Dear Juan Weisz , I think that you can consider as the start point of this discuss the answer of GPT-4: " In your question, you're touching on several interesting mathematical topics, such as the Leibniz rule (also known as the product rule in calculus), differential geometry, concepts of infinitesimals, and algebraic structures.

The Leibniz rule, or the product rule, is a fundamental theorem in calculus that states the derivative of the product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function. The rule can be symbolically represented as:

D(fg) = f Dg + g Df.

As you mention, if you consider differentials as geometric areas of infinitesimally small rectangles, then the product rule can be visualized by subtracting the area of two slightly different rectangles. The missing term dx dy indeed appears in this geometric interpretation, but this term is typically omitted because it is of second order (i.e., it becomes negligible as dx and dy approach zero).

The idea of a smallest positive number brings us to the realm of infinitesimals, which are quantities so small that there is no real number greater than zero that is less than them. These were historically problematic but have been made rigorous in the framework of non-standard analysis via hyperreal numbers, or in synthetic differential geometry using nilpotent infinitesimals.

In your question, you suggest "fixing" the problem with an ideal in algebra, using a nilpotent element e where e^2 = 0. This idea is in the spirit of synthetic differential geometry where nilpotent infinitesimals are used. In this framework, the product rule holds "on the nose", without need of a dx dy term. However, to fully develop this would require quite a bit of abstract mathematics, including the use of category theory and topos theory. "

Valdym

Thanks for the coment.

So far i thought of using only the theory of rings. Im not familiar with the areas you mention. Any reason?

X+ey is invertible as long as x not zero

And note f(x+ey)=f(x)+yeDf(x)

gives the derivative.

Of coutse the correct expredsion is

d(xy) =xdy+ydx

With perfect infinitesimals.

Proofs

(x+he)(y+ke)-xy =x ke + y he

On the nose

Also

F(x+ey)G(x+ey)=F(x)G(x)+ey(F DG+G DF)

from above.

If x and h are vactors the similar holds for functions of several variables

There is dot prodct of h with gradient.

Ok assuming topos relevant

Which relevant simpest connection can be given?

There are two very different

interpretations of

x+ey

One is over the whole plane with

singularity along a vertical at x=0

Then this expresion is a ring with non

zero non invertible elements.

The other interpretation is along the

x axis alone and y is just a parameter

not a coordinate.

Then the interpretstion is field like

x alone.

What do people think?

Im rather disturbed with the

Synthetical differential crowd

now that I read.

All their results could easily been drawn

from Lagrange and or Taylor.

This old wine from new bottles...