All Notations
Latex Equations
Logarithms & Exponentials
\[ \ln(x) \text{ }log_{10}(x) \text{ }a^b \]
Sigma notation with limits:
\[ \sum_{i=1}^{n} a_i \]
Integration with limits:
\[ \int_{a}^{b} f(x) \, dx \]
Product notation with limits:
\[ \prod_{i=1}^{n} a_i \]
AM-GM-HM Inequality
\[\frac{\sum_{i=1}^{n}a_i}{n} \geqslant\sqrt[n]{\prod_{i=1}^{n}}a_i\geqslant\frac{n}{\sum_{i=1}^{n}\frac{1}{a_i}}\]
Matrix example:
\[ A = \begin{bmatrix} 1 & 2 & 21 \\ 3 & 4 & 43 \\ 5 & 6 & 65 \end{bmatrix}=\begin{vmatrix} 1 & 2 & 21 \\ 3 & 4 & 43 \\ 5 & 6 & 65 \end{vmatrix}=\begin{Bmatrix} 1 & 2 & 21 \\ 3 & 4 & 43 \\ 5 & 6 & 65 \end{Bmatrix}\]
Permutations & Combinations
\[ P(n, k) = \frac{n!}{(n-k)!} \ =\text{ }^{n}\text{P}_ {r} \] \[ C(n, r) = \frac{n!}{(n-k)!(k)!} = \binom{n}{r} \ =\text{ }^{n}\text{C}_{r} \]
Derivative notation with limits:
\[ f'(x) = \lim_{{h \to \infty}} \frac{{f(x + h) - f(x)}}{{h}} \]
\[ \frac{{\partial f}}{{\partial x}}(a, b) = \lim_{{h \to \infty}} \frac{{f(a + h, b) - f(a, b)}}{{h}} \]
De Morgan's First Theorem for Sets:
\[ \overline{\text{A} \cup \text{B}} = \overline{\text{A}} \cap \overline{\text{B}} \]
De Morgan's Second Theorem for Sets:
\[ \overline{\text{A} \cap \text{B}} = \overline{\text{A}} \cup \overline{\text{B}} \]
Binomial Expansion
\[(a+b)^{\text{n}}=\sum_{i=1}^{n}\text{ }^{n}C_{i}a^{n-i}b^{i}\]
General solution of a quadratic equation:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
\[ x = \frac{-b \mp \sqrt{b^2 - 4ac}}{2a} \]
Cubic roots of unity:
\[ x = \sqrt[3]{Z} = 1 , x = 1\]
\[ w = x = \frac{-1- \sqrt{3}i}{2}\]
\[ w^{2} = x = \frac{-1+ \sqrt{3}i}{2}\]
Linear homogeneous differential equation
\[ a_3 \frac{d^3y}{dx^3} + a_2 \frac{d^2y}{dx^2} + a_1 \frac{dy}{dx} + a_0y = 0 = \alpha \]
Scalar Product
\[\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}||\hat{A}\cdot\hat{B}|\]
\[\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}||cos\text{ }\theta\text{ }|\]
Vector Product
\[\vec{A}\times\vec{B}=|\vec{A}||\vec{B}||\hat{A}\times\hat{B}|\hat{n}\]
\[\vec{A}\times\vec{B}=|\vec{A}||\vec{B}||sin\text{ }\theta\text{ }|\hat{n}\]
Maxwell's equations
First equation:
\[\oint\vec{E}.\vec{dA}=\frac{Q_{enclosed}}{\varepsilon_\circ}\]
Second equation:
\[\oint\vec{B}.\vec{dA}=0\]
Third equation:
\[\oint\vec{E}.\vec{dl}=-\frac{d}{dt}\Phi_{_{B}}\]
Fourth equation:
\[\oint\vec{B}.\vec{dl}=\mu_{\circ}i_{c}+\mu_{\circ}\varepsilon_\circ\frac{d}{dt}\oint\vec{E}.\vec{dA}\]
Important Constants:
\[\text{Atomic Numbers}\]
H=1, He = 2, Li=3, Be=4, B=5, C=6, N=7, O=8,
N=9, Na=11, Mg=12, Si=14, Al=13, P=15, S=16,
Cl=17, Ar=18, K =19, Ca=20, Cr=24, Mn=25,
Fe=26, Co=27, Ni=28, Cu = 29, Zn=30, As=33,
Br=35, Ag=47, Sn=50, I=53, Xe=54, Ba=56,
Pb=82, U=92.
\[\text{Atomic Masses}\]
H=1, He=4, Li=7, Be=9, B=11, C=12, N=14, O=16,
F=19, Na=23, Mg=24, Al = 27, Si=28, P=31, S=32,
Cl=35.5, K=39, Ca=40, Cr=52, Mn=55, Fe=56, Co=59,
Ni=58.7, Cu=63.5, Zn=65.4, As=75, Br=80, Ag=108,
Sn=118.7, I=127, Xe=131, Ba=137, Pb=207, U=238.
\[h\text{=}6.626\times10^{-34}\text{ }Js;\text{ }\hbar\text{=}1.055\times10^{-34}\]
\[G\text{=}6.67\times10^{-11}\text{ }Nm^{2}kg^{-2};\text{ }k\text{=}9\times10^{9}\text{ }Nm^{2}C^{-2}\]
\[\varepsilon_\circ\text{=}8.85\times10^{-12}\text{ }C^{2}N^{-1}m^{-2}\text{ };\mu_{\circ}\text{=}4\pi\times10^{-7}\text{ }Hm^{-2}\]
\[e\text{=}1.6\times10^{-19}V\text{ };m_{e}\text{=}9.1\times10^{-31}\text{ }kg\]
\[R\text{=}8.314\text{ }J\text{ }K^{-1}mol^{-1}\]\[=2\text{ }Cal\text{ }K^{-1}mol^{-1}\]\[=0.0821 Lit\text{ }atm\text{ }K^{-1}mol^{-1}\]
\[\text{ }1calorie=4.2J\]