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
π(π΅)
π¦(πΉ)
Succession
A Succession π is the ordered set of numbers where the elements πβ are well defined.
π΅πππππππππ:= { πβ=0, πβ=1, πβ=πβββ+πβββ }
Limit
πππ
π₯βΆπΆ
What is the average speed, or more technically, βvelocityβ, of our vehicle? It is given by the distance travelled, β³x, divided by the time taken to travel it, β³t
. Or:
β³x / β³t = f(t + β³t) -f(t) / β³t
Starting with a curve for velocity v(t)
, we imagine its area as filled up by very thin strips. Each has base β³t
, and height v(t)
.
Distance travelled over each time-interval β³t = v.t = Area of strip v.β³t Total distance x(t) = sum of areas v.β³t
βSum {all the strips v(t).β³t}β
πCalculus
Discovered by Gottfried Leibniz and Isaac Newton.
π Differentiation & Integration
- π’ Concerned with change, rates, accumulation, and optimization.
- β°οΈ The two main concepts are differentiation (rates of change) and integration (accumulation).
π Infinite Sequences & Series
- Convergenceβdivergence
π Advanced Analysis (Real & Complex)
- Strange integrals, contour-integration, fractals in the complex plane.