We have also covered two other forms of calculus over the the last week:
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.
There is the theory of containers (unrelated to the docker kind). They represent functors on sets and the differential operator on them satisfies the usual laws.
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.
Of course LLMs probably have sparked an interest in calculus. I have dusted of decades old knowledge, relearning the difference between a d and a δ, or what is df/dx ln(x).
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).
One cool application of the discrete laplace operator is that we can use it to calculate the number of spanning trees of the graph by considering the determinants of the submatrices we get my removing 1 row and 1 column from the matrix!
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.
It generalizes easily. The main difference is the construction of the Laplacian matrix, which won't be symmetric in a directed graph, but you can still do spectral analysis and standard matrix derivatives, diagonalization, factorizations, etc.
love getting down voted for sharing joy in mutual overlapping discovery. This happens all the time in science and in math... I was using these sort of constructs for information theoretic explorations that attempted to extend physical laws to new domains. To put it loosely I was attempting to find a simpler explanation for some of our understanding of fundamental physics. Some of it stuck, some of it sucked. It was a lot of fun though
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