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I have been reading the Classical Field theory part from The Quantum field theory book of Lewis H Ryder. After defining classical field $\phi(x^\mu)$ he says something about adding variations on both the field, and the coordinates. $$\phi(x^{\mu}) \longrightarrow \phi(x^{\prime\mu})=\phi(x^{\mu})+\delta\phi(x^{\mu})$$ $$x^{ \mu}\longrightarrow x^{\prime \mu}=x^{ \mu}+\delta x^{ \mu} $$ Now, my question is how this $\delta x$ is different from $dx$? Clearly $\delta x$ is not just an infinitesimal number like $dx$. How does a Variation defined in such an independent variable $x$? What is exactly the definition of $variation - \delta$?

Qmechanic
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Baibhab Bose
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  • There is no such thing as an infinitasimal number. If there was, you could take half of it and have a more infinitesimal number. – Gyro Gearloose Sep 17 '20 at 13:51
  • @GyroGearloose https://en.wikipedia.org/wiki/Nonstandard_analysis – Daniel Sep 17 '20 at 14:04
  • @Daniel I know, but is physics based on that? – Gyro Gearloose Sep 17 '20 at 14:05
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    @GyroGearloose Then what is $\delta x$ here? and How does it differ from dx ? – Baibhab Bose Sep 17 '20 at 14:14
  • @BaibhabBose I have studies maths but nowhere saw a sound and rigid definition. I'm myself curios of the answers here. As for the $dx$ it is just a symbol used in derivatives and integrals. From math point of view all boils down on taking limits. – Gyro Gearloose Sep 17 '20 at 14:17
  • To reopen this question (v2), consider to give your definition of $dx$. – Qmechanic Sep 17 '20 at 15:00
  • @Qmechanic physics and maths people use same symbols with different meanings. My interpretation of $dx$ for any expression like $f(dx)$ is to replace it with $\lim_{\epsilon\to 0}f(\epsilon)$. – Gyro Gearloose Sep 17 '20 at 15:16
  • Possible duplicates: https://physics.stackexchange.com/q/65724/2451 , https://physics.stackexchange.com/q/153791/2451 and links therein. – Qmechanic Sep 17 '20 at 15:19
  • "Closed. This question needs details or clarity." Of course, I guess that's the main reason the OP posted it. There have been more questions about "infinitesimal" values and such shortly, and the way physics use math is, quite, astonishing. – Gyro Gearloose Sep 17 '20 at 18:00

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