I am not sure whether this needs to be transported in another topic (as academic guidance). I have some preliminary knowledge on tensor analysis, which helps me being more confident with indices notation etc... Also I'm accustomed to the definition of tensors, which tells us that a tensor is an object which has a given transformation law... Like a tensor of rank [itex](n,k)[/itex] transforms under a coordinate transformation (diffeomorphism) [itex]\phi^{a}(x^b)[/itex] as:(adsbygoogle = window.adsbygoogle || []).push({});

[itex] (T')^{a_{1}...a_{n}}_{b_{1}...b_{k}}= \frac{\partial \phi^{a_{1}}}{\partial x^{w_{1}}}...\frac{\partial \phi^{a_{n}}}{\partial x^{w_{n}}} \frac{\partial x^{r_{1}}}{\partial \phi^{b_{1}}}...\frac{\partial x^{r_{k}}}{\partial \phi^{b_{k}}} (T)^{w_{1}...w_{n}}_{r_{1}...r_{k}}[/itex]

However, a tensor is not only defined as such... I learned that a (n,k) tensor can also be a mapping... In particular it takes vectors from a vector space (n times) and its dual (k times) and maps them to the real numbers:

[itex]T: V \otimes V \otimes ... \otimes V \otimes V^{*}\otimes ... \otimes V^{*} \rightarrow R [/itex]

so in that case instead of indices one can use arguments of the corresponding vectors in V....

eg the metric is a tensor [itex]g: V \otimes V \rightarrow R[/itex] which gets translated to [itex]g_{ab}x^{a}x^{b}= g(x^{a},x^{b})= ds^{2}[/itex]

But wouldn't also the metric send me from [itex]V \rightarrow V^{*}[/itex] like an object which lowers or raises indices? In that case: [itex]g(x^{a})=x^{*b}[/itex]

Is there any good book that can give me a good insight on this way of defining things? Thanks...

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# Preliminary knowledge on tensor analysis

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