Provide at least two other examples of mathematical notation or wording denoting the same process.?

Answer 1

1. In calculus, dx/dt and x with a dot over it mean the same thing, the derivative of x with repsect to time.

2. In differential geometry, the covariant derivative and the Levi-Civita connection are the same thing, the derivative of a vector field.

I am not sure if this is what you had in mind. But your question reminds me of an important concept in mathematics, that of duality.

There are many kinds of mathematical duality. The general idea is that there is often more than one way to understand the same mathematical problem or idea.

Let's look at an example from the field of optimization. The main problem (called the primal problem) is to maximize a company's profits, subject to some constraints (limited amount of cash, limited number of people).

Now look at a related problem, that of miminizing a company's costs, subject to some constraints (wages, cost of materials).

Depending upon the equation relating the profits to the costs, it may turn out that these two problems have exactly the same solution.

If so, we say that the second problem is the dual problem.

This is a powerful idea, because it sometimes turns out that the primal problem is very hard to solve, and the dual problem is easier to solve. Since they both give the same answer, solving one is as good as solving the other.