formulas
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Biot-Savart Law
$$B(r) = \frac{\mu_0}{4\pi} \int \frac{I d\ell \times r'}{|r'|^3}$$
Vector Calculus Product Rules
$$\nabla (fg) = f(\nabla g) + g(\nabla f)$$
Canonical Partition Function
$$Z = \sum_{i} e^{-\beta E_i}$$
Grand Canonical Partition Function
$$\Xi = \sum_{i} e^{\beta ( \mu N_i - E_i)}$$
Hydraulic Diameter
$$D_h = \frac{4A_c}{P}$$
Reynolds Number
$$Re = \frac{\rho v L}{\mu}$$
Kinematic Viscosity
$$
u = \frac{\mu}{\rho}$$
Sine Law
$$\frac{\sin(\text{A})}{\text{a}} = \frac{\sin(\text{B})}{\text{b}} = \frac{\sin(\text{C})}{\text{c}}$$
Snell's Law
$$n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$$
Error Function
$$\text{erf}(z) = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} dt$$
Bernoulli's Principle
$$P + \frac{1}{2} \rho v^2 + \rho g h = \text{const}$$
Hydrostatic Equation
$$\frac{dP}{dz} = - \rho g$$
Specific Gravity
$$SG = \frac{\rho}{\rho_{H_2O}}$$
Specific Weight
$$\gamma = \rho g$$
Viscosity
$$\tau = \mu \frac{\partial u}{\partial y}$$
Finite Difference Approximations
$$\frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$
Greek Letters
$$\Delta \Omega \lambda$$
Log Laws
$$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$$
Coulomb's Law
$$F = \frac{k q_1 q_2}{r^2}$$
Capacitance
$$C = \frac{\epsilon A}{d}$$
Capacitor Charge
$$V = \frac{Q}{C}$$
Capacitor Current
$$I = C \frac{dV}{dt}$$
Capacitor Energy
$$E = \frac{1}{2} C V^2$$
Inductor Energy
$$E = \frac{1}{2} L I^2$$
Inductor Voltage
$$V = L \frac{dI}{dt}$$
Relativistic Addition of Velocities
$$u = \frac{v + u'}{1 + \frac{v \cdot u'}{c^2}}$$
Maxwell's Equations
$$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0},\ \nabla \cdot \mathbf{B} = 0,...$$
Newton's Law of Cooling
$$\frac{dT}{dt} = -k(T - T_{\text{env}})$$
Boltzmann Entropy
$$S=k_B\ln\Omega$$
De Moivre's Theorem
$$\small (\cos x+ i \sin x)^n=\cos(nx)+i\sin(nx)$$