Difference between $\mathrm {d} x$ and $\delta x $

Question

Are $\mathrm {d} x $ and $\delta x $ the same mathematical object from the point of view of the nonstandard analysis?

Answer 1

I will interpret the question as relating not to little-delta but rather to capital-delta. The notation $\Delta x$ is used in Keisler's textbook to denote an infinitesimal increment of the independent variable $x$. If $y=f(x)$, one also defines the change of the dependent variable $\Delta y=f(x+\Delta x)-f(x)$. The ratio $\frac{\Delta y}{\Delta x}$, rounded off to the nearest real, gives the derivative $f'(x)$. To write the derivative in Leibniz's differential notation as the ratio $\frac{dy}{dx}$, one sets $dx=\Delta x$ and $dy=f'(x)dx$. Then the product rule can be expressed by $d(uv)=udv+vdu$, etc. For additional details see Keisler.

As far as little-delta is concerned, the notation $\delta x$ is often used in the calculus of variations. If you are interested in hyperreal formalisations of related theories in physics you might be interested in consulting the volume by Albeverio et al:

Albeverio, Sergio; Høegh-Krohn, Raphael; Fenstad, Jens Erik; Lindstrøm, Tom. Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, 122. Academic Press, Inc., Orlando, FL, 1986.