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