Which of the following quantities are meaningful and which are not?
a) $A_{ij}=B_{ik}A_{kj}$ (Meaningful, because $k$ indexes cancel each other. How can I describe this rule, I don't know.)
b) $A_{ij}=B_{ik}C_{kl}$ (Not meaningful, because free indexes on the LHS are $i$ and $j$ but on RHS $i$ ...
I don't think anyone except your teacher calls doing math with indices "Newtonian World Index Algebra". That notation (implying summation by repeating indices) is usually called Einstein notation
Yes: there is not enough information given. The answers depend on whether you're trying to define the l.h.s. of these expressions by the r.h.s. or whether $A,B,C,D$ all exist a priori and these are just supposed to be relations between them
@ICCQBE As I said, for things like your c), it is not decidable without knowing whether it's supposed to be a definition of a new quantity or a relation between existing quantities
@Knight Can you not click on the close(1) and see for yourself?
If you can't, I won't tell you since other users who do not ask a mod about it also don't get to know that.
If $(ijk)$ double permutation, then $\epsilon_{ijk}$ equals to $+1$ and if It's single permutation, then $\epsilon_{ijk}$ equals to $-1$ and if it has got two same indexes then $\epsilon_{ijk}$ equas to $0$.
I have a question about defining Levi-Civita tensor on 2-dimensions
In two dimensions, you can just have an anti-symmetric quantity $\epsilon_{ij} = -\epsilon_{ji} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$. Livi-Civita isn't a real tensor though
@danielunderwood Sure it is, the $n$-dimensional version being $\epsilon\equiv e^1\wedge e^2\wedge\cdot\cdot\cdot\wedge e^n\in\Lambda^n\mathbb{R}^{n*}$; an antisymmetric rank-$n$ tensor with a one-dimensional basis.
@dsm What daniel presumably means is that it's not a "real tensor" in the physicist's (well, general relativist's) sense, since $\epsilon_{ijk}$ does not define a coordinate-independent tensor field on a manifold
When physicists say "tensor", they often really mean what a mathematician calls a "tensor field"
I think I've seen a version of that with "Hilbert space (physics) = finite-dimensional Hilbert space (math)"
which, while not -quite- true (given how much we deal with the Schrodinger equation) is still a good description of our attitudes towards Hilbert space
hmm. trying to figure out how to tackle the following problem I just devised in thermodynamics
An application of equilibrium thermodynamics outside the context of gases is that of a rubber band, which (ideally) behaves like an entropic spring
The typical experiments for that are either with respect to the variables (T,f) (temperature and tension) or (T,L) (temperature and length). (compare with the gas cases of (T,P) or (T,V).)
For the first one, I hang a weight on a rubber band (constant temperature and tension). For the second, I place the weight on a scale and use the rubber band to reduce the scale weight (constant temperature and length).
What I'm thinking about now is: What happens if I start from the first case and then pull the weight down slightly, to see oscillations?
Clearly that's not constant length, but I'm not seeing it as constant tension either.
looking around online, rubber bands seem sufficiently non-Hookean that I probably should leave this well enough alonen
Which of the following quantities are meaningful and which are not?
a) $A_{ij}=B_{ik}A_{kj}$ (Meaningful, because $k$ indexes cancel each other. How can I describe this rule, I don't know.)
b) $A_{ij}=B_{ik}C_{kl}$ (Not meaningful, because free indexes on the LHS are $i$ and $j$ but on RHS $i$ ...
