Formulas

Every formula links to its topic note. Grown continuously — /teach and /process-lecture-notes append here.

Calculus — MATH 1A03 / 1AA3

Formula	Topic
$\frac{d}{dx}[x^n]=n{x}^{n-1}$	differentiation-rules
$\frac{d}{dx}[f(g(x))]=f^{\prime }(g(x))g^{\prime }(x)$	differentiation-rules
$(fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }$	differentiation-rules
${\left(\frac{f}{g}\right)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$	differentiation-rules
$\frac{d}{dx}[\sin x]=\cos x,\frac{d}{dx}[e^x]=e^x,\frac{d}{dx}[\ln x]=\frac{1}{x}$	differentiation-rules
${\int }_a^{b}f(x)dx=F(b)-F(a)$	antiderivatives
$\int udv=uv-\int vdu$	integration-by-parts
$f(x)={\sum }_{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$	taylor-series
$\frac{dy}{dx}=g(x)h(y)\Rightarrow \int \frac{dy}{h(y)}=\int g(x)dx$	separable-odes

Kinematics & Mechanics — PHYSICS 1C03

Formula	Topic
$v(t)=\frac{dx}{dt},a(t)=\frac{dv}{dt}$	kinematics-with-calculus
$x=x_0+v_{0}t+\frac{1}{2}a{t}^2$	kinematics-with-calculus
$\vec{F}=m\vec{a}$	newtons-laws-2d
$W=\int \vec{F}\cdot d\vec{s},W_{net}=\Delta KE$	work-energy-theorem
$\vec{p}=m\vec{v},\vec{J}=\Delta \vec{p}$	momentum-and-collisions
$x(t)=A\cos (\omega t+\varphi ),\omega =\sqrt{k/m}$	oscillations
$v=f\lambda$	wave-properties
$F=G\frac{m_1{m}_2}{r^2}$	newtons-laws-2d · inverse square: see interdisciplinary

Thermodynamics — PHYSICS 1C03 / CHEM 1A03

Formula	Topic
$PV=nRT$	gas-laws
$\Delta U=Q-W$	thermodynamics
$q=mc\Delta T$	thermochemistry-hess-law
$\Delta H_{rxn}=\sum \Delta H_f^{\circ }(\text{products})-\sum \Delta H_f^{\circ }(\text{reactants})$	thermochemistry-hess-law
$\Delta S=\frac{Q_{rev}}{T}$	thermodynamics

Equilibrium & Kinetics — CHEM 1AA3

Formula	Topic
$K_c=\frac{[\text{C}{]}^c[\text{D}{]}^d}{[\text{A}{]}^a[\text{B}{]}^b}$	equilibrium-constants
$K_a=\frac{[{\text{H}}^{+}][{\text{A}}^{-}]}{[\text{HA}]},\text{pH}=-\log [{\text{H}}^{+}]$	acid-base-equilibria
$\text{pH}=\text{p}{K}_a+\log \frac{[{\text{A}}^{-}]}{[\text{HA}]}$ (Henderson–Hasselbalch)	buffers
$\ln [A{]}_t=\ln [A{]}_0-kt$ (1st order)	reaction-kinetics
$k=A{e}^{-E_a/RT}$ (Arrhenius)	reaction-kinetics · Boltzmann bridge: see interdisciplinary
$N(t)=N_0{e}^{-\lambda t},t_{1/2}=\frac{\ln 2}{\lambda }$	nuclear-decay

Electrochemistry — CHEM 1AA3

Formula	Topic
$\Delta G^{\circ }=-nF{E}_{cell}^{\circ }$	electrochemical-cells
$E=E^{\circ }-\frac{RT}{nF}\ln Q$ (Nernst)	electrochemical-cells

Relativistic Kinematics — PHYSICS 1CC3

Formula	Topic
$\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$	lorentz-transformations
$\Delta t=\gamma \Delta t_0,L=L_0/\gamma$	lorentz-transformations
$E=\gamma m{c}^2,E^2=(pc{)}^2+(m{c}^2{)}^2$	mass-energy-equivalence
$\vec{p}=\gamma m\vec{v}$	mass-energy-equivalence

Quantum — PHYSICS 1CC3

Formula	Topic
$E=hf=\frac{hc}{\lambda }$	photoelectric-effect
$K{E}_{max}=hf-\varphi$	photoelectric-effect
$\lambda =\frac{h}{p}$	de-broglie-wavelength
$E_n=-\frac{13.6\text{eV}}{n^2}$	bohr-model
$-\frac{{\hslash }^2}{2m}\frac{{\partial }^2\psi }{\partial x^2}+V\psi =E\psi$ (time-independent, 1D)	schrodinger-equation
$\Delta x\Delta p\ge \frac{\hslash }{2}$	schrodinger-equation

Medical Physics — MEDPHYS 1A03

Formula	Topic
$I=I_0{e}^{-\mu x}$ (attenuation)	xray-ct-imaging
$I\propto \frac{1}{r^2}$	radiation-types · see interdisciplinary
$\tau =rF\sin \theta$	biomechanical-forces

Astronomy — ASTRON 1F03

Formula	Topic
$d(\text{pc})=\frac{1}{p{(}^{\prime \prime })}$ (parallax)	distance-measurement
$m-M=5{\log }_{10}(d)-5$ (distance modulus)	distance-measurement
$L=4\pi R^2\sigma T^4$ (Stefan–Boltzmann)	stellar-life-cycles
${\lambda }_{max}T=2.898\times 10^{-3}\text{mcdotpK}$ (Wien)	hr-diagram