Currently (Sage 8.7), the definition of a tensor field on a differentiable manifold is a 2-step operation. For instance, for a vector field:

sage: M = Manifold(2, 'M')
sage: X.<x,y> = M.chart()
sage: v = M.vector_field()  # step 1: declaration
sage: v[:] = -y, x          # step 2: initialization of components
sage: v.display()
-y d/dx + x d/dy

This ticket adds the possibility to perform the definition in a single step:

sage: v = M.vector_field(-y, x)
sage: v.display()
-y d/dx + x d/dy

Moreover, some flexibility is introduced in passing the components: it can be a list:

sage: M.vector_field([-y, x]).display()
-y d/dx + x d/dy

or more generally any iterable, like a vector of symbolic expressions:

sage: M.vector_field(vector([-y, x])).display()
-y d/dx + x d/dy

The components can also be provided in a vector frame distinct from the default one:

sage: f = M.vector_frame('f')
sage: M.vector_field(y^2, -1, frame=f).display(f)
y^2 f_0 - f_1

An alternative is passing a dictionary, the keys of which are the vector frames in which the components are defined:

sage: M.vector_field({f: [y^2, -1]}).display(f)
y^2 f_0 - f_1

The dictionary is mandatory if the components are given in various frames at once:

sage: M.vector_field({X.frame(): [-y, x], f: [y^2, -1]}).display(f)
y^2 f_0 - f_1

Note that the possibility of initializing the components while declaring a vector field was introduced on Euclidean spaces in #24623. This ticket extends this to any kind of differentiable manifold and any kind of tensor field. Accordingly, the redefinition of the method vector_field in the class EuclideanSpace has been suppressed: it falls back now to the method vector_field of the mother class DifferentiableManifold.

Basically the (optional) component initialization is performed by the method TensorField._init_components, which is invoked by all the end-user methods devoted to the creation of tensor fields on manifolds, i.e. the methods automorphism_field, diff_form, multivector_field, one_form, sym_bilin_form_field, tensor_field and vector_field of class DifferentiableManifold.

CC: @tscrim

Component: geometry

Keywords: tensor field

Author: Eric Gourgoulhon

Branch/Commit: 990a858

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/27581