#30318: Improve dot and cross product

Author: egourgoulhon

src/sage/manifolds/differentiable/curve.py

@@ -315,8 +315,10 @@ class DifferentiableCurve(DiffMap):
 
     The Frenet-Serret frame is orthonormal::
 
-        sage: [[u.dot(v).expr() for v in FS] for u in FS]
-        [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
+        sage: matrix([[u.dot(v).expr() for v in FS] for u in FS])
+        [1 0 0]
+        [0 1 0]
+        [0 0 1]
 
     The derivative vectors `N'` and `B'`::
 

src/sage/manifolds/differentiable/pseudo_riemannian_submanifold.py

@@ -724,8 +724,7 @@ def calc_normal(chart):
             n_comp = (n_form.contract(*args) / factorial(self._dim)).contract(
                 self.ambient_metric().inverse().along(self._immersion))
             if self._ambient._dim - self._dim == 1:
-                n_comp = n_comp / n_comp.norm(
-                    self.ambient_metric().along(self._immersion))
+                n_comp = n_comp / n_comp.norm(self.ambient_metric())
 
             norm_rst = self._normal.restrict(chart.domain())
             norm_rst.add_comp(max_frame.restrict(chart.domain()))[:] = n_comp[:]

src/sage/manifolds/differentiable/vectorfield.py

@@ -1147,13 +1147,25 @@ def dot_product(self, other, metric=None):
             u.v: (0, 2*pi) --> R
                t |--> sin(t)^2
 
+        Scalar product between a vector field along the curve and a vector
+        field on the ambient Euclidean plane::
+
+            sage: e_x = M.cartesian_frame()[1]
+            sage: s = u.dot_product(e_x); s
+            Scalar field u.e_x on the Real interval (0, 2*pi)
+            sage: s.display()
+            u.e_x: (0, 2*pi) --> R
+               t |--> cos(t)
+
         """
         default_metric = metric is None
         if default_metric:
             metric = self._ambient_domain.metric()
         dest_map = self.parent().destination_map()
-        if dest_map is not self._domain.identity_map():
+        if dest_map != metric.parent().base_module().destination_map():
             metric = metric.along(dest_map)
+        if dest_map != other.parent().destination_map():
+            other = other.along(dest_map)
         resu = metric(self, other)
         # From the above operation the name of resu is "g(u,v')" where
         # g = metric._name, u = self._name, v = other._name
@@ -1248,7 +1260,7 @@ def norm(self, metric=None):
         if default_metric:
             metric = self._ambient_domain.metric()
         dest_map = self.parent().destination_map()
-        if dest_map is not self._domain.identity_map():
+        if dest_map != metric.parent().base_module().destination_map():
             metric = metric.along(dest_map)
         resu = metric(self, self).sqrt()
         if self._name is not None:
@@ -1357,6 +1369,16 @@ def cross_product(self, other, metric=None):
             sage: w.display()
             -sin(t) e_x - cos(t) e_y + (2*cos(t)^2 - 1) e_z
 
+        Cross product between a vector field along the curve and a vector field
+        on the ambient Euclidean space::
+
+            sage: e_x = M.cartesian_frame()[1]
+            sage: w = u.cross_product(e_x); w
+            Vector field C' x e_x along the Real interval (0, 2*pi) with values
+             on the Euclidean space E^3
+            sage: w.display()
+            C' x e_x = e_y - cos(t) e_z
+
         """
         if self._ambient_domain.dim() != 3:
             raise ValueError("the cross product is not defined in dimension " +
@@ -1365,10 +1387,12 @@ def cross_product(self, other, metric=None):
         if default_metric:
             metric = self._ambient_domain.metric()
         dest_map = self.parent().destination_map()
-        if dest_map is self._domain.identity_map():
+        if dest_map == metric.parent().base_module().destination_map():
             eps = metric.volume_form(1)
         else:
             eps = metric.volume_form(1).along(dest_map)
+        if dest_map != other.parent().destination_map():
+            other = other.along(dest_map)
         resu = eps.contract(1, 2, self.wedge(other), 0, 1) / 2
         # The result is named "u x v" only for a default metric:
         if (default_metric and self._name is not None and