Basic Math
Frequently used notations
⚠ $\equiv $
- defined as, an equivalent of- Example 1:: We define
⚠ $N $
as a number of items::⚠ $N \mathbf \equiv}$
"a number of items". - Example 2:: We define
⚠ $"Many"$
as a number of items greater than 10:⚠ $"Many" \mathbf{\equiv} N > 10$
.
- Example 1:: We define
⚠ $\infty$
- the infinity symbol.⚠ ${x\rightarrow \infty }$
means that x grows without bound.⚠ $\approx$
- approximately equal to- The age of the Universe is
⚠ $\approx$
14 billions years
- The age of the Universe is
⚠ $\lim$
- a limit of- A limit of a function f(x) = 1/x is zero when x is running to infinity,
⚠ $\lim_{x \rightarrow \inf } \frac {1}{x} = 0 $
- A limit of a function f(x) = 1/x is zero when x is running to infinity,
⚠ $\pi$
- the "pi" constant, is the ratio between the circumference and diameter of a circle⚠ $\pi \approx 3.1415926$
⚠ $e$
- the Euler's number.the e =⚠ ${\lim _{x=0,\inf} \left( {1 + \frac{1}{x}} \right)^x$
,⚠ $e \approx 1.718281828$
⚠ $\Delta$
- a difference, change between values⚠ $\Delta X \equiv x_{2} - x_{1}$
⚠ $\sum_{k=0}^{N} A_k$
- a sum of⚠ $N$
values,⚠ $A_{k}$
, where index⚠ $k $
runs from 0 to⚠ $N$
.- Many more symbols here
Algebra
Equations ( rearranging equations rules)
- if a=b, then a+c = b+c ; a* c = b* c.
- if a=b and e=d , then a-e = b - d
Exponents
Exponentiation == "vozvedenie v stepenx":
- f(x) =
⚠ $x^n$
, if x - integer, than⚠ $x^n = {x * x * ...* x}_{n times}$
- Rules
⚠ $x^0$
= 1;⚠ $x^1$
= x⚠ $x^{-a} = \frac{1}{x^a}$
⚠ $x^{a}*{x}^b= x^{a+b}$
⚠ $\frac{x^{a}}{{x}^b} = x^{a-b}$
Polynomials
⚠ $y^{n}(x) \equiv {\sum_{k=0}^{n} a_{k}*x^{k} = a_{0}+a_{1}*x+ a_{2} * x^{2} + ... + a_{n}* x^{n}$
;- Distributive law:: a * (b + c) = a*b + a*c
Logarithms
The logarithm is the inverse function to exponentiation.
- y =
⚠ $\log_{b} (x)$
, if⚠ $ y^b = x$
.- y =
⚠ $\log_{10} (x)$
- decimal logarithm, x=10^y; - y =
⚠ $\log_{e}(x) \equiv \ln(x) $
- the natural logarithm, x= e^y;
- y =
- Rules
⚠ $\log_{b} (x*y) = \log_{b} (x) + \log_{b}(y) $
⚠ $\log_{b} (x/y) = \log_{b} (x) - \log_{b}(y) $
⚠ $\log_{b} (x^y) = y*\log_{b} (x) $
Trigonometry
- HS ( sin(x), cos(x), tg(x) only, Pythagorean Theorem)
Vectors
- Definition, Examples, Application
- Components and Resultants ( sum of vectors)
- Scalar multiplication
- Basis vectors
- Magnitude, projections
- Vector resolution ( a process where one vector is broken down into two or more smaller vectors).
- Vextor multiplication ( cross product)
- HS++
Differential
Derivative of f(x) ⚠ $== \lim_{\Delta x \rightarrow 0} \frac {\Delta f} {\Delta x}$
- Properties
- Applications
Integrals
- Properties
- Applications