The Amazing Relationship Between Integration And Euler’s Number

[fresh_42]

We use integration to measure lengths, areas, or volumes. This is a geometrical interpretation, but we want to examine an analytical interpretation that leads us to Integration reverses differentiation. Hence let us start with differentiation.

Weierstraß Definition of Derivatives​

$f$ is differentiable at $x$ if there is a linear map $D_{x}f$, such that
\begin{equation*}
\underbrace{D_{x}(f)}_{\text{Derivative}}\cdot \underbrace{v}_{\text{Direction}}=\left(\left. \dfrac{df(t)}{dt}\right|_{t=x}\right)\cdot v=\underbrace{f(x+v)}_{\text{location plus change}}-\underbrace{f(x)}_{\text{location}}-\underbrace{o(v)}_{\text{error}}
\end{equation*}
where the error $o(v)$ increases slower than linear (cp. Landau symbol). The derivative can be the Jacobi-matrix, a gradient, or simply a slope. It is always an array of numbers. If we speak of derivatives as functions, then we mean $f'\, : \,x\longmapsto D_{x}f.$ Integration is the problem to compute $f$ from $f'$ or $f$ from
$$
\dfrac{f(x+v)-f(x)}{|v|}+o(1)
$$
The quotient in this expression is linear in $f$ so
$$
D_x(\alpha f+\beta g)=\alpha D_x(f)+\beta D_x(g)
$$
If we add the Leibniz rule
$$
D_x(f\cdot g)=D_{x}(f)\cdot g(x) + f(x)\cdot D_{x}(g)
$$
and the chain rule
$$
D_x(f\circ g)=D_{g(x)}(f)\circ g \cdot D_x(g)
$$
then we have the main properties of differentiation.

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[Klystron]

PF has abundant threads extolling the virtues of the irrational number π, various methods for approximating or calculating its digits after the decimal point, and π's appearance in various formulae.

This Insights article succinctly derives my favorite transcendental number e, its importance as the base of the natural logarithms and the exciting exp and ln_e inverse functions, and e's appearance in formulae modelling exponential growth. While mildly disappointed at not encountering specific examples of e used in electronic vector theory where I first encountered it, the author amply illustrates the relationship between derivatives and integration.