ljy-002 / DiffNotes.jl

Just some notes taken when I am studying Calculus, still updating.

DiffNotes

Just some notes taken when I am studying Calculus, still updating.

Highlights

These codes will make it more convenient while programming on Calculus.

Usage

Add dependencies

julia> ]
(@v1.10) pkg> add Calculus
(@v1.10) pkg> add ForwardDiff

Download and include the codes.

(Actually I don't want to upload these codes to Julia Packages, since they are too simple)

julia> include("desktop/diffnotes.jl") # Here I put the codes on Desktop
Main.DiffNotes

julia> using .DiffNotes

Select a library (Calculus/ForwardDiff)

Note: The reason I made the macro is that both those two library have their advantages and disadvantages. Therefore, when you find it wrong using one lib, you can change the library you use at any time.

This macro (@uselib) will define (or redefine) the function d, which all the other functions depend on.

(As reference, I found that the library Calculus support more kinds of functions, while the library ForwardDiff is more accurate)

julia> @uselib Calculus
d (generic function with 2 method)

Now use it!

The codes contains functions providing the calculation of derivative, derivative function, second and higher derivative (and their corresponding functions), partial derivative (function), curvature and l'Hospital's rule.

Note: the derivative function calculator will not do any symbolic calculations, as it is just a abbreviation for x -> d(f, x)

julia> @uselib ForwardDiff
d (generic function with 2 methods)

julia> f = d(x -> x)
#1 (generic function with 1 method)

julia> f(8)
1

julia> d(x -> x, Inf)
1.0

julia> d²(sin, π)
-1.2246467991473532e-16

julia> d²(sin, π/2)
-1.0

julia> dⁿ(sin, 4, π/2)
1.0

julia> ∂((x, y) -> x^2 + y^2, 2)(6) # Same as `∂((x, y) -> x^2 + y^2, 2; n=1)(6)`, which `(6)` is the other variable(s).
4

julia> κ(x -> x - sin(x), x -> 1 - cos(x), π) # Curvature
-0.25

julia> lHôpital(x -> sind(180*x), x -> x^2-1, 1)
-1.5707963267948966

julia> @uselib Calculus
d (generic function with 2 methods)

julia> lHôpital(x -> exp(-1/x^2), x -> x^2, 0)
0.0

Need to improve

2. L'Hospital's Rule calculator sometimes give different answers using different libs.