# tangent space vector mapping

Very simple question.

I am going through the SM_tutorial and branched off into a sidestream; trying to understand and put things in a context that I already know.

The tutorial defines a function f() on 3 space and defines the associated tangent_space. I have a couple of questions

1) How do I take a vector in the tangent_space

Say: v = Tp.an_element(); print(v)

"Tangent vector at Point p on the 3-dimensional differentiable manifold M"

or vxx = Tp((-2,1,5), name='vxx')

and apply it to f() (or f(p) although the TM is only defined at p so far)?

If I define a vector in the base space it works::

v = U.vector_field(name='v')

s = v(f)

I know in standard texts the mapping of vectors in TM_p to the base is defined, but couldn't find it in the Sage Manifold documentation.

2) Taking : v = Tp.an_element(); print(v)

I get something that looks like a vector (with a different ancestry) but has a value

v.display()

∂/∂x+2∂/∂y+3∂/∂z

Where did this value come from? In one of the documentation it (sort of) implies it's an example; is this true?

Why isn't it left undefined?

Ray

Regarding (2), I couldn't reproduce this. In particular, when I create a manifold (and define some coordinates on it) and then run

p = theManifold.point(name='p')

TpM = theManifold.tangent_space(p)

v = TpM.an_element()

v.dipslay(),

I get the error message

ValueError: no basis could be found for computing the components in the None

@my_screen_name: I cannot reproduce your issue. Can you provide the full code, starting from the definition of the manifold?