Riemannian metric, the Hodge star is idempotent:
\n",
"
To get the list of functions (methods) that can be called on a object, type the name of the object, followed by a dot and the TAB key, e.g.
\n",
"
sa.
\n",
"
To get information on an object or a method, use the question mark:
\n",
"Using a double question mark leads directly to the **Python source code** (SageMath is **open source**, isn't it?):
"
]
},
{
"cell_type": "code",
"execution_count": 288,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"\u001b[0;31mSignature:\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mricci_scalar\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mname\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlatex_name\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mDocstring:\u001b[0m\n",
" Return the metric's Ricci scalar.\n",
"\n",
" The Ricci scalar is the scalar field r defined from the Ricci\n",
" tensor Ric and the metric tensor g by\n",
"\n",
" r = g^{ij} Ric_{ij}\n",
"\n",
" INPUT:\n",
"\n",
" * \"name\" -- (default: \"None\") name given to the Ricci scalar; if\n",
" none, it is set to \"r(g)\", where \"g\" is the metric's name\n",
"\n",
" * \"latex_name\" -- (default: \"None\") LaTeX symbol to denote the\n",
" Ricci scalar; if none, it is set to \"r(g)\", where \"g\" is the\n",
" metric's name\n",
"\n",
" OUTPUT:\n",
"\n",
" * the Ricci scalar r, as an instance of \"DiffScalarField\"\n",
"\n",
" EXAMPLES:\n",
"\n",
" Ricci scalar of the standard metric on the 2-sphere:\n",
"\n",
" sage: M = Manifold(2, 'S^2', start_index=1)\n",
" sage: U = M.open_subset('U') # the complement of a meridian (domain of spherical coordinates)\n",
" sage: c_spher. = U.chart(r'th:(0,pi):\\theta ph:(0,2*pi):\\phi')\n",
" sage: a = var('a') # the sphere radius\n",
" sage: g = U.metric('g')\n",
" sage: g[1,1], g[2,2] = a^2, a^2*sin(th)^2\n",
" sage: g.display() # standard metric on the 2-sphere of radius a:\n",
" g = a^2 dth⊗dth + a^2*sin(th)^2 dph⊗dph\n",
" sage: g.ricci_scalar()\n",
" Scalar field r(g) on the Open subset U of the 2-dimensional\n",
" differentiable manifold S^2\n",
" sage: g.ricci_scalar().display() # The Ricci scalar is constant:\n",
" r(g): U → ℝ\n",
" (th, ph) ↦ 2/a^2\n",
"\u001b[0;31mSource:\u001b[0m \n",
" \u001b[0;32mdef\u001b[0m \u001b[0mricci_scalar\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mlatex_name\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;34mr\"\"\"\u001b[0m\n",
"\u001b[0;34m Return the metric's Ricci scalar.\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m The Ricci scalar is the scalar field `r` defined from the Ricci tensor\u001b[0m\n",
"\u001b[0;34m `Ric` and the metric tensor `g` by\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m .. MATH::\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m r = g^{ij} Ric_{ij}\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m INPUT:\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m - ``name`` -- (default: ``None``) name given to the Ricci scalar;\u001b[0m\n",
"\u001b[0;34m if none, it is set to \"r(g)\", where \"g\" is the metric's name\u001b[0m\n",
"\u001b[0;34m - ``latex_name`` -- (default: ``None``) LaTeX symbol to denote the\u001b[0m\n",
"\u001b[0;34m Ricci scalar; if none, it is set to \"\\\\mathrm{r}(g)\", where \"g\"\u001b[0m\n",
"\u001b[0;34m is the metric's name\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m OUTPUT:\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m - the Ricci scalar `r`, as an instance of\u001b[0m\n",
"\u001b[0;34m :class:`~sage.manifolds.differentiable.scalarfield.DiffScalarField`\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m EXAMPLES:\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m Ricci scalar of the standard metric on the 2-sphere::\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m sage: M = Manifold(2, 'S^2', start_index=1)\u001b[0m\n",
"\u001b[0;34m sage: U = M.open_subset('U') # the complement of a meridian (domain of spherical coordinates)\u001b[0m\n",
"\u001b[0;34m sage: c_spher. = U.chart(r'th:(0,pi):\\theta ph:(0,2*pi):\\phi')\u001b[0m\n",
"\u001b[0;34m sage: a = var('a') # the sphere radius\u001b[0m\n",
"\u001b[0;34m sage: g = U.metric('g')\u001b[0m\n",
"\u001b[0;34m sage: g[1,1], g[2,2] = a^2, a^2*sin(th)^2\u001b[0m\n",
"\u001b[0;34m sage: g.display() # standard metric on the 2-sphere of radius a:\u001b[0m\n",
"\u001b[0;34m g = a^2 dth⊗dth + a^2*sin(th)^2 dph⊗dph\u001b[0m\n",
"\u001b[0;34m sage: g.ricci_scalar()\u001b[0m\n",
"\u001b[0;34m Scalar field r(g) on the Open subset U of the 2-dimensional\u001b[0m\n",
"\u001b[0;34m differentiable manifold S^2\u001b[0m\n",
"\u001b[0;34m sage: g.ricci_scalar().display() # The Ricci scalar is constant:\u001b[0m\n",
"\u001b[0;34m r(g): U → ℝ\u001b[0m\n",
"\u001b[0;34m (th, ph) ↦ 2/a^2\u001b[0m\n",
"\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m \"\"\"\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_ricci_scalar\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mmanif\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_ambient_domain\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mric\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mricci\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mig\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0minverse\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mframe\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mig\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcommon_basis\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mric\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mcric\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mric\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_components\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mframe\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mcig\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mig\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_components\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mframe\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mrsum1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mmanif\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mirange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mrsum1\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0mcig\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mcric\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mrsum2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;36m0\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mmanif\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mirange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0mj\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mmanif\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mirange\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mstart\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mrsum2\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0mcig\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m*\u001b[0m \u001b[0mcric\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mj\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_ricci_scalar\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mrsum1\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mrsum2\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mname\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_ricci_scalar\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_name\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m\"r(\"\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_name\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34m\")\"\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_ricci_scalar\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_name\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mname\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mlatex_name\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_ricci_scalar\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_latex_name\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34mr\"\\mathrm{r}\\left(\"\u001b[0m \u001b[0;34m+\u001b[0m \\\n",
" \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_latex_name\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0;34mr\"\\right)\"\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_ricci_scalar\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_latex_name\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mlatex_name\u001b[0m\u001b[0;34m\u001b[0m\n",
"\u001b[0;34m\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_ricci_scalar\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mFile:\u001b[0m /private/var/tmp/sage-10.0-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/manifolds/differentiable/metric.py\n",
"\u001b[0;31mType:\u001b[0m method\n"
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"source": [
"## さらに先へ：Going further\n",
"\n",
"[ページ:SageManifolds の例](https://sagemanifolds.obspm.fr/examples.html)をご覧ください。とくに、平行化不能な多様体（スカラー場は少なくとも二つの座標系で定義される必要があり、\n",
"モジュール $\\mathfrak{X}(\\mathcal{M})$はもはやフリーではなく、\n",
"テンソル場は少なくとも二つのフレームで定義される必要があります）\n",
"での使い方については、\n",
"[2-次元 球面](https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_sphere_S2.ipynb)\n",
"が役に立つでしょう。\n",
"\n",
"Christian Bär氏による[チュートリアル ビデオ](https://www.youtube.com/playlist?list=PLnrOCYZpQUuJlnQbQ48zgGk-Ks1t145Yw)もご覧ください。\n",
"\n",
"Have a look at the [examples on SageManifolds page](https://sagemanifolds.obspm.fr/examples.html), especially the \n",
"[2-dimensional sphere](https://nbviewer.jupyter.org/github/sagemanifolds/SageManifolds/blob/master/Notebooks/SM_sphere_S2.ipynb) for usage on a non-parallelizable manifold (each scalar field has to be defined in at least two coordinate charts, the $C^\\infty(\\mathcal{M})$-module $\\mathfrak{X}(\\mathcal{M})$ is no longer free and each tensor field has to be defined in at least two vector frames).\n",
"\n",
"You may also take a look at the [tutorial videos](https://www.youtube.com/playlist?list=PLnrOCYZpQUuJlnQbQ48zgGk-Ks1t145Yw) by Christian Bär.
"
]
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