Merge #30473

src/doc/en/reference/misc/index.rst

@@ -168,6 +168,7 @@ Formatted Output
    sage/typeset/character_art_factory
    sage/typeset/ascii_art
    sage/typeset/unicode_art
+   sage/typeset/unicode_characters
    sage/misc/repr
    sage/misc/sage_input
    sage/misc/table

src/doc/en/thematic_tutorials/vector_calculus/vector_calc_advanced.rst

@@ -58,9 +58,9 @@ while there are five vector frames defined on `\mathbb{E}^3`::
 
     sage: E.frames()
     [Coordinate frame (E^3, (e_x,e_y,e_z)),
-     Coordinate frame (E^3, (d/dr,d/dth,d/dph)),
+     Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph)),
      Vector frame (E^3, (e_r,e_th,e_ph)),
-     Coordinate frame (E^3, (d/drh,d/dph,d/dz)),
+     Coordinate frame (E^3, (∂/∂rh,∂/∂ph,∂/∂z)),
      Vector frame (E^3, (e_rh,e_ph,e_z))]
 
 Indeed, there are two frames associated with each of the three coordinate
@@ -85,9 +85,9 @@ On the other side, the coordinate frames `\left(\frac{\partial}{\partial r},
 coordinate charts::
 
     sage: spherical.frame()
-    Coordinate frame (E^3, (d/dr,d/dth,d/dph))
+    Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph))
     sage: cylindrical.frame()
-    Coordinate frame (E^3, (d/drh,d/dph,d/dz))
+    Coordinate frame (E^3, (∂/∂rh,∂/∂ph,∂/∂z))
 
 
 Charts as maps `\mathbb{E}^3 \rightarrow \mathbb{R}^3`
@@ -128,7 +128,7 @@ The default metric tensor of `\mathbb{E}^3` is::
     sage: g
     Riemannian metric g on the Euclidean space E^3
     sage: g.display()
-    g = dx*dx + dy*dy + dz*dz
+    g = dx⊗dx + dy⊗dy + dz⊗dz
     sage: g[:]
     [1 0 0]
     [0 1 0]
@@ -139,7 +139,7 @@ Cartesian one. Of course, we may ask for display with respect to other
 frames::
 
     sage: g.display(spherical_frame)
-    g = e^r*e^r + e^th*e^th + e^ph*e^ph
+    g = e^r⊗e^r + e^th⊗e^th + e^ph⊗e^ph
     sage: g[spherical_frame, :]
     [1 0 0]
     [0 1 0]
@@ -159,20 +159,20 @@ contrary, in the coordinate frame
 which is not orthonormal, some components differ from 0 or 1::
 
     sage: g.display(spherical.frame())
-    g = dr*dr + (x^2 + y^2 + z^2) dth*dth + (x^2 + y^2) dph*dph
+    g = dr⊗dr + (x^2 + y^2 + z^2) dth⊗dth + (x^2 + y^2) dph⊗dph
 
 Note that the components are expressed in terms of the default chart, namely
 the Cartesian one. To have them displayed in terms of the spherical chart, we
 have to provide the latter as the second argument of the method
 ``display()``::
 
     sage: g.display(spherical.frame(), spherical)
-    g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
+    g = dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph
 
 Since SageMath 8.8, a shortcut is::
 
     sage: g.display(spherical)
-    g = dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
+    g = dr⊗dr + r^2 dth⊗dth + r^2*sin(th)^2 dph⊗dph
 
 The matrix view of the components is obtained via the square bracket operator::
 
@@ -184,9 +184,9 @@ The matrix view of the components is obtained via the square bracket operator::
 Similarly, for cylindrical coordinates, we have::
 
     sage: g.display(cylindrical_frame)
-    g = e^rh*e^rh + e^ph*e^ph + e^z*e^z
+    g = e^rh⊗e^rh + e^ph⊗e^ph + e^z⊗e^z
     sage: g.display(cylindrical)
-    g = drh*drh + rh^2 dph*dph + dz*dz
+    g = drh⊗drh + rh^2 dph⊗dph + dz⊗dz
     sage: g[cylindrical.frame(), :, cylindrical]
     [   1    0    0]
     [   0 rh^2    0]
@@ -232,38 +232,38 @@ Levi-Civita tensor (also called *volume form*) associated with `g`
     sage: epsilon is E.volume_form()
     True
     sage: epsilon.display()
-    epsilon = dx/\dy/\dz
+    epsilon = dx∧dy∧dz
     sage: epsilon.display(spherical)
-    epsilon = r^2*sin(th) dr/\dth/\dph
+    epsilon = r^2*sin(th) dr∧dth∧dph
     sage: epsilon.display(cylindrical)
-    epsilon = rh drh/\dph/\dz
+    epsilon = rh drh∧dph∧dz
 
 Checking that all orthonormal frames introduced above are right-handed::
 
     sage: ex, ey, ez = E.cartesian_frame()[:]
     sage: epsilon(ex, ey, ez).display()
-    epsilon(e_x,e_y,e_z): E^3 --> R
-       (x, y, z) |--> 1
-       (r, th, ph) |--> 1
-       (rh, ph, z) |--> 1
+    epsilon(e_x,e_y,e_z): E^3 → ℝ
+       (x, y, z) ↦ 1
+       (r, th, ph) ↦ 1
+       (rh, ph, z) ↦ 1
 
 ::
 
     sage: epsilon(*spherical_frame)
     Scalar field epsilon(e_r,e_th,e_ph) on the Euclidean space E^3
     sage: epsilon(*spherical_frame).display()
-    epsilon(e_r,e_th,e_ph): E^3 --> R
-       (x, y, z) |--> 1
-       (r, th, ph) |--> 1
-       (rh, ph, z) |--> 1
+    epsilon(e_r,e_th,e_ph): E^3 → ℝ
+       (x, y, z) ↦ 1
+       (r, th, ph) ↦ 1
+       (rh, ph, z) ↦ 1
 
 ::
 
     sage: epsilon(*cylindrical_frame).display()
-    epsilon(e_rh,e_ph,e_z): E^3 --> R
-       (x, y, z) |--> 1
-       (r, th, ph) |--> 1
-       (rh, ph, z) |--> 1
+    epsilon(e_rh,e_ph,e_z): E^3 → ℝ
+       (x, y, z) ↦ 1
+       (r, th, ph) ↦ 1
+       (rh, ph, z) ↦ 1
 
 
 Vector fields as derivations
@@ -273,20 +273,20 @@ Let `f` be a scalar field on `\mathbb{E}^3`::
 
     sage: f = E.scalar_field(x^2+y^2 - z^2, name='f')
     sage: f.display()
-    f: E^3 --> R
-       (x, y, z) |--> x^2 + y^2 - z^2
-       (r, th, ph) |--> -2*r^2*cos(th)^2 + r^2
-       (rh, ph, z) |--> rh^2 - z^2
+    f: E^3 → ℝ
+       (x, y, z) ↦ x^2 + y^2 - z^2
+       (r, th, ph) ↦ -2*r^2*cos(th)^2 + r^2
+       (rh, ph, z) ↦ rh^2 - z^2
 
 Vector fields act as derivations on scalar fields::
 
     sage: v(f)
     Scalar field v(f) on the Euclidean space E^3
     sage: v(f).display()
-    v(f): E^3 --> R
-       (x, y, z) |--> -2*z^3
-       (r, th, ph) |--> -2*r^3*cos(th)^3
-       (rh, ph, z) |--> -2*z^3
+    v(f): E^3 → ℝ
+       (x, y, z) ↦ -2*z^3
+       (r, th, ph) ↦ -2*r^3*cos(th)^3
+       (rh, ph, z) ↦ -2*z^3
     sage: v(f) == v.dot(f.gradient())
     True
 
@@ -346,9 +346,9 @@ vector frames defined on `\mathbb{E}^3`::
 
     sage: XE.bases()
     [Coordinate frame (E^3, (e_x,e_y,e_z)),
-     Coordinate frame (E^3, (d/dr,d/dth,d/dph)),
+     Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph)),
      Vector frame (E^3, (e_r,e_th,e_ph)),
-     Coordinate frame (E^3, (d/drh,d/dph,d/dz)),
+     Coordinate frame (E^3, (∂/∂rh,∂/∂ph,∂/∂z)),
      Vector frame (E^3, (e_rh,e_ph,e_z))]
 
 
@@ -385,9 +385,9 @@ The bases on `T_p\mathbb{E}^3` are inherited from the vector frames of
 
     sage: Tp.bases()
     [Basis (e_x,e_y,e_z) on the Tangent space at Point p on the Euclidean space E^3,
-     Basis (d/dr,d/dth,d/dph) on the Tangent space at Point p on the Euclidean space E^3,
+     Basis (∂/∂r,∂/∂th,∂/∂ph) on the Tangent space at Point p on the Euclidean space E^3,
      Basis (e_r,e_th,e_ph) on the Tangent space at Point p on the Euclidean space E^3,
-     Basis (d/drh,d/dph,d/dz) on the Tangent space at Point p on the Euclidean space E^3,
+     Basis (∂/∂rh,∂/∂ph,∂/∂z) on the Tangent space at Point p on the Euclidean space E^3,
      Basis (e_rh,e_ph,e_z) on the Tangent space at Point p on the Euclidean space E^3]
 
 For instance, we have::

src/doc/en/thematic_tutorials/vector_calculus/vector_calc_cartesian.rst

@@ -152,8 +152,8 @@ which admits ``dot()`` as a shortcut alias::
     sage: s
     Scalar field u.v on the Euclidean space E^3
     sage: s.display()
-    u.v: E^3 --> R
-       (x, y, z) |--> -y*u_x(x, y, z) + x*u_y(x, y, z) + sin(x*y*z)*u_z(x, y, z)
+    u.v: E^3 → ℝ
+       (x, y, z) ↦ -y*u_x(x, y, z) + x*u_y(x, y, z) + sin(x*y*z)*u_z(x, y, z)
 
 It maps points of `\mathbb{E}^3` to real numbers::
 
@@ -181,8 +181,8 @@ It is a scalar field on `\mathbb{E}^3`::
     sage: s
     Scalar field |u| on the Euclidean space E^3
     sage: s.display()
-    |u|: E^3 --> R
-       (x, y, z) |--> sqrt(u_x(x, y, z)^2 + u_y(x, y, z)^2 + u_z(x, y, z)^2)
+    |u|: E^3 → ℝ
+       (x, y, z) ↦ sqrt(u_x(x, y, z)^2 + u_y(x, y, z)^2 + u_z(x, y, z)^2)
     sage: s.expr()
     sqrt(u_x(x, y, z)^2 + u_y(x, y, z)^2 + u_z(x, y, z)^2)
 
@@ -272,8 +272,8 @@ function of `(x,y,z)`::
 
     sage: F = E.scalar_field(function('f')(x,y,z), name='F')
     sage: F.display()
-    F: E^3 --> R
-       (x, y, z) |--> f(x, y, z)
+    F: E^3 → ℝ
+       (x, y, z) ↦ f(x, y, z)
 
 The value of `F` at a point::
 
@@ -287,8 +287,8 @@ The gradient of `F`::
     sage: grad(F).display()
     grad(F) = d(f)/dx e_x + d(f)/dy e_y + d(f)/dz e_z
     sage: norm(grad(F)).display()
-    |grad(F)|: E^3 --> R
-       (x, y, z) |--> sqrt((d(f)/dx)^2 + (d(f)/dy)^2 + (d(f)/dz)^2)
+    |grad(F)|: E^3 → ℝ
+       (x, y, z) ↦ sqrt((d(f)/dx)^2 + (d(f)/dy)^2 + (d(f)/dz)^2)
 
 
 Divergence
@@ -298,8 +298,8 @@ The divergence of the vector field `u`::
 
     sage: s = div(u)
     sage: s.display()
-    div(u): E^3 --> R
-       (x, y, z) |--> d(u_x)/dx + d(u_y)/dy + d(u_z)/dz
+    div(u): E^3 → ℝ
+       (x, y, z) ↦ d(u_x)/dx + d(u_y)/dy + d(u_z)/dz
 
 For `v` and `w`, we have::
 
@@ -360,8 +360,8 @@ The curl of a gradient is always zero::
 The divergence of a curl is always zero::
 
     sage: div(curl(u)).display()
-    div(curl(u)): E^3 --> R
-       (x, y, z) |--> 0
+    div(curl(u)): E^3 → ℝ
+       (x, y, z) ↦ 0
 
 An identity valid for any scalar field `F` and any vector field `u` is
 
@@ -382,8 +382,8 @@ The Laplacian `\Delta F` of a scalar field `F` is another scalar field::
 
     sage: s = laplacian(F)
     sage: s.display()
-    Delta(F): E^3 --> R
-       (x, y, z) |--> d^2(f)/dx^2 + d^2(f)/dy^2 + d^2(f)/dz^2
+    Delta(F): E^3 → ℝ
+       (x, y, z) ↦ d^2(f)/dx^2 + d^2(f)/dy^2 + d^2(f)/dz^2
 
 The following identity holds:
 

src/doc/en/thematic_tutorials/vector_calculus/vector_calc_change.rst

@@ -204,7 +204,7 @@ At this stage, `\mathbb{E}^3` is endowed with three vector frames::
 
     sage: E.frames()
     [Coordinate frame (E^3, (e_x,e_y,e_z)),
-     Coordinate frame (E^3, (d/dr,d/dth,d/dph)),
+     Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph)),
      Vector frame (E^3, (e_r,e_th,e_ph))]
 
 The second one is the *coordinate* frame `\left(\frac{\partial}{\partial r},
@@ -254,9 +254,9 @@ chart ``spherical``)::
 
     sage: for vec in spherical_frame:
     ....:     vec.display(spherical.frame(), spherical)
-    e_r = d/dr
-    e_th = 1/r d/dth
-    e_ph = 1/(r*sin(th)) d/dph
+    e_r = ∂/∂r
+    e_th = 1/r ∂/∂th
+    e_ph = 1/(r*sin(th)) ∂/∂ph
 
 
 Introducing cylindrical coordinates
@@ -299,9 +299,9 @@ There are now five vector frames defined on `\mathbb{E}^3`::
 
     sage: E.frames()
     [Coordinate frame (E^3, (e_x,e_y,e_z)),
-     Coordinate frame (E^3, (d/dr,d/dth,d/dph)),
+     Coordinate frame (E^3, (∂/∂r,∂/∂th,∂/∂ph)),
      Vector frame (E^3, (e_r,e_th,e_ph)),
-     Coordinate frame (E^3, (d/drh,d/dph,d/dz)),
+     Coordinate frame (E^3, (∂/∂rh,∂/∂ph,∂/∂z)),
      Vector frame (E^3, (e_rh,e_ph,e_z))]
 
 The orthonormal frame associated with cylindrical coordinates is
@@ -359,9 +359,9 @@ chart ``cylindrical``)::
 
     sage: for vec in cylindrical_frame:
     ....:     vec.display(cylindrical.frame(), cylindrical)
-    e_rh = d/drh
-    e_ph = 1/rh d/dph
-    e_z = d/dz
+    e_rh = ∂/∂rh
+    e_ph = 1/rh ∂/∂ph
+    e_z = ∂/∂z
 
 
 How to evaluate the coordinates of a point in various systems
@@ -423,19 +423,19 @@ transformations, the expression of `f` in terms of other coordinates is
 automatically computed::
 
     sage: f.display()
-    f: E^3 --> R
-       (x, y, z) |--> x^2 + y^2 - z^2
-       (r, th, ph) |--> -2*r^2*cos(th)^2 + r^2
-       (rh, ph, z) |--> rh^2 - z^2
+    f: E^3 → ℝ
+       (x, y, z) ↦ x^2 + y^2 - z^2
+       (r, th, ph) ↦ -2*r^2*cos(th)^2 + r^2
+       (rh, ph, z) ↦ rh^2 - z^2
 
 We can limit the output to a single coordinate system::
 
     sage: f.display(cartesian)
-    f: E^3 --> R
-       (x, y, z) |--> x^2 + y^2 - z^2
+    f: E^3 → ℝ
+       (x, y, z) ↦ x^2 + y^2 - z^2
     sage: f.display(cylindrical)
-    f: E^3 --> R
-       (rh, ph, z) |--> rh^2 - z^2
+    f: E^3 → ℝ
+       (rh, ph, z) ↦ rh^2 - z^2
 
 The coordinate expression in a given coordinate system is obtained via the
 method :meth:`~sage.manifolds.scalarfield.ScalarField.expr`::
@@ -468,10 +468,10 @@ The computation of the expressions of `g` in the other coordinate systems is
 triggered by the method ``display()``::
 
     sage: g.display()
-    g: E^3 --> R
-       (x, y, z) |--> x^2 + y^2 + z^2
-       (r, th, ph) |--> r^2
-       (rh, ph, z) |--> rh^2 + z^2
+    g: E^3 → ℝ
+       (x, y, z) ↦ x^2 + y^2 + z^2
+       (r, th, ph) ↦ r^2
+       (rh, ph, z) ↦ rh^2 + z^2
 
 
 How to express a vector field in various frames