What is Addition?
Any binary operation that takes two elements from a set and the result is also an element from that set.
What is Multiplixation?
Any binary operation that takes a something else and an element from a set and the result is also an element from that set.
What is an Algebra?
A set with three operations and the axioms implied by “vector space” and “bilinear”.
The operations:
- Addition
- Multiplication
- Scalar Multiplication
We might consider learning the basics of Boolean Algebra to give us an example of an algebra.
What is Calculus?
A calculus contains four things: a discrete limit, a continuous limit, a discrete iteration, a continuous iteration.
Elementary Calculus, the calculus taught in high school and college, lacks a discrete limit but it has the other three things:
- Derivative (continuous limit)
- Summation (discrete iteration)
- Integral (continuous iteration)
Appendix A
These questions about basic definitions was motivated by our noticing that there is Tensor Algebra and Tensor Calculus and we also noticed Vector Calculus.
We are leaving the door open for the possibility that different authors writing about a particular topic will cover different things. It would be surprising if we had difficulty deciding where tensor algebra and tensor calculus begins, and likewise for vector algebra infector calculus. Also, attempts to define a “no-man’s-land” between the two might have 60 or 70% of the disciplined existing as “no-man’s-land”.
Take a swig of your drink and then read this:
Vector Algebra is the set of Algebraic Operations in Vector Calculus (one is a subset in the other).
We will not be daunted. At night we will parachute drop you into Vector Algebra and, after you have secured everything there, you will send patrols into Vector Calculus.
Vectoreverything 