I'm sure this question has been asked before, but I haven't been able to find a good answer using the search function.(adsbygoogle = window.adsbygoogle || []).push({});

I have a differentiable manifold M and some coordinate chart. I have vector and covector fields on this manifold, but no metric. At some point P, I know I can write the tangent vectors as a linear combination of directional derivative operators

[tex]v = \sum v_i \frac{\partial}{\partial x^i}[/tex]

and I can write the covectors as a linear combination of one-forms

[tex]\omega = \sum \omega_j \, dx^j[/tex]

Furthermore, I ought to be able to take the contraction [itex]\langle \omega, v \rangle[/itex] algebraically, using linearity and the relations

[tex]\langle dx^i, \frac{\partial}{\partial x^j} \rangle = \delta^i_j[/tex]

Now, by definition, the one-form basis lies in the dual space of linear functionals on V, so the contraction above really means that the linear functionals dx^i must "act upon", in some sense, the basis vectors d/dx^j of V; i.e.

[tex]dx^i \left( \frac{\partial}{\partial x^j} \right) = \delta^i_j[/tex]

But my question is, what in the world does this mean? I thought that this "canonical" set of bases was supposed to lead to some natural way to compute the contraction, but I don't see it.

What does it mean for dx( ) to act upon some other object?

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# The one-form basis and contraction

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