Skip to content

Basics

Equations

  • Linear Equation
    • \(y = ax + b\)
    • Solution: $$ x = \frac{-b}{a} $$
  • Quadratic Equation
    • \(y = ax^2 + bx + c\)
    • Solution: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
  • Cubic Equation
    • \(y = ax^3 + bx^2 + cx + d\)
    • Solution, let
    • \(\Delta _{0}=b^{2}-3ac\)
    • \(\Delta _{1}=2b^{3}-9abc+27a^{2}d\)
\[ C={\sqrt[{3}]{\frac {\Delta_{1}\pm {\sqrt {\Delta _{1}^{2}-4\Delta_{0}^{3}}}}{2}}} \]
\[ \xi = \frac{-1 + \sqrt{-3}}{2} \]
\[ x_{k}=-{\frac {1}{3a}}\left(b+\xi ^{k}C+{\frac {\Delta _{0}}{\xi ^{k}C}}\right),\qquad k\in \{0,1,2\} \]

Factorization

  • \(a^2 - b^2 = (a-b)(a+b)\)
  • \(a^3 - b^3 = (a-b)(a^2 + ab -b^2)\)
  • \(a^3 + b^3 = (a+b)(a^2 - ab -b^2)\)

Logarithms

Definition

  • \(\log_b c = a \implies b^a = c, b \not= 0\)

Properties

  • \(\log_a m^n = n \log_a m\)
  • \(\log_a (mn) = \log_a m + \log_a n\)
  • \(\log_a \frac{m}{n} = \log_a m - \log_a n\)
  • \(\log_a 1 = 0\)
  • \(\log_a a = 1\)
  • \(\log_{a^b} m = \frac{1}{b}\log_a m\)
  • \(\log_{a^b} m^n = \frac{n}{b}\log_a m\)
  • \(\log_a b = \frac{\log_c b}{\log_c a}\)
  • \(\log_a b = \frac{1}{\log_b a}\)
  • \(a^{\log_a x} = x\)

Notation

  • \(\log a\) refers to the base being 10
  • \(\ln a\) refers to base being \(e\)

Complex Numbers

\[i = \sqrt{-1}\]

Exponential Notation

\[ e^{i\theta} = \cos \theta + i \sin \theta \]

De Moivre's theorem

\[ (r(\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta) \]

\(n^{\text{th}}\) root of complex numbers

If, \(z = re^{i\theta} = r(\cos \theta + i \sin \theta)\) then

\[ z^{1/n} = \sqrt[n]{r} e^{i(\theta + 2\pi k)/n}, k = 0, \pm1, \pm 2, \dots \]

Vector

Scalar product

\[ \textbf{a} \cdot \textbf{b} = \|\mathbf {a} \|\ \|\mathbf {b} \|\cos \theta = a_1b_1 + a_2b_2 + a_3b_3 \]

Vector product, cross product

\[ \textbf{a} \times \textbf{b} = (\|\mathbf {a} \|\ \|\mathbf {b} \|\sin \theta) \hat{\textbf{n}}= \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{j} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} \]

Triple product

\[[\textbf{a}, \textbf{b}, \textbf{c}] = (\textbf{a} \times \textbf{b}) \cdot \textbf{c} = \textbf{a} \cdot (\textbf{b} \times \textbf{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \\ \end{vmatrix} \]

two time cross product

\[ \textbf{a} \times (\textbf{b} \times \textbf{c}) = (\textbf{a} \cdot \textbf{c}) \textbf{b} - (\textbf{a} \cdot \textbf{b}) \textbf{c} \]