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For a typical real manifold of $n$ dimensions, the basis of a tangent vector space $T$ at point $p$ is $$\frac{\partial}{\partial x_i},\ldots,\frac{\partial}{\partial x_j}$$ So is the general basis of the $k$-vector

$$\frac{\partial}{\partial x_{i}}\otimes\frac{\partial}{\partial x_j}\otimes\frac{\partial}{\partial x_{l}}\otimes\cdots\otimes\frac{\partial}{\partial x_t}$$

where $i,j,l,\ldots,t$ are $k$ indices, run from $1$ to $n$ and $\otimes$ is tensor multiplication?

Also for tensor field at $p$ of valence $(a,b)$ is the basis

$$\frac{\partial}{\partial x_i}\otimes\cdots\otimes\frac{\partial}{\partial x_t}\otimes dx_q\otimes dx_r\otimes\cdots\otimes dx_y$$

where $i,...,t$ are $a$ indices and $q,r,...,y$ are $b$ indices?

Michael Hardy
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orange_soda
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  • ... to read up things, chapter 8-16 of Roger Penrose's Road to Reality gives quite a good introductary overview of modern geometry (while already mentioning some applications), and the rest of the book then deals with actualy applying these concepts to modern physics. – Dilaton Oct 03 '13 at 16:03
  • Ah I understand. Did you already ask for such a book here on maths? Maybe you could even restate your "Optics in curved spacetime" question here. That was a very good and well defined questions (it should absolutely not have been closed) and physics is the only science site I know with such a silly anti-book/references/study material policy... To help improve the book policy there, it would be very helpful if you could opt in here too. The books you mention look interesting to me too. – Dilaton Oct 04 '13 at 08:30

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Yes, although most modern books will use subscripts with indices (which look scary) but make the counting clearer than writing letters. For example, a $(2,1)$ tensor should look like $$\sum a^i_{j_1j_2}\frac{\partial}{\partial x_i}\otimes dx_{j_1}\otimes dx_{j_2}\,.$$ Most of us think of a $k$-vector as dual to a $k$-form (so it will be an alternating $(0,k)$-tensor), so I'd write it as linear combinations of $$\frac{\partial}{\partial x_{i_1}}\wedge\frac{\partial}{\partial x_{i_2}}\wedge \dots \wedge\frac{\partial}{\partial x_{i_k}}\,.$$

Ted Shifrin
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  • @dj_mummy I think the books ignore them for a few reasons. 1) These bases are implicit in the definition of the corresponding bundles: For example, $k$-forms (at a point) are the $k$-forms of the tangent space as a vector space, so the natural basis is the one inherited from normal linear algebra. 2) One of the great tenants of differential geometry is that it is coordinate independent. It makes concrete calculations hard, but it's nice to know you don't actually need to specify a basis. 3) We are usually more interested in fields, in which case we only get a frame in at most a local chart. – Tyler Holden May 26 '13 at 16:30
  • Well, I don't think serious graduate-level differential geometry books ignore these issues at all. We are all about working with tensor fields (like the second fundamental form -- which is in fancy terms a normal-bundle-valued $(2,0)$-tensor), both abstractly and in local coordinates. And those of us raised on Cartan's method of moving frames use differential forms far more than we use vector fields :) – Ted Shifrin May 26 '13 at 16:57