A few minor changes in doctests of Euclidean spaces

egourgoulhon · egourgoulhon · commit 220726c07bc0 · 2018-04-30T18:18:13.000+02:00
diff --git a/src/sage/manifolds/differentiable/euclidean.py b/src/sage/manifolds/differentiable/euclidean.py
@@ -28,7 +28,7 @@
 - :class:`EuclideanPlane` for `n = 2`,
 - :class:`Euclidean3dimSpace` for `n = 3`.
 
-The user interface is provided by the generic function :func:`EuclideanSpace`.
+The user interface is provided by :class:`EuclideanSpace`.
 
 .. _EuclideanSpace_example1:
 
@@ -245,7 +245,7 @@
 We start by declaring the 3-dimensional Euclidean space ``E``, with
 ``(x,y,z)`` as Cartesian coordinates::
 
-    sage: E.<x,y,z> = EuclideanSpace(3)
+    sage: E.<x,y,z> = EuclideanSpace()
     sage: E
     Euclidean space E^3
 
@@ -460,7 +460,10 @@ class EuclideanSpace(PseudoRiemannianManifold):
 
         sage: E.<x,y> = EuclideanSpace()
 
-    The coordinate symbols can be customized::
+    Note that providing the dimension as an argument of ``EuclideanSpace`` is
+    not necessary in that case, since it can be deduced from the number of
+    coordinates within ``<,>``. Besides, the coordinate symbols can be
+    customized::
 
         sage: E.<X,Y> = EuclideanSpace()
         sage: E.cartesian_coordinates()
@@ -585,12 +588,12 @@ def __classcall_private__(cls, n=None, name=None, latex_name=None,
             sage: type(E2)
             <class 'sage.manifolds.differentiable.euclidean.EuclideanPlane_with_category'>
 
-            sage: E3.<x,t,p> = EuclideanSpace(coordinates='spherical'); E3
+            sage: E3.<r,t,p> = EuclideanSpace(coordinates='spherical'); E3
             Euclidean space E^3
             sage: type(E3)
             <class 'sage.manifolds.differentiable.euclidean.Euclidean3dimSpace_with_category'>
             sage: E3.default_frame()._latex_indices
-            (x, {\theta}, {\phi})
+            (r, {\theta}, {\phi})
         """
         if n is None:
             if names is None:
@@ -754,16 +757,16 @@ def _first_ngens(self, n):
 
         This is useful only for the use of Sage preparser::
 
-            sage: preparse("E.<x,y,z> = EuclideanSpace(3)")
-            "E = EuclideanSpace(Integer(3), names=('x', 'y', 'z',));
+            sage: preparse("E.<x,y,z> = EuclideanSpace()")
+            "E = EuclideanSpace(names=('x', 'y', 'z',));
              (x, y, z,) = E._first_ngens(3)"
 
         TESTS::
 
             sage: E = EuclideanSpace(2)
             sage: E._first_ngens(2)
             (x, y)
-            sage: E.<u,v> = EuclideanSpace(2)
+            sage: E.<u,v> = EuclideanSpace()
             sage: E._first_ngens(2)
             (u, v)
 
@@ -919,7 +922,7 @@ def vector_field(self, *args, **kwargs):
 
         A vector field in the Euclidean plane::
 
-            sage: E.<x,y> = EuclideanSpace(2)
+            sage: E.<x,y> = EuclideanSpace()
             sage: v = E.vector_field(x*y, x+y)
             sage: v.display()
             x*y e_x + (x + y) e_y
@@ -1457,7 +1460,7 @@ def polar_coordinates(self, symbols=None, names=None):
             sage: E.coord_change(E.cartesian_coordinates(),
             ....:                E.polar_coordinates()).display()
             r = sqrt(x^2 + y^2)
-            ph = arctan(y/x)
+            ph = arctan2(y, x)
 
         The coordinate variables are returned by the square bracket operator::
 
@@ -2199,7 +2202,7 @@ def spherical_coordinates(self, symbols=None, names=None):
             ....:                E.spherical_coordinates()).display()
             r = sqrt(x^2 + y^2 + z^2)
             th = arctan2(sqrt(x^2 + y^2), z)
-            ph = arctan(y/x)
+            ph = arctan2(y, x)
 
         The coordinate variables are returned by the square bracket operator::
 
@@ -2347,7 +2350,7 @@ def cylindrical_coordinates(self, symbols=None, names=None):
             sage: E.coord_change(E.cartesian_coordinates(),
             ....:                E.cylindrical_coordinates()).display()
             rh = sqrt(x^2 + y^2)
-            ph = arctan(y/x)
+            ph = arctan2(y, x)
             z = z
 
         The coordinate variables are returned by the square bracket operator::
@@ -2484,7 +2487,7 @@ def scalar_triple_product(self, name=None, latex_name=None):
 
         EXAMPLES::
 
-            sage: E.<x,y,z> = EuclideanSpace(3)
+            sage: E.<x,y,z> = EuclideanSpace()
             sage: triple_product = E.scalar_triple_product()
             sage: triple_product
             3-form epsilon on the Euclidean space E^3
diff --git a/src/sage/manifolds/differentiable/scalarfield.py b/src/sage/manifolds/differentiable/scalarfield.py
@@ -1093,7 +1093,7 @@ def gradient(self, metric=None):
 
         Gradient of a scalar field in the Euclidean plane::
 
-            sage: M.<x,y> = EuclideanSpace(2)
+            sage: M.<x,y> = EuclideanSpace()
             sage: f = M.scalar_field(cos(x*y), name='f')
             sage: v = f.gradient(); v
             Vector field grad(f) on the Euclidean plane E^2
@@ -1104,7 +1104,7 @@ def gradient(self, metric=None):
 
         Gradient in polar coordinates::
 
-            sage: M.<r,phi> = EuclideanSpace(2, coordinates='polar')
+            sage: M.<r,phi> = EuclideanSpace(coordinates='polar')
             sage: f = M.scalar_field(r*cos(phi), name='f')
             sage: f.gradient().display()
             grad(f) = cos(phi) e_r - sin(phi) e_phi
@@ -1191,7 +1191,7 @@ def laplacian(self, metric=None):
 
         Laplacian of a scalar field on the Euclidean plane::
 
-            sage: M.<x,y> = EuclideanSpace(2)
+            sage: M.<x,y> = EuclideanSpace()
             sage: f = M.scalar_field(function('F')(x,y), name='f')
             sage: s = f.laplacian(); s
             Scalar field Delta(f) on the Euclidean plane E^2
diff --git a/src/sage/manifolds/differentiable/tensorfield.py b/src/sage/manifolds/differentiable/tensorfield.py
@@ -3427,7 +3427,7 @@ def divergence(self, metric=None):
 
         Divergence of a vector field in the Euclidean plane::
 
-            sage: M.<x,y> = EuclideanSpace(2)
+            sage: M.<x,y> = EuclideanSpace()
             sage: v = M.vector_field(x, y, name='v')
             sage: s = v.divergence(); s
             Scalar field div(v) on the Euclidean plane E^2
@@ -3559,7 +3559,7 @@ def laplacian(self, metric=None):
 
         Laplacian of a vector field in the Euclidean plane::
 
-            sage: M.<x,y> = EuclideanSpace(2)
+            sage: M.<x,y> = EuclideanSpace()
             sage: v = M.vector_field(x^3 + y^2, x*y, name='v')
             sage: Dv = v.laplacian(); Dv
             Vector field Delta(v) on the Euclidean plane E^2
diff --git a/src/sage/manifolds/differentiable/vectorfield.py b/src/sage/manifolds/differentiable/vectorfield.py
@@ -1011,7 +1011,7 @@ def curl(self, metric=None):
 
         Curl of a vector field in the Euclidean 3-space::
 
-            sage: M.<x,y,z> = EuclideanSpace(3)
+            sage: M.<x,y,z> = EuclideanSpace()
             sage: v = M.vector_field(-y, x, 0, name='v')
             sage: v.display()
             v = -y e_x + x e_y
@@ -1102,7 +1102,7 @@ def dot_product(self, other, metric=None):
 
         Scalar product in the Euclidean plane::
 
-            sage: M.<x,y> = EuclideanSpace(2)
+            sage: M.<x,y> = EuclideanSpace()
             sage: u = M.vector_field(x, y, name='u')
             sage: v = M.vector_field(y, x, name='v')
             sage: s = u.dot_product(v); s
@@ -1188,7 +1188,7 @@ def norm(self, metric=None):
 
         Norm in the Euclidean plane::
 
-            sage: M.<x,y> = EuclideanSpace(2)
+            sage: M.<x,y> = EuclideanSpace()
             sage: v = M.vector_field(-y, x, name='v')
             sage: s = v.norm(); s
             Scalar field |v| on the Euclidean plane E^2
@@ -1276,7 +1276,7 @@ def cross_product(self, other, metric=None):
 
         Cross product in the Euclidean 3-space::
 
-            sage: M.<x,y,z> = EuclideanSpace(3)
+            sage: M.<x,y,z> = EuclideanSpace()
             sage: u = M.vector_field(-y, x, 0, name='u')
             sage: v = M.vector_field(x, y, 0, name='v')
             sage: w = u.cross_product(v); w
diff --git a/src/sage/manifolds/differentiable/vectorfield_module.py b/src/sage/manifolds/differentiable/vectorfield_module.py
@@ -1362,8 +1362,10 @@ def __init__(self, domain, dest_map=None):
         self._induced_bases = {}
         if self._dest_map != self._domain.identity_map():
             for frame in self._ambient_domain._top_frames:
-                basis = self.basis(from_frame=frame)
-                self._induced_bases[frame] = basis
+                if (frame.destination_map() ==
+                    self._ambient_domain.identity_map()):
+                    basis = self.basis(from_frame=frame)
+                    self._induced_bases[frame] = basis
 
         # Initialization of the components of the zero element:
         zero = self.zero()
diff --git a/src/sage/manifolds/operators.py b/src/sage/manifolds/operators.py
@@ -78,7 +78,7 @@ def grad(scalar):
 
     Gradient of a scalar field in the Euclidean plane::
 
-        sage: E.<x,y> = EuclideanSpace(2)
+        sage: E.<x,y> = EuclideanSpace()
         sage: f = E.scalar_field(sin(x*y), name='f')
         sage: from sage.manifolds.operators import grad
         sage: grad(f)
@@ -149,7 +149,7 @@ def div(tensor):
 
     Divergence of a vector field in the Euclidean plane::
 
-        sage: E.<x,y> = EuclideanSpace(2)
+        sage: E.<x,y> = EuclideanSpace()
         sage: v = E.vector_field(cos(x*y), sin(x*y), name='v')
         sage: v.display()
         v = cos(x*y) e_x + sin(x*y) e_y
@@ -218,7 +218,7 @@ def curl(vector):
 
     Curl of a vector field in the Euclidean 3-space::
 
-        sage: E.<x,y,z> = EuclideanSpace(3)
+        sage: E.<x,y,z> = EuclideanSpace()
         sage: v = E.vector_field(sin(y), sin(x), 0, name='v')
         sage: v.display()
         v = sin(y) e_x + sin(x) e_y
@@ -271,7 +271,7 @@ def laplacian(field):
 
     Laplacian of a scalar field on the Euclidean plane::
 
-        sage: E.<x,y> = EuclideanSpace(2)
+        sage: E.<x,y> = EuclideanSpace()
         sage: f = E.scalar_field(sin(x*y), name='f')
         sage: from sage.manifolds.operators import laplacian
         sage: Df = laplacian(f); Df
