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comment Relation between Vector space $V$ and its dual $V^{*}$
@Ben: well, if you do not make the distinction between row- and column vectors, you can do that, but I'd advise against it (vectors are traditionally represented as column vectors, and covectors as row vectors, ie $n\times 1$ and $1\times n$ matrices)
Mar
2
comment Relation between Vector space $V$ and its dual $V^{*}$
@Ben: personally, I thinks this is misleading, in particular the Riemannian case, both in terms of differential geometry (vectors are equivalence classes of curves $\mathbb R\to M$, whereas covectors are equivalence classes of functions $M\to \mathbb R$) and physics (velocities vs momenta)
Mar
2
comment Relation between Vector space $V$ and its dual $V^{*}$
@Ben: it's abuse of notation, and I do not believe I've seen any literature going quite that far; the basic idea is that given a non-degenerate bilinear form (resp. a metric tensor in case of (pseudo-)Riemannian geometry), $V$ and $V^*$ become canonically isomorphic and can be 'identified'; if you prefer, you could introduce a new space $V^\blacktriangle$ isomorphic to both, and a single geometric object from $V^\blacktriangle$ can be represented by elements of $V$ and $V^*$ both; the culmination of that idea is that there are only (4-)vectors with co- and contravariant components
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Feb
27
revised Is the accelerated expansion of the universe consistent with conservation of energy?
stop mixing densities and totals
Feb
26
revised Is the accelerated expansion of the universe consistent with conservation of energy?
photon energy, not energy density
Feb
26
comment Is the accelerated expansion of the universe consistent with conservation of energy?
quoting said article: “energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on.” [...] There’s nothing incorrect about that way of thinking about it
Feb
26
revised Is the accelerated expansion of the universe consistent with conservation of energy?
added 1 character in body
Feb
26
revised Is the accelerated expansion of the universe consistent with conservation of energy?
added 82 characters in body