Algebraic graph calculus (2021)
gabarro.org
gabarro.org
Alien calculus [0][1] Matrix Calculus [2][3]
Might as well throw in some other generalizations of the derivative [4] , it is amazing how these concepts apply to all sorts of mathematical structures once you generalize them, whlie you are probably fairly interested in math if you are reading the comments here, but on the offi chance you have not done so, look up measure theory, it is a fun concept that allows you to generalize the integral in neat ways that you'll notice has a lot of application to computer science especially vision related applications.
[0] https://news.ycombinator.com/item?id=35476236 [1] https://www.quantamagazine.org/alien-calculus-could-save-par... [2] https://news.ycombinator.com/item?id=35568311 [3] https://www.matrixcalculus.org/ [4] https://math.stackexchange.com/a/1209684
Since this may be a misunderstanding, and it's very common among my students, let me suggest that you almost certainly mean df/dx, where f(x) = ln(x); or d(ln(x))/dx; or (d/dx)(ln(x)). All of these indicate taking the derivative of ln(x) with respect to x (which is 1/x). df/dx ln(x) would be the product of the derivative of some unspecified function f with ln(x).
In short a container is a set of shapes and a set of positions for data in each shape. The derivative of a container is the container you get by removing one position in each shape.
An example of a container is the list container, which maps any set to the sets of lists of elements from the set. The derivative of the list is the pair of lists.
As for multisets, the bag functor is better defined on groupoids than sets. But there is a generalised notion of container for groupoids, with a similar calculus. The bag functor is special because it is its own derivative, just like the exponential function.
The curl(u) operator, which confines vectors and vector fields to R^3, is an outer product of nabla with u [1, page 9] in geometric algebra. And as such it is defined for arbitrary number of dimensions, not just 2, 3 and 7.
[1] https://hal.sorbonne-universite.fr/hal-00920544v2/document
Matrices (and more) can also be described in GA, because they are linear operators.
The idea is that k-forms are functions defined on the k-cliques of the graph. TFA is just the 1-dimensional case of this.
The 2-dimensional case would be:
0-forms: functions defined on vertices
1-forms: functions defined on edges
2-forms: functions defined on triangles
The exterior derivative is defined in a natural way, by taking differences along signed boundaries.
In the case of a triangulated surface, the Hodge dual has a nice interpretation via the dual triangulation.