Edwin,
The concept is not an algebraic equivalent for calculus, it is algebra working in harmony with calculus. I do not know about newer texts, but if you look up "Mathematical Methods for Physicists" by Arfkin, in chapter one on vector analysis, he mentions an integral definition for gradient, divergence and curl as limits for a volume with the point of application an interior point going to zero of the ratio of a surface integral divided by enclosed volume integral. The surface integral is over the differential surface normal vector respectively multiplied by a scalar function, an inner product with a vector function, and cross product with a vector function; for gradient, divergence and curl. He uses this to demonstrate for example spherical-polar representations of these three forms.
I look at this not as an alternate description for n dimensional differentiation, but instead its fundamental definition. Algebra comes into play because multiplication is its dominion. The multiplication on the differential surface normal is an algebraic expression, and if you are working with Quaternions, the three forms of scalar multiplication, scalar result vector -vector products and vector result vector - vector products are all covered by a single operation, the Quaternion product of two algebraic elements, here a 4D differential surface normal and a 4D function. If you were to leg out the Quaternion Ensemble Derivative for a transformation between rectilinear native coordinates and spatial spherical-polar coordinates, you will find proper representations of spherical-polar gradient, divergence and curl, which you may individually isolate with the resultant basis element products. Do it again, you get the second order forms. We all know what they are, so there are no mysteries on whether or not the result is correct as some may argue if the work was done in Octonion 8D space.
There still is the notion of a difference, not simply between two arbitrary points but instead over the full (n-1) dimensional surface, but also over the full set of algebraic products between the surface normal and function to differentiate in order to come up with something transformable. The limit is as the volume approaches 0, arbitrarily close but never touching the point at which we wish to define the differentiation. So there is always a definable surface and a difference between functional values at the point of application and values in a coordinate neighborhood defined by the surface.
This is the genesis of the Ensemble Derivative.
Hope this helps.
Rick