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: $"Many" \mathbf{\equiv} N > 10$.
• $\infty$ - 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
• $\lim$ - 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 $
• $\pi$ - 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;
• 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)

1. HS++

Differential

Derivative of f(x) $== \lim_{\Delta x \rightarrow 0} \frac {\Delta f} {\Delta x}$

• Properties
• Applications

Integrals

• Properties
• Applications

Vectors