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Boltzmann Entropy

$$S=k_B\ln\Omega$$

De Moivre's Theorem

$$\small \cos x+ i \sin x)^n=\cos(nx)+i\sin(nx)$$

Bayes' Theorem

$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$

Conditional Probability

$$P(E|F) = \frac{P(E \cap F)}{P(F)}$$

Gaussian/Normal Distribution

$$f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\
mu}{\sigma}\right)^2}$$

Curl

$$\scriptsize \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} &
\mathbf{j} & \mathbf{k} \\\\ \frac{\partial}{\partial x} & \frac{\
partial}{\partial y} & \frac{\partial}{\partial z} \\\\ F_x & F_y &
F_z \end{vmatrix}$$

Divergence

$$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\
partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$$

Gradient

$$\nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\
partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \
mathbf{k}$$

Matrix Multiplication

$$\scriptsize \begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin
{pmatrix} w & x \\ y & z \end{pmatrix} = \begin{pmatrix} aw + by & ax
+ bz \\ cw + dy & cx + dz \end{pmatrix}$$

Quadric Surfaces

$$\\ {x^2 \over a^2} + {y^2 \over b^2} + {z^2 \over c^2} = 1$$

Equation of a Plane

$$ax + by + cz = d$$

Equation of a Sphere

$$\small R^2 = (x-x_0)^2 + (y-y_0)^2 + (z - z_0)^2$$

Fourier + Inverse Fourier Transform

$$X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}\,dt$$

Modulus of Resilience

$$U_r = \int_{0}^{\epsilon_Y}\sigma\mathop{d\epsilon} \approx \frac{\
sigma_{YS}^2}{2E}$$

Modulus of Rigidity/Shear Modulus

$$G = \frac{\tau}{\gamma}$$

Modulus of Toughness

$$U_t = \int_{0}^{\epsilon_f}\sigma\mathop{d\epsilon}$$

Poisson's Ratio

$$\nu = - \frac{\epsilon_x}{\epsilon_z} = -\frac{\epsilon_y}{\
epsilon_z}$$

Resistance in a straight conductor

$$R = \frac{\rho l}{A}$$

Thermal expansion

$$\Delta L = L_0 \alpha \Delta T$$

Arithmetic Gradients

$$\scriptsize P = A \left[\frac{(1+i)^n-1}{i(1+i)^n}\right] + G \left
[\frac{(1+i)^n-in-1}{i^2(1+i)^n}\right]$$

Equivalent Uniform Annual Cost

$$A = P \left[\frac{i(1 + i)^n}{(1 + i)^n - 1}\right]$$

Present and Future Value

$$F = P(1 + i)^n$$

Light Wavelength and Frequency Relationship

$$c = \lambda f$$

Photon Energy

$$E = hf$$

Feynmann's Trick For Exponential Integrals

$$\int_0^{\infty} x^n e^{-tx} dx = \frac{n!}{t^{n+1}}$$

Finite and Infinite Geometric Series

$$S_n = \frac{a(1-r^n)}{1-r}$$

Convolution

$$(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)d\tau$$

Shockley Diode Model

$$I = I_S \cdot (e^{\frac{V_D}{nV_T}} - 1)$$

Schrodinger's Equation

$$\scriptsize i\hbar\frac{\partial}{\partial t}\Psi(x,t) = \left[-\
frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right]\Psi
(x,t)$$

L'Hopital's Rule

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'
(x)}$$

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