Using infinitesimals in multivariable calculus

Question

In ordinary calculus,if we don't make any absurd notation,the use of infinitesimals is extremely useful and intuitive which ended up giving correct results. For example $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ are obvious if we treat $dy,dx,dt$ as infinitesimals. Can we do the same in case of multivariable calculus? For example if $V$ is a function of $x$ and $y$ and $x,y$ are both functions of $r,\theta$,can we still do stuffs like $\frac{\delta V}{\delta x}=\frac{\frac{\delta V}{\delta \theta}}{\frac{\delta x}{\delta \theta}}$? Or are there scenarios where such use of infinitesimals lead to incorrect results?

Answer 1

There are pitfalls with using notation that is too abbreviated when working in multivariate calculus, whether or not one uses infinitesimals. Partial derivatives are treated carefully in Keisler's textbook Elementary calculus with infinitesimals. While in single-variable calculus, it is obviously an advantage to be able to express chain rule as $\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}$ as literally involving fractions, there are some subtleties here, as well. Thus, $\frac{dy}{dx}$ is a bit more complicated than "the y-increment over the x-increment". The precise justification is in terms of the standard part. This is also explained very well in Keisler.