What is the difference between $\delta x$ and $dx$ Sometimes I see the $\delta x$ and $dx$ but I don't know exactly what is the difference between them.
 A: $dx$ is always a differential. Informally this corresponds to an infinitesimal change in $x$. Taken formally it is just a symbol that says that a preceding integration is to be taken with respect to $x$. It is also always an exact differential: if you integrate $dx$ around a closed loop in the state space, you get the change in $x$ which is zero.
$\delta x$ is sometimes an inexact differential. This again corresponds informally to an infinitesimal change in $x$, but it changes in a way which does not add up to zero over a closed loop in the state space. One of the most important physical examples is the infinitesimal change in heat $\delta q$ in thermodynamics.
Other times $\delta x$ is actually just a finite change in the variable $x$, synonymous with $\Delta x$. This depends somewhat on context.
If you meant $\partial x$, that is used to denote a partial derivative with respect to a variable. For example, $\frac{\partial}{\partial t} f(t,x(t))$ is only a derivative with respect to $t$, while $\frac{d}{dt} f(t,x(t))$ is a total derivative, which contains a term corresponding to the change in $x$.
A: You need to clarify the context. For instance, if we are talking about differential forms, $\delta x$ could be the co-differential and ${\rm d}x$ the differential form (among other differences, ${\rm d}$ maps from $k$ to $k+1$ forms while $\delta$ works in the other direction).
In other contexts, $\delta$ could mean an improper differential, a variation, a finite difference or something similar.
