I have spent the last year trying to learn Clifford Algebra in the form of Geometric Algebra. (Have I turned you off yet.) You may ask why bother. This is the correct way to apply Calculus to Vectors. You know vectors are the little "arrows" people use in Physics. However the problem with vectors as taught is that you can't divide within the vector math itself. It is not a complete algebra to solve problems. The American Physicists Gibbs gave us a work-around method we use in vector study and Calculus much later taught by the small book "The Div, Grad, Curl and all that stuff" that is known to most physical scientists and others. But here is a story about a new, fresh look at Quantum Mechanics:
https://www.simonsfoundation.org/qua...antum-physics/
The Grassman Algebra is front and center in Geometric Algebra. In this article the emphasis is upon area as representing answers. Vectors most people have used are called directed line segments. In order to extend the algebra to allow division operations, you have to include directed areas, volumes and beyond up to the dimensions under study.
You can get a feel for this in looking at any parallelogram formed between two parallel lines. If you move the upper line segment the shape of the parallelogram is skewed in ever highly angled parallelograms. But with the top and bottom sides remaining the same and the height between the parallel lines the same, the area of any shape formed remains the same. The answer to why areas are important in this article.
http://www.homeschoolmath.net/teaching/g/area_3.php
And using Calculus making ever smaller parallelograms to give us the area, we get an algebra for vectors in any dimension we can think of.
Of course this isn't all of it (for instance you have to have something like the imaginary i for square roots of minus numbers which exists in Geometric Algebra). But I hope it can give you a glimpse of the tools for the next iteration in understanding coming in the not to distant future.