# Tagged Questions

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### Why can't we do some basic algebra in tensor calculus?

I have a very, very stupid question on the basics of tensor calculus. Consider $R_{ij} = 0$. 1)If I expand the ricci tensor $R_{ij}= g^{lm}R_{iljm}=0$. Now, my question is that, why can't we divide ...
480 views

### Why isn't invariant notation common?

In principle, one can write quantities in a manifestly invariant - rather than covariant - fashion in e.g. special relativity. For example, rather than writing just $x^\mu$, we could write the basis ...
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### Tensor algebra doubt

Is it possible to take a tensor to the other side of the equation, and the tensor becomes its inverse(i.e contravariant becomes covariant and vice versa)? It is a stupid question, but It confuses me. ...
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### Index Notation Double Curl

My question is about Einstein notation. It does not matter the specifics of this example (the del operator could be another random vector), I just want to know if my assumption about notation is ...
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### What is the difference between $\nabla _{\sigma}$ and $\nabla^{\sigma}$?

What is the difference between: $\nabla _{\sigma}$ and $\nabla^{\sigma}$? I've been told that the first is the covariant derivative, however I'm just starting a course on spacetime geometry and ...
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### Error in books of conformal field theory?

If you look at the book Conformal Field Theory (by Philippe Francesco, Pierre Mathieu and David Senechal) or the lecture notes Applied Conformal Field Theory (by Paul Ginsparg), and many other places: ...
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### On Einstein notation with multiple indices

On Einstein notation with multiple indices: For example, consider the expression: $$a^{ij} b_{ij}.$$ Does the notation signify, $$a^{00} b_{00} + a^{01} b_{01} + a^{02} b_{02} + ...$$ i.e. you ...
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### Kronecker delta confusion

I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same): ...
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### Difference between slanted indices on a tensor

In my class, there is no distinction made between, $$C_{ab}{}^{b}$$ and $$C^{b}{}_{ab}.$$ All I know, and read about so far, is the distinction of covariant and contravariant, form/vector, etc. ...
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### A tensor summation question

With the definition of the tensor: $$J_{ij} = I_{ij} - \tfrac{1}{3}\delta_{ij}I^{k}_{k}, \qquad i,j,k\in\{1,2,3\},$$ I have seen the quantity: ...
Let’s consider this equation for a scalar quantity $f$ as a function of a 3D vector $a$ as: $$f(\vec a) = S_{ijkk} a_i a_j$$ where $S$ is a tensor of rank 4. Now, I’m not sure what to make of the ...