Peter4075
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 Oct 18 awarded Benefactor Oct 18 accepted Basis one-form and basis vector confusion Oct 17 comment Basis one-form and basis vector confusion So are you saying, using my example of $z=r\cos\phi$, that $v\left(f\right)=dz$ where $v=\frac{\partial}{\partial x^{a}}$ and $f=z$? Oct 17 comment Basis one-form and basis vector confusion I was taken aback to read in your answer that $\partial_{\nu}\mathrm{d}x^{\mu}$ “definitely doesn't mean differentiate $\mathrm{d}x^{\mu}$.” That was the one thing I thought I understood in the whole problem, ie differentiating $dx^{1}$ wrt $dx^{1}$ gives 1, wrt to $dx^{a}$ where $a\neq1$ gives zero (because the coordinates are independent of each other). So are are saying $e_{\nu}\omega^{\mu}=\partial_{\nu}dx^{\mu}=\delta_{\nu}^{\mu}$ only because it is defined to do so and that I'm going to have to forget about my explanation in terms of coordinate independence? Oct 17 comment Basis one-form and basis vector confusion Sorry, I meant to use the total differential of $z$ $(\mathrm{d}z=\cos\phi\mathrm{d}r-r\sin\phi\mathrm{d}\phi)$. But at least now that I'm using a simple example, the language of coordinate functions is becoming a little easier to visualise. I used $a=4$ in $\mathrm{d}x^{4}(\partial_{a})=\delta_{a}^{4}$ because any other value would give zero. Does that mean time component of $v$ in this case equals 1? Oct 17 comment Basis one-form and basis vector confusion Thanks. In your para 2, could an example of these $x^{1}$ etc coordinates on the manifold be Cartesian coordinates $x,y,z$ (assuming we're in Euclidean space). If these coordinates were defined in terms of spherical coordinates $r,\theta,\phi$, we could say $z=r\cos\phi$ and thus $\frac{dz}{dr}=\cos\phi$ and $dz=\cos\phi dr$. Is $dz$ a one-form? Also, in para 4 you say vector $v=\frac{\partial}{\partial x^{a}}$. In para 3 you say if we plug $v$ into $dx^{4}$ we get the time component of $v$. But surely $\frac{\partial}{\partial x^{a}}dx^{4}=\frac{\partial}{\partial x^{4}}dx^{4}=1$? Oct 15 awarded Promoter Oct 15 comment Basis one-form and basis vector confusion Thanks to Amitai Yuval, but the second paragraph loses me. What are these $n$ real valued functions? Are they coordinate transformations from one chart to another? The reason for my question was trying to understand how the product of a one-form and a contravariant vector gives a scalar. Hence my trying to see how the product of the basis vectors $e_{\nu}\omega^{\mu}$ gives $\delta_{\nu}^{\mu}$. As is painfully obvious, my knowledge of manifolds is via a very basic introduction to the physics of spacetime. Oct 14 comment Basis one-form and basis vector confusion I'm afraid that's above my head. Any chance of a more high school level type answer? I'm OK with contravariant vector components as tangent vectors, and one-form components as gradients and that these things live at a point on a manifold (4d for spacetime). I understand that $$e_{\nu}\omega^{\mu}=\delta_{\nu}^{\mu}.$$ But "charts", "bundles" and "$n$ real valued functions" lose me. Oct 14 awarded Informed Oct 13 asked Basis one-form and basis vector confusion Sep 30 revised Tensor equations. Can I change an equation from covariant to contravariant? Pointing to the answer to my query. Sep 30 comment Raising and lowering indices - any chance of checking my work? @JackLee - Thanks very much. I know it's really basic stuff, but when you're teaching yourself it's hard to find step by step derivations that are pitched at the beginner's level. Sep 29 revised Raising and lowering indices - any chance of checking my work? Said "lowering", meant "raising". Sep 29 asked Raising and lowering indices - any chance of checking my work? Sep 28 comment Tensor equations. Can I change an equation from covariant to contravariant? I'm still learning index manipulation. Any chance of checking that my answer to this question is correct? Thanks. Sep 28 answered Tensor equations. Can I change an equation from covariant to contravariant? Aug 15 awarded Notable Question Jul 19 comment Order of evaluation of a simple term Well, the two methods give slightly different answers (to three decimal places). Jul 19 asked Order of evaluation of a simple term