Peter4075
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 Nov 8 asked Time taken to cook frozen peas Nov 4 awarded Notable Question Oct 30 accepted Derivation of one-form/vector equation in Carroll confusion Oct 30 comment Derivation of one-form/vector equation in Carroll confusion Hmmm, there's a lot to take in here for a plodder like me. You've been very patient. Thanks again for your help. Oct 29 comment Derivation of one-form/vector equation in Carroll confusion Could this be a more straightforward answer (at my level) to my question of whether $\frac{df}{d\lambda}=\mathrm{d}f\left(\frac{d}{d\lambda}\right)$ can be derived without assuming $\mathrm{\mathrm{d}x^{i}}\left(\frac{\partial}{\partial x^{j}}\right)=\delta_{j}^{i}$? Much further on (p203) they actually use Eqn 2.17 to derive $\mathrm{\mathrm{d}x^{i}}\left(\frac{\partial}{\partial x^{j}}\right)=\delta_{j}^{i}$. Oct 29 comment Derivation of one-form/vector equation in Carroll confusion Equation 2.17 on p60 of Gravitation is $\partial_{v}f=\left\langle \mathrm{d}f,v\right\rangle$ (equivalent, I pray, to $\frac{df}{d\lambda}=\mathrm{d}f\left(\frac{d}{d\lambda}\right)$). As they explain, Eqn 2.17 is itself derived (by applying $\frac{d}{d\lambda}$) from Eqn 2.15 (p59): $f\left(P\right)=f\left(P_{0}\right)+\left\langle \mathrm{d}f,P-P_{0}\right\rangle +\textrm{(non linear terms)}$. Oct 29 comment Derivation of one-form/vector equation in Carroll confusion Also, (take a deep breath, try to stay calm, do NOT kick the cat!), if the answers to my latest questions are all “yes”, I cannot see much in the way of difference between your definition of a one-form $d_{x}f(v)=\partial_{v}f(x)$ and my original equation $\mathrm{d}f\left(\frac{d}{d\lambda}\right)=\frac{df}{d\lambda}$. Apart from your definition applying to a particular position $x$. Oct 29 comment Derivation of one-form/vector equation in Carroll confusion So when you write $d_{x}f(v)=\partial_{v}f(x)$ you mean the one-form $d_{x}f$ acting on the vector $v$ equals the derivative of $f$ with respect to $\lambda$ at position $x$? So the two sets of brackets mean different things. In the case of $\left(v\right)$, they mean “acting on”, and in the case of $\left(x\right)$, they mean “at the position”? Oct 28 comment Derivation of one-form/vector equation in Carroll confusion Thanks for you patience, but I still don't see (hence the bounty) whether $\mathrm{d}f\left(\frac{d}{d\lambda}\right)=\frac{df}{d\lambda}$ can be derived without assuming $\mathrm{\mathrm{d}x^{i}}\left(\frac{\partial}{\partial x^{j}}\right)=\delta_{j}^{i}$. Is that a yes or no? I've just noticed that on page 60 of Gravitation by Misner, Thorne and Wheeler they appear to derive the equation (2.17) from some kind of Taylor series (equation 2.15 on the previous page), but I can't follow that derivation either. Oct 26 comment Derivation of one-form/vector equation in Carroll confusion But isn't $\partial_{v}f(x)=d_{x}f(v)$ the equivalent of $\frac{df}{d\lambda}\left(x\right)= \textrm{d}f\left(\frac{dx^{\mu}}{d\lambda}\right)$, which is equivalent to $\frac{df}{d\lambda}=\textrm{d}f\left(\frac{d}{d\lambda}\right)$, in other words, the very equation we are trying to derive? Oct 26 comment Derivation of one-form/vector equation in Carroll confusion So I need to think of everything in your derivation being evaluated at a point $x$? Are you saying $\partial_{\mu}f(x)=df$? I can't see why that is. Doesn't $df=\frac{\partial f}{\partial x^{\mu}}dx^{\mu}$? Oct 26 comment Derivation of one-form/vector equation in Carroll confusion Apologies for my limited (high school) maths, but what's the difference between my $\frac{df}{d\lambda}$ and your $\frac{df}{d\lambda}\left(x\right)$? Thanks. Oct 25 comment Derivation of one-form/vector equation in Carroll confusion If you can see it, here's the link to p56 of the Schutz book: books.google.co.uk/… Oct 25 comment Derivation of one-form/vector equation in Carroll confusion So to speak.... Oct 25 revised Derivation of one-form/vector equation in Carroll confusion added 2 characters in body Oct 25 comment Derivation of one-form/vector equation in Carroll confusion I've edited my question to try to make my confusion clearer. Oct 25 revised Derivation of one-form/vector equation in Carroll confusion Reaction to comment by @ACuriousMind Oct 25 asked Derivation of one-form/vector equation in Carroll confusion Sep 28 awarded Popular Question Sep 21 awarded Custodian