Improve documentation for SymPy calculus on manifolds

src/sage/manifolds/continuous_map.py

@@ -1163,26 +1163,22 @@ def coord_functions(self, chart1=None, chart2=None):
             sage: Phi1.coord_functions(c_uv.restrict(A), c_xyz)
             Coordinate functions (u*v, u/v, u + v) on the Chart (A, (u, v))
 
-        The same test with ``sympy``::
+        Same example with SymPy as the symbolic calculus engine::
 
-            # sage: X.set_calculus_method('sympy')
-            sage: M = Manifold(2, 'M', structure='topological')
-            sage: N = Manifold(3, 'N', structure='topological')
-            sage: c_uv.<u,v> = M.chart(calc_method='sympy')
-            sage: c_xyz.<x,y,z> = N.chart(calc_method='sympy')
+            sage: M.set_calculus_method('sympy')
+            sage: N.set_calculus_method('sympy')
             sage: Phi = M.continuous_map(N, (u*v, u/v, u+v), name='Phi',
             ....:                        latex_name=r'\Phi')
-            sage: Phi.display()
-            Phi: M --> N
-               (u, v) |--> (x, y, z) = (u*v, u/v, u + v)
             sage: Phi.coord_functions(c_uv, c_xyz)
             Coordinate functions (u*v, u/v, u + v) on the Chart (M, (u, v))
-            sage: Phi.coord_functions() # equivalent to above since 'uv' and 'xyz' are default charts
-            Coordinate functions (u*v, u/v, u + v) on the Chart (M, (u, v))
-            sage: type(Phi.coord_functions())
-            <class 'sage.manifolds.chart_func.MultiCoordFunction'>
-
-
+            sage: Phi.coord_functions(c_UV, c_xyz)
+            Coordinate functions (-U**2/4 + V**2/4, -(U + V)/(U - V), V)
+             on the Chart (M, (U, V))
+            sage: Phi.coord_functions(c_UV, c_XYZ)
+            Coordinate functions ((-U**3 + U**2*V + U*V**2 + 2*U*V + 6*U - V**3
+             - 2*V**2 + 6*V)/(2*(U - V)), (U**3/4 - U**2*V/4 - U*V**2/4 + U*V
+             - U + V**3/4 - V**2 - V)/(U - V), (U**3 - U**2*V - U*V**2 - 4*U*V
+             - 8*U + V**3 + 4*V**2 - 8*V)/(4*(U - V))) on the Chart (M, (U, V))
 
         """
         dom1 = self._domain; dom2 = self._codomain
@@ -1826,7 +1822,6 @@ def __invert__(self):
         if not self._is_isomorphism:
             raise ValueError("the {} is not an isomorphism".format(self))
         coord_functions = {} # coordinate expressions of the result
-
         for (chart1, chart2) in self._coord_expression:
             coord_map = self._coord_expression[(chart1, chart2)]
             n1 = len(chart1._xx)

src/sage/manifolds/differentiable/affine_connection.py

@@ -293,20 +293,11 @@ class AffineConnection(SageObject):
         sage: nab.restrict(U)(a.restrict(V)) == da.restrict(W)
         True
 
-    Same tests with ``sympy``::
+    Same examples with SymPy as the engine for symbolic calculus::
 
         sage: M.set_calculus_method('sympy')
-
         sage: nab = M.affine_connection('nabla', r'\nabla')
-
-    The connection is first defined on the open subset U by means of its
-    coefficients w.r.t. the frame eU (the manifold's default frame)::
-
         sage: nab[0,0,0], nab[1,0,1] = x, x*y
-
-    The coefficients w.r.t the frame eV are deduced by continuation of the
-    coefficients w.r.t. the frame eVW on the open subset `W=U\cap V`::
-
         sage: for i in M.irange():
         ....:     for j in M.irange():
         ....:         for k in M.irange():
@@ -338,7 +329,6 @@ class AffineConnection(SageObject):
          manifold M
         sage: da.display(eU)
         nabla(a) = -x*y d/dx*dx - d/dx*dy + d/dy*dx - x*y**2 d/dy*dy
-
         sage: da.display(eV)
         nabla(a) = (-u**3/16 + u**2*v/16 - u**2/8 + u*v**2/16 - v**3/16 + v**2/8) d/du*du
          + (u**3/16 - u**2*v/16 - u**2/8 - u*v**2/16 + v**3/16 + v**2/8 + 1) d/du*dv

src/sage/manifolds/differentiable/diff_form.py

@@ -207,21 +207,16 @@ class DiffForm(TensorField):
         1/2*u^2*v du - 1/2*u^3 dv
 
 
-    Same tests with ``sympy``::
+    .. RUBRIC:: Examples with SymPy as the symbolic engine
+
+    From now on, we ask that all symbolic calculus on manifold `M` are
+    performed by SymPy::
 
         sage: M.set_calculus_method('sympy')
-        sage: U.set_calculus_method('sympy')
-        sage: V.set_calculus_method('sympy')
-        sage: W.set_calculus_method('sympy')
-        sage: a = M.diff_form(2, name='a') ; a
-        2-form a on the 2-dimensional differentiable manifold M
-        sage: a.parent()
-        Module Omega^2(M) of 2-forms on the 2-dimensional differentiable
-         manifold M
-        sage: a.degree()
-        2
 
-    Setting the components of ``a``::
+    We define a 2-form `a` as above::
+
+        sage: a = M.diff_form(2, name='a')
         sage: a[eU,0,1] = x*y^2 + 2*x
         sage: a.add_comp_by_continuation(eV, W, c_uv)
         sage: a.display(eU)
@@ -231,27 +226,17 @@ class DiffForm(TensorField):
 
     A 1-form on ``M``::
 
-        sage: a = M.one_form('a') ; a
-        1-form a on the 2-dimensional differentiable manifold M
-        sage: a.parent()
-        Module Omega^1(M) of 1-forms on the 2-dimensional differentiable
-         manifold M
-        sage: a.degree()
-        1
-
-    Setting the components of the 1-form in a consistent way::
-
+        sage: a = M.one_form('a')
         sage: a[eU,:] = [-y, x]
         sage: a.add_comp_by_continuation(eV, W, c_uv)
         sage: a.display(eU)
         a = -y dx + x dy
         sage: a.display(eV)
         a = v/2 du - u/2 dv
 
-    The exterior derivative of the 1-form is a 2-form::
+    The exterior derivative of ``a``::
 
-        sage: da = a.exterior_derivative() ; da
-        2-form da on the 2-dimensional differentiable manifold M
+        sage: da = a.exterior_derivative()
         sage: da.display(eU)
         da = 2 dx/\dy
         sage: da.display(eV)
@@ -263,35 +248,31 @@ class DiffForm(TensorField):
         sage: b[eU,:] = [1+x*y, x^2]
         sage: b.add_comp_by_continuation(eV, W, c_uv)
 
-    Adding two 1-forms results in another 1-form::
+    Adding two 1-forms::
 
-        sage: s = a + b ; s
-        1-form a+b on the 2-dimensional differentiable manifold M
+        sage: s = a + b
         sage: s.display(eU)
         a+b = (x*y - y + 1) dx + x*(x + 1) dy
         sage: s.display(eV)
         a+b = (u**2/4 + u*v/4 + v/2 + 1/2) du + (-u*v/4 - u/2 - v**2/4 + 1/2) dv
 
-    The exterior product of two 1-forms is a 2-form::
+    The exterior product of two 1-forms::
 
-        sage: s = a.wedge(b) ; s
-        2-form a/\b on the 2-dimensional differentiable manifold M
+        sage: s = a.wedge(b)
         sage: s.display(eU)
         a/\b = -x*(2*x*y + 1) dx/\dy
         sage: s.display(eV)
         a/\b = (u**3/8 + u**2*v/8 - u*v**2/8 + u/4 - v**3/8 + v/4) du/\dv
 
-    Multiplying a 1-form by a scalar field results in another 1-form::
+    Multiplying a 1-form by a scalar field::
 
         sage: f = M.scalar_field({c_xy: (x+y)^2, c_uv: u^2}, name='f')
-        sage: s = f*a ; s
-        1-form on the 2-dimensional differentiable manifold M
+        sage: s = f*a
         sage: s.display(eU)
         -y*(x**2 + 2*x*y + y**2) dx + x*(x**2 + 2*x*y + y**2) dy
         sage: s.display(eV)
         u**2*v/2 du - u**3/2 dv
 
-
     """
     def __init__(self, vector_field_module, degree, name=None, latex_name=None):
         r"""

src/sage/manifolds/differentiable/levi_civita_connection.py

@@ -720,7 +720,8 @@ def ricci(self, name=None, latex_name=None):
             sage: g[0,0], g[1,1] = -(1-2*m/r), 1/(1-2*m/r)
             sage: g[2,2], g[3,3] = r^2, (r*sin(th))^2
             sage: g.display()
-            g = (2*m/r - 1) dt*dt - 1/(2*m/r - 1) dr*dr + r^2 dth*dth + r^2*sin(th)^2 dph*dph
+            g = (2*m/r - 1) dt*dt - 1/(2*m/r - 1) dr*dr + r^2 dth*dth
+             + r^2*sin(th)^2 dph*dph
             sage: nab = g.connection() ; nab
             Levi-Civita connection nabla_g associated with the Lorentzian
              metric g on the 4-dimensional differentiable manifold M

src/sage/manifolds/manifold.py

@@ -1655,20 +1655,6 @@ def scalar_field(self, coord_expression=None, chart=None, name=None,
             F: U --> R
                (x, y, z) |--> cos(y)*sin(x) + z
 
-
-        Same tests with ``sympy``::
-
-            sage: c_xyz.set_calculus_method('sympy')
-            sage: f = U.scalar_field(sin(x)*cos(y) + z, name='F'); f
-            Scalar field F on the Open subset U of the 3-dimensional topological manifold M
-            sage: f.display()
-            F: U --> R
-               (x, y, z) |--> z + sin(x)*cos(y)
-
-            sage: type(f.coord_function(c_xyz).expr())
-               <class 'sympy.core.add.Add'>
-
-
         See the documentation of class
         :class:`~sage.manifolds.scalarfield.ScalarField` for more
         examples.

src/sage/manifolds/scalarfield.py

@@ -602,7 +602,7 @@ class ScalarField(CommutativeAlgebraElement):
 
     .. RUBRIC:: Examples with SymPy as the symbolic engine
 
-    From now on, we ask that all symbolic calculus on manifold `M` is
+    From now on, we ask that all symbolic calculus on manifold `M` are
     performed by SymPy::
 
         sage: M.set_calculus_method('sympy')
@@ -1544,7 +1544,6 @@ def coord_function(self, chart=None, from_chart=None):
                     new_expr = self._express[known_chart].expr()
                     self._express[chart] = chart.function(new_expr)
                     return self._express[chart]
-
             # If this point is reached, the expression must be computed
             # from that in the chart from_chart, by means of a
             # change-of-coordinates formula:
@@ -1567,15 +1566,12 @@ def coord_function(self, chart=None, from_chart=None):
                 if not found:
                     raise ValueError("no starting chart could be found to " +
                                      "compute the expression in the {}".format(chart))
-
             change = self._domain._coord_changes[(chart, from_chart)]
-
             # old coordinates expressed in terms of the new ones:
             coords = [ change._transf._functions[i].expr()
                        for i in range(self._manifold.dim()) ]
             new_expr = self._express[from_chart](*coords)
             self._express[chart] = chart.function(new_expr)
-
             self._del_derived()
         return self._express[chart]
 
@@ -1902,7 +1898,6 @@ def _display_expression(self, chart, result):
         result._latex = r"\begin{array}{llcl} " + symbol + r"&" + \
                         latex(self._domain) + r"& \longrightarrow & " + \
                         field_latex_name + r" \\"
-
         if chart is None:
             for ch in self._domain._top_charts:
                 _display_expression(self, ch, result)