Fundamental Mathematics/Calculus

Calculus

Mathematical operations that perform on function, equation

Calculus mathematics

Change in variables

For any function f(x) . Over the interval of ${\displaystyle x}$  to ${\displaystyle x+\Delta x}$

Change in variable x

${\displaystyle \Delta x=(x+\Delta x)-x}$

Change in function f(x)

${\displaystyle \Delta f(x)=f(x+\Delta x)-f(x)}$

Rate of change

Rate of change

${\displaystyle {\frac {\Delta f(x)}{\Delta x}}={\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$

Limit

${\displaystyle \lim _{x\to a}f(x)}$

Finite Limit We call ${\displaystyle L}$  the limit of ${\displaystyle f(x)}$  as ${\displaystyle x}$  approaches ${\displaystyle c}$  if ${\displaystyle f(x)}$  becomes arbitrarily close to ${\displaystyle L}$  whenever ${\displaystyle x}$  is sufficiently close (and not equal) to ${\displaystyle c}$  .

When this holds we write

${\displaystyle \lim _{x\to c}f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to c}$

Infinite Limit We call ${\displaystyle L}$  the limit of ${\displaystyle f(x)}$  as ${\displaystyle x}$  approaches infinity if ${\displaystyle f(x)}$  becomes arbitrarily close to ${\displaystyle L}$  whenever ${\displaystyle x}$  is sufficiently large.

When this holds we write

${\displaystyle \lim _{x\to \infty }f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to \infty }$

Similarly, we call ${\displaystyle L}$  the limit of ${\displaystyle f(x)}$  as ${\displaystyle x}$  approaches negative infinity if ${\displaystyle f(x)}$  becomes arbitrarily close to ${\displaystyle L}$  whenever ${\displaystyle x}$  is sufficiently negative.

When this holds we write

${\displaystyle \lim _{x\to -\infty }f(x)=L}$

or

${\displaystyle f(x)\to L\quad {\mbox{as}}\quad x\to -\infty }$

Differentiation

Let ${\displaystyle f(x)}$  be a function. Then

${\displaystyle {\frac {d}{dt}}f(t)=f'(x)=\sum \lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}}$  wherever this limit exists.

In this case we say that ${\displaystyle f}$  is differentiable at ${\displaystyle x}$  and its derivative at ${\displaystyle x}$  is ${\displaystyle f'(x)}$  .

Integration

Mathematics operation on a continuous function to find its area under graph . There are 2 types of integration

${\displaystyle \int f(x)dx=F(x)+C}$

Where ${\displaystyle F}$  satisfies ${\displaystyle F'(x)=f(x)}$

Suppose ${\displaystyle f}$  is a continuous function on ${\displaystyle [a,b]}$  and ${\displaystyle \Delta x={\frac {b-a}{n}}}$  . Then the definite integral of ${\displaystyle f}$  between ${\displaystyle a}$  and ${\displaystyle b}$  is

${\displaystyle \int \limits _{a}^{b}f(x)dx=\lim _{n\to \infty }A_{n}=\lim _{n\to \infty }\sum _{i=1}^{n}f(x_{i}^{*})\Delta x}$

Where ${\displaystyle x_{i}^{*}}$  are any sample points in the interval ${\displaystyle [x_{i-1},x_{i}]}$  and ${\displaystyle x_{k}=a+k\cdot \Delta x}$  for ${\displaystyle k=0,\dots ,n}$  .}}

Solving differential equations

Given

${\displaystyle {\frac {d}{dt}}f(t)=-sf(t)}$
${\displaystyle {\frac {df(t)}{f(t)}}=-sdt}$
${\displaystyle \int {\frac {df(t)}{f(t)}}=\int -sdt}$
${\displaystyle Lnf(t)=-st+c}$
${\displaystyle f(t)=e^{-st+c}=Ae^{-st}}$
${\displaystyle A=e^{c}}$

In summary

 Ordered differential equation Equation of the form Root of equation 1st ordered differential equation ${\displaystyle {\frac {d}{dt}}f(t)=-sf(t)}$ ${\displaystyle f(t)=Ae^{-st}}$ 2nd ordered differential equation ${\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)=-{\sqrt {s}}f(t)}$ ${\displaystyle f(t)=Ae^{\pm j{\sqrt {s}}t}}$ ${\displaystyle n}$ th ordered differential equation ${\displaystyle {\frac {d^{n}}{dt^{n}}}f(t)=-n{\sqrt {s}}f(t)}$ ${\displaystyle f(t)=Ae^{\pm jn{\sqrt {s}}t}}$

Solving Ordered Differential Equations

1st ordered differential equation

Equation of general form

${\displaystyle A{\frac {d}{dx}}f(x)+Bf(x)=0}$

After arrangement, equation above becomes

${\displaystyle {\frac {d}{dx}}f(x)=-sf(x)}$  Where ${\displaystyle s={\frac {B}{A}}}$

Equation has a root

${\displaystyle f(x)=Ae^{-st}}$

2nd ordered differential equation
${\displaystyle A{\frac {d^{2}}{dx^{2}}}f(x)+B{\frac {d}{dx}}f(x)+C=0}$
${\displaystyle {\frac {d^{2}}{dx^{2}}}f(x)+{\frac {B}{A}}{\frac {d}{dx}}f(x)+{\frac {C}{A}}=0}$
${\displaystyle s^{2}f(x)+2\alpha sf(x)+\beta f(x)=0}$
${\displaystyle s^{2}f(x)=-2\alpha sf(x)-\beta f(x)}$

The solution of the 2nd ordered polynomial equation above

 One real root ${\displaystyle s=-\alpha }$ ${\displaystyle i(t)=Ae^{-\alpha t}}$ Two real roots ${\displaystyle s=-\alpha \pm {\sqrt {\beta -\alpha }}}$ ${\displaystyle i(t)=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}}$ One complex root ${\displaystyle s=-\alpha \pm j{\sqrt {\beta -\alpha }}}$ ${\displaystyle i(t)=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha )}}t}}$

With

${\displaystyle A(\alpha )=Ae^{-\alpha t}}$
${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$
${\displaystyle \beta ={\frac {1}{T}}={\frac {1}{LC}}}$
${\displaystyle \alpha =\gamma \beta }$
${\displaystyle T=LC}$
${\displaystyle \gamma =RC}$

Partial Differential Equations

${\displaystyle {\frac {\partial }{\partial t}}f(t)=-sf(t)}$

Integral Transformation

Any function f(t) can be transform into Laplace function or Fourier function by using Laplace transform or Fourier transform

 ${\displaystyle f(t)}$ ${\displaystyle F(s)}$ ${\displaystyle F(j\omega )}$ ${\displaystyle f(t)}$ ${\displaystyle \int f(t)e^{-st}dt}$ ${\displaystyle \int f(t)e^{-j\omega t}dt}$ ${\displaystyle {\frac {d}{dt}}}$ ${\displaystyle s}$ ${\displaystyle j\omega }$ ${\displaystyle \int dt}$ ${\displaystyle {\frac {1}{s}}}$ ${\displaystyle {\frac {1}{j\omega }}}$

Example

 ${\displaystyle f(t)}$ ${\displaystyle F(s)}$ ${\displaystyle F(j\omega )}$ ${\displaystyle L{\frac {d}{dt}}}$ ${\displaystyle sL}$ ${\displaystyle j\omega L}$ ${\displaystyle {\frac {1}{L}}\int dt}$ ${\displaystyle {\frac {1}{sL}}}$ ${\displaystyle {\frac {1}{j\omega L}}}$

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