Formulas for IPhO

$$F(x) = F(x_0) + \sum \frac{F^{(n)}(x_0)}{n!}(x - x_0)^n$$

$$x = x^{(0)} + x^{(1)} + x^{(2)} + \cdots$$

$$ay'' + by' + cy = 0 \quad\Rightarrow\quad y = Ae^{\lambda_1 x} + Be^{\lambda_2 x}$$

$$z = a + bi = |z|e^{i\varphi}$$

$$\vec{a}\cdot\vec{b} = ab\cos\varphi = a_x b_x + a_y b_y + a_z b_z$$ $$|\vec{a}\times\vec{b}| = ab\sin\varphi$$

$$c^2 = a^2 + b^2 - 2ab\cos\varphi$$ $$\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = 2R$$

$$\sin(\alpha\pm\beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$$ $$\cos(\alpha\pm\beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$

$$\theta_{\text{inscribed}} = \tfrac{1}{2}\,\theta_{\text{central}}$$

$$S = \tfrac{1}{2}\,a\,h_a = pr = \sqrt{p(p-a)(p-b)(p-c)} = \frac{abc}{4R}$$

$$\vec{r}_G = \frac{\vec{r}_A + \vec{r}_B + \vec{r}_C}{3}$$

$$(fg)' = fg' + f'g, \qquad f[g(x)]' = f'[g(x)]\,g'(x)$$

$$\int x^n\,dx = \frac{x^{n+1}}{n+1}$$

$$a_{11}x^2 + 2a_{12}xy + a_{22}y^2 + a_1 x + a_2 y + a_0 = 0$$

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

$$\frac{d\vec{v}}{dt} = \frac{dv_x}{dt}\vec{e}_x + \frac{dv_y}{dt}\vec{e}_y + \frac{dv_z}{dt}\vec{e}_z$$

$$[F] = [m][a] = \text{kg}\cdot\text{m/s}^2 = \text{N}$$

$$\text{If two quantities are given equal} \Rightarrow \text{look there for the key}$$

$$\text{Algebra first, numbers last}$$

$$\text{Difficult problem} \Rightarrow \text{simple solution and simple answer}$$

$$\vec{v} = \frac{d\vec{x}}{dt}, \quad \vec{x} = \int\vec{v}\,dt$$ $$\vec{a} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{x}}{dt^2}, \quad \vec{v} = \int\vec{a}\,dt$$

$$\omega = \frac{d\varphi}{dt}, \quad \varepsilon = \frac{d\omega}{dt}$$ $$\vec{a} = \vec{\tau}\frac{dv}{dt} + \vec{n}\frac{v^2}{R}$$

$$v_A\cos\alpha = v_B\cos\beta$$

$$\vec{v}_2 = \vec{v}_0 + \vec{v}_1$$ $$\vec{a}_2 = \vec{a}_0 + \vec{a}_1 + \omega^2\vec{R} + \vec{a}_{\text{Cor}}$$

$$y \le \frac{v_0^2}{2g} - \frac{gx^2}{2v_0^2}$$

$$\delta\!\int \frac{ds}{v} = 0$$

$$\sum F_x = 0, \quad \sum F_y = 0, \quad \sum M = 0$$

$$\vec{F}_{\text{contact}} = \vec{N} + \vec{f}, \quad \tan\theta_{\max} = \mu$$

$$\vec{F} = m\vec{a}, \qquad \vec{M} = I\vec{\varepsilon}$$

$$K = \tfrac{1}{2}\mu\dot{\xi}^2, \quad \mu\ddot{\xi} = -\frac{d\Pi(\xi)}{d\xi}$$

$$\vec{r}_c = \frac{\sum m_i \vec{r}_i}{\sum m_j}, \quad \vec{P} = \sum m_i \vec{v}_i$$ $$\vec{L} = \sum m_i \vec{r}_i \times \vec{v}_i, \quad K = \sum \tfrac{1}{2}m_i v_i^2$$

$$\vec{L} = \vec{L}_c + M_\Sigma \vec{R}_c \times \vec{v}_c$$ $$K = K_c + \tfrac{1}{2}M_\Sigma v_c^2$$

$$I = I_c + mb^2$$

$$\vec{F}_\Sigma = \frac{d\vec{P}}{dt}, \qquad \vec{M}_\Sigma = \frac{d\vec{L}}{dt}$$

$$\omega^2(l) = \frac{g}{l + I/(ml)}$$ $$\tilde{l} = l + I/(ml)$$

$$I = \alpha\, m R^2$$

$$E = \text{const} \quad (\text{elastic, no friction})$$ $$\vec{P} = \text{const} \quad (\text{no net external force})$$ $$\vec{L} = \text{const} \quad (\text{no net external torque})$$

$$\vec{F}_{\text{inertial}} = -m\vec{a}$$ $$\vec{F}_{\text{centrifugal}} = m\omega^2\vec{R}$$ $$\vec{F}_{\text{Cor}} = 2m\vec{v}\times\vec{\Omega}$$

$$\sum \vec{p}_{\text{before}} = \sum \vec{p}_{\text{after}}$$ $$\sum E_{\text{before}} = \sum E_{\text{after}} \quad (\text{elastic})$$

$$v_P = \omega \cdot CP$$

$$N = \frac{T}{R}$$

$$p = p_0 - w$$

$$p + \tfrac{1}{2}\rho v^2 + \rho\varphi = \text{const}$$

$$p + \rho v^2 = \text{const}$$

$$\oint p\,dx = \text{const}$$

$$\frac{d^2 U}{dx^2}\bigg|_{x_0} > 0 \quad\Rightarrow\quad \text{stable}$$

$$F \propto |\vec{r}|: \quad \langle K\rangle = \langle\Pi\rangle$$ $$F \propto |\vec{r}|^{-2}: \quad 2\langle K\rangle = -\langle\Pi\rangle$$

$$\Delta v = u\ln\frac{M}{m}$$

$$\ddot{x} + 2\gamma\dot{x} + \omega_0^2 x = 0 \quad (\gamma < \omega_0)$$ $$x = x_0 e^{-\gamma t}\sin\!\left(t\sqrt{\omega_0^2 - \gamma^2} - \varphi_0\right)$$

$$\ddot{x}_i = \sum_j a_{ij} x_j$$

$$x_j = \sum_i X_i x_{j0}\sin(\omega_i t + \varphi_{ij} + \phi_i)$$

$$\omega^2 = \frac{\kappa}{\mu}, \quad \kappa = \Pi''(0)$$

$$\varphi = kx - \omega t + \varphi_0$$ $$v_f = \nu\lambda = \omega/k, \quad v_g = d\omega/dk$$

$$e^{i(kx-\omega t)} + e^{i(-kx-\omega t)} = 2e^{-i\omega t}\cos kx$$

$$c_s = \sqrt{(\partial p/\partial\rho)_{\text{adiab}}} = \sqrt{\gamma p/\rho} = \bar{v}\sqrt{\gamma/3}$$

$$c_s = \sqrt{E/\rho}$$

$$v = \sqrt{gh} \quad (\text{shallow water, } h \ll \lambda)$$ $$v = \sqrt{T/\rho_{\text{lin}}} \quad (\text{string})$$

$$\nu = \nu_0\,\frac{1 + v_\parallel/c_s}{1 - u_\parallel/c_s}$$

$$\Delta x = c_s \Delta t$$

$$\delta\!\int n\,ds = 0$$

$$\frac{\sin\alpha_1}{\sin\alpha_2} = \frac{n_2}{n_1} = \frac{v_1}{v_2}$$

$$n' = n/r$$

$$k_x, k_y = \text{const}, \quad |\vec{k}|/n = \text{const}$$

$$\frac{1}{a} + \frac{1}{b} = \frac{1}{f} \equiv D$$

$$x_1 x_2 = f^2$$

$$\text{4 rules for ray tracing through lenses}$$

$$I = \Phi/\Omega \quad [\text{cd}]$$ $$E = \Phi/S \quad [\text{lx}]$$

$$\Phi = 4\pi\sum I_i, \qquad E = I/r^2$$

$$A = \sum_i A_i e^{i\varphi_i}$$

$$\varphi_{\max} = \arcsin\!\left(\frac{n\lambda}{a}\right), \quad n \in \mathbb{Z}$$ $$I \propto \cos^2\!\left(k\frac{a}{2}\sin\varphi\right)$$

$$\varphi_{\min} = \arcsin\!\left(\frac{n\lambda}{d}\right), \quad n \in \mathbb{Z},\; n \neq 0$$

$$\frac{\lambda}{\Delta\lambda} = nN$$

$$\frac{\lambda}{\Delta\lambda} = \frac{L}{\lambda}$$

$$\frac{\lambda}{\Delta\lambda} = a\frac{dn}{d\lambda}$$

$$\varphi = 1.22\,\lambda/d$$

$$2a\sin\alpha = k\lambda$$

$$\Delta\varphi = \pi \quad (\text{reflection from denser medium})$$

$$\frac{\nu}{\Delta\nu} \approx \frac{2a}{\lambda(1-r)}$$

$$I = cn\varepsilon_0 E^2 = \frac{c}{n\mu_0}B^2$$

$$I = I_0\cos^2\varphi$$

$$\Delta\varphi = \pi/2$$

$$\tan\varphi_B = n$$

$$U = IR, \quad P = UI$$ $$R_{\text{series}} = \sum R_i, \quad R_{\parallel}^{-1} = \sum R_i^{-1}$$

$$\sum_{\text{node}} I = 0, \qquad \sum_{\text{contour}} U = 0$$

$$\text{Node potentials} \quad | \quad \text{Loop currents} \quad | \quad \Delta \leftrightarrow Y$$

$$R_{\text{chain}} = \frac{R}{2} + \sqrt{\frac{R^2}{4} + RR'}$$

$$Z_R = R, \quad Z_C = \frac{1}{i\omega C}, \quad Z_L = i\omega L$$

$$\tau_{RC} = RC, \quad \tau_{LR} = L/R, \quad \omega_{LC} = 1/\sqrt{LC}$$

$$\Delta W + Q = Uq$$

$$W_C = \tfrac{1}{2}CU^2, \qquad W_L = \tfrac{1}{2}LI^2$$

$$\mathcal{E} = -\frac{d\Phi}{dt} = -\frac{d(LI)}{dt}, \quad \Phi = BS$$

$$\text{U-I curve} \cap \text{load line} = \text{operating point}$$

$$t \gg \tau: \quad I_C \approx 0,\; \mathcal{E}_L \approx 0$$ $$t \ll \tau: \quad C \text{ short-circuited},\; L \text{ broken}$$

$$I_L(t^+) = I_L(t^-)$$

$$\Phi = \text{const}, \quad LI = \text{const}$$

$$\Phi_1 = L_1 I_1 + L_{12} I_2$$ $$L_{12} = L_{21} \equiv M, \quad M \le \sqrt{L_1 L_2}$$

$$F = \frac{kq_1 q_2}{r^2}, \qquad \Pi = \frac{kq_1 q_2}{r}$$

$$\oint \vec{B}\cdot d\vec{S} = 0$$ $$\oint \varepsilon\varepsilon_0 \vec{E}\cdot d\vec{S} = Q$$ $$\oint \vec{g}\cdot d\vec{S} = -4\pi GM$$

$$\oint \vec{E}\cdot d\vec{l} = 0 \;(= -\dot\Phi)$$ $$\oint \frac{\vec{B}\cdot d\vec{l}}{\mu\mu_0} = I$$ $$\oint \vec{g}\cdot d\vec{l} = 0$$

$$d\vec{B} = \frac{\mu\mu_0 I}{4\pi}\frac{d\vec{l}\times\vec{e}_r}{r^2}$$

$$\vec{F} = e(\vec{v}\times\vec{B} + \vec{E})$$ $$\vec{F} = I\vec{l}\times\vec{B}$$

$$\text{Wire:}\; E = \frac{\sigma}{2\pi\varepsilon_0 r},\quad B = \frac{I\mu_0}{2\pi r}$$ $$\text{Surface:}\; E = \frac{\sigma}{2\varepsilon_0},\quad B = \frac{\mu_0 j}{2}$$

$$B = In\mu\mu_0 \quad (\text{inside, long})$$ $$L = Vn^2\mu\mu_0$$

$$\Pi = k\sum_{i>j}\frac{q_i q_j}{r_{ij}} = \frac{1}{2}\int\varphi(\vec{r})\,dq$$

$$\vec{E}_{\text{inside}} = 0$$

$$C_{\text{plane}} = \frac{\varepsilon\varepsilon_0 S}{d}, \quad C_{\text{sphere}} = 4\pi\varepsilon\varepsilon_0 r$$ $$C_{\text{coax}} = \frac{2\pi\varepsilon\varepsilon_0 l}{\ln(R/r)}$$

$$\vec{d}_e = \sum q_i\vec{r}_i = \vec{l}\,q, \qquad \vec{d}_\mu = I\vec{S}$$

$$W = -\vec{d}\cdot\vec{E}\;(\vec{B})$$ $$\vec{M} = \vec{d}\times\vec{E}\;(\vec{B})$$

$$\varphi = k\frac{\vec{d}\cdot\vec{e}_r}{r^2}, \qquad E, B \propto r^{-3}$$

$$F = (\vec{E}\,\vec{d}_e)', \quad F = (\vec{B}\,\vec{d}_\mu)'$$

$$q' = -q \quad (\text{grounded plane at distance } d)$$

$$\vec{d} \propto \vec{E}$$

$$P \sim \frac{B^2 v^2}{\rho}$$ $$F\tau \sim \frac{B^2 a^3 d}{\rho}$$

$$v_{\text{drift}} = E/B = F/(eB)$$

$$\mathcal{E} = vBa, \quad r = \rho a/(bc)$$

$$W_{\text{cycle}} = \oint B\,dH$$

$$\vec{D} = \varepsilon\varepsilon_0\vec{E} = \varepsilon_0\vec{E} + \vec{P}$$ $$\vec{H} = \vec{B}/(\mu\mu_0) = \vec{B}/\mu_0 - \vec{J}$$

$$E_t,\; D_n,\; H_t,\; B_n \quad \text{are continuous}$$

$$W = \tfrac{1}{2}\left(\varepsilon\varepsilon_0 E^2 + \frac{B^2}{\mu\mu_0}\right)$$

$$\vec{j} = ne\vec{v} = \sigma\vec{E} = \vec{E}/\rho$$

$$\text{System responds so as to oppose changes}$$

$$pV = \frac{m}{\mu}RT$$

$$U = \frac{i}{2}RT \quad (\text{per mole})$$

$$V_m = 22{,}4\;\text{L}$$

$$pV^\gamma = \text{const}, \qquad TV^{\gamma-1} = \text{const}$$

$$\gamma = \frac{c_p}{c_v} = \frac{i+2}{i}$$

$$\rho = \rho_0 e^{-\mu gh/(RT)} = \rho_0 e^{-U/(kT)}$$

$$f(v) \propto e^{-mv^2/(2kT)}$$

$$\Delta p = \rho g\Delta h \quad (\text{if } \Delta p \ll p)$$

$$p = \tfrac{1}{3}mn\bar{v}^2 = nkT$$ $$\bar{v} = \sqrt{3kT/m}, \quad \nu = \tfrac{1}{4}n\bar{v}S$$

$$\eta = \frac{T_1 - T_2}{T_1}$$

$$\eta = \frac{T_1}{T_1 - T_2}$$

$$dS = \frac{dQ}{T}$$

$$\delta U = \delta Q + \delta A$$

$$\Delta S \ge 0, \qquad \eta_{\text{real}} \le \eta_{\text{Carnot}}$$

$$A = \int p\,dV, \quad A_{\text{adiab}} = \frac{i}{2}\Delta(pV)$$

$$p = \sum p_i$$

$$p_v = p_0$$

$$P = \frac{kS\Delta T}{l}$$

$$Q = \int c(T)\,dT$$

$$U = S\sigma, \quad F = l\sigma, \quad p = 2\sigma/R$$

$$P = \varepsilon\sigma T^4$$

$$f_{\max} = Ak_B T/h \quad (A \approx 2.8)$$ $$\lambda_{\max} = hc/(A'k_B T) \quad (A' \approx 5)$$

$$\vec{p} = \hbar\vec{k} \quad (|\vec{p}| = h/\lambda), \qquad E = \hbar\omega = h\nu$$

$$\Delta p\,\Delta x \ge \frac{\hbar}{2}, \quad \Delta E\,\Delta t \ge \frac{\hbar}{2}, \quad \Delta\omega\,\Delta t \ge \frac{1}{2}$$

$$h\nu = E_n - E_m$$

$$E_n = \left(n + \tfrac{1}{2}\right)h\nu_0$$

$$\Gamma\tau \approx \hbar, \quad \tau = l/\sqrt{\Gamma/m}$$

$$E_n \propto -1/n^2$$

$$\Delta\lambda = \lambda_C(1 - \cos\theta)$$

$$A + \tfrac{1}{2}mv^2 = h\nu$$

$$F = \frac{GMm}{r^2}, \qquad \Pi = -\frac{GMm}{r}$$

$$\text{trajectory} = \text{ellipse, parabola, or hyperbola}$$

$$\frac{dA}{dt} = \frac{L}{2m} = \text{const}$$

$$\frac{T_1^2}{T_2^2} = \frac{a_1^3}{a_2^3}$$

$$E = -\frac{GMm}{2a}$$

$$\varepsilon = d/a \ll 1$$

$$l_1 + l_2 = 2a, \quad \alpha_1 = \alpha_2, \quad S = \pi ab$$

$$\vec{\varepsilon} = \frac{\vec{L}\times\vec{v}}{GMm} + \vec{e}_r = \text{const}$$

$$x' = \gamma(x - vt), \quad y' = y, \quad t' = \gamma\!\left(t - \frac{vx}{c^2}\right)$$ $$p'_x = \gamma(p_x - mv), \quad m' = \gamma\!\left(m - \frac{p_x v}{c^2}\right)$$

$$s^2 = c^2 t^2 - x^2 - y^2 - z^2$$ $$m_0^2 c^2 = m^2 c^2 - p_x^2 - p_y^2 - p_z^2$$

$$w = \frac{u + v}{1 + uv/c^2}$$

$$\nu' = \nu_0\sqrt{\frac{1 - v/c}{1 + v/c}}$$

$$l' = l_0/\gamma$$

$$t' = t_0\gamma$$

$$\Delta t = -\gamma v\Delta x / c^2$$

$$\vec{F} = \frac{d\vec{p}}{dt} = \frac{d}{dt}(m\vec{v}), \quad m = m_0\gamma$$

$$v \approx c, \quad p \approx mc, \quad \sqrt{1 - v^2/c^2} \approx \sqrt{2(1 - v/c)}$$

$$\vec{B}'_\parallel = \vec{B}_\parallel, \quad \vec{E}'_\parallel = \vec{E}_\parallel$$ $$\vec{E}'_\perp = \gamma(\vec{E}_\perp + \vec{v}\times\vec{B}_\perp)$$ $$\vec{B}'_\perp = \gamma\!\left(\vec{B}_\perp - \vec{v}\times\frac{\vec{E}_\perp}{c^2}\right)$$