Calculus

Differentiation

The way differential calculus is implemented is that it works on pure functions such as Sin instead of their instances like Sin(x). Calculus works on two levels. There is the level in terms of the limits. Then, up one level, there are the abstract rules for manipulating derivatives. To find the derivative of something like Sin(Cos(x)), we use the compose operator to write it as Sin o Cos We can check the multiplication rule for example: To find the derivative of a function like x^2 Sin(x^3) we write:

Integration

We define the integral of a function, f, as the integral of f(x) taken between 0 and x. The limits are implicitly assumed to be from 0 to x. Which means that the integral of 1/(x+a) is undefined for a=0 and integral of sin(x) is 1-cos(x)

Multi Variable Calculus

First let us define a real scalar field in 3 dimensional real space: We can do: And: And: We would also like to have \(\nabla\) and such as pure vector operators

Diff wrt x

Can we also define differentiation with explicit variables instead of pure functions? diff($x*y,$x)=y, Any reason not to?

Notes

We might also consider differention w.r.t. a vector. This would return a tensor (field) one rank higher. $$\sqrt{1}$$