6. Calculus - JulTob/Mathematics GitHub Wiki

Calculus

$\text{\color{gold}To see a World in a Grain of Sand}$
$\text{\color{gold}And a Heaven in a Wild Flower,}$
$\text{\color{gold}Hold Infinity in the palm of your hand}$
$\text{\color{gold}And Eternity in an hour}$
$\text{\color{gold}- William Blake}$

What is Calculus?

Functions

𝑓(𝘡)
𝑦(𝘹)

Succession

A Succession πš‚ is the ordered set of numbers where the elements πšœβ‚™ are well defined.

π™΅πš’πš‹πš˜πš—πš—πšŠπšŒπšŒπš’:= { πšβ‚€=0, πšβ‚=1, πšβ‚™=πšβ‚™β‚‹β‚+πšβ‚™β‚‹β‚‚ }

Limit

πš•πš’πš–
π‘₯⟢𝟢

What is the average speed, or more technically, β€œvelocity”, of our vehicle? It is given by the distance travelled, β–³x, divided by the time taken to travel it, β–³t. Or: β–³x / β–³t = f(t + β–³t) -f(t) / β–³t

Starting with a curve for velocity v(t), we imagine its area as filled up by very thin strips. Each has base β–³t, and height v(t).

Distance travelled over each time-interval β–³t = v.t = Area of strip v.β–³t Total distance x(t) = sum of areas v.β–³t

β€œSum {all the strips v(t).β–³t}”

πŸ“ŠCalculus

Discovered by Gottfried Leibniz and Isaac Newton.

πŸ“Š Differentiation & Integration

  • 🎒 Concerned with change, rates, accumulation, and optimization.
  • ⛰️ The two main concepts are differentiation (rates of change) and integration (accumulation).

πŸ“‰ Infinite Sequences & Series

  • Convergence–divergence

πŸ“‹ Advanced Analysis (Real & Complex)

  • Strange integrals, contour-integration, fractals in the complex plane.