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I know that:

1) Change in $x$ ie., $Δx$, when $\lim Δx→0$, then $Δx$ is replaced by $dx$.

2) I also know that $∂x$ is used in partial derivative.

Then what is $δx$? Is $dx$ and $δx$ is just the same or something different. I want some mathematical explanation.

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    $\begingroup$ It all depends on the context. As much as we want to believe, there is not a universal mathematical notation for certain things. Please give the context as to where this occurs so others can answer the question in the best way possible. $\endgroup$
    – BioPhysicist
    Commented Sep 5, 2018 at 13:28
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    $\begingroup$ It depends, but one widespread usage of the $\delta$ convention is in calculus of variations problems. $\endgroup$
    – Avantgarde
    Commented Sep 5, 2018 at 13:38
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    $\begingroup$ It would be easier if you provide an example, a formula where you have seen this notation and you are not sure about the meaning. I would be temped to say that dx and δx are the same things most of the cases, but you also have functions like the Kronecker delta which use that symbol. $\endgroup$
    – user190081
    Commented Sep 5, 2018 at 13:50
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/65724/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Sep 5, 2018 at 14:12
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    $\begingroup$ See also physics.stackexchange.com/q/153791/25301 $\endgroup$
    – Kyle Kanos
    Commented Sep 5, 2018 at 14:16

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Generally, I've seen "$\delta x$" to mean a tiny change in $x$, without necessarily invoking all of the mathematical machinery of a derivative. In a derivation, you might start with variables $ \delta x $ and $\delta t$ as ordinary quantities, and then later put in an explicit step that takes the limits as they approach $0$.

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