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Quantum mechanics

Wavefunctions

A fundamental physical constant occurring in quantum mechanics is the Planck constant, h. A common abbreviation is ħ = h/2π, also known as the reduced Planck constant or Dirac constant.

Quantity (Common Name/s)	(Common) Symbol/s	Defining Equation	SI Units	Dimension
Wavefunction	ψ, Ψ	To solve from the Schrödinger equation	varies with situation and number of particles
Wavefunction probability density	ρ	$\rho =\left|\Psi \right|^{2}=\Psi ^{*}\Psi$	m−3	[L]−3
Wavefunction probability current	j	Non-relativistic, no external field: ${\begin{aligned}\mathbf {j} &={\frac {-i\hbar }{2m}}\left(\Psi ^{*}\nabla \Psi -\Psi \nabla \Psi ^{*}\right)\\&={\frac {\hbar }{m}}\operatorname {Im} \left(\Psi ^{*}\nabla \Psi \right)=\operatorname {Re} \left(\Psi ^{*}{\frac {\hbar }{im}}\nabla \Psi \right)\end{aligned}}$ denotes the complex conjugate	m−2 s−1	[T]−1 [L]−2

The general form of wavefunction for a system of particles, each with position r_i and z-component of spin s_{z i}. Sums are over the discrete variable s_z, integrals over continuous positions r.

For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is mathematically necessary). Following are general mathematical results, used in calculations.

Property or effect	Nomenclature	Equation
Wavefunction for N particles in 3d	r = (r1, r2... rN) sz = (sz 1, sz 2, ..., sz N)	In function notation: $\Psi =\Psi \left(\mathbf {r} ,\mathbf {s_{z}} ,t\right)$ in bra–ket notation: $|\Psi \rangle =\sum _{s_{z1}}\sum _{s_{z2}}\cdots \sum _{s_{zN}}\int _{V_{1}}\int _{V_{2}}\cdots \int _{V_{N}}\mathrm {d} \mathbf {r} _{1}\mathrm {d} \mathbf {r} _{2}\cdots \mathrm {d} \mathbf {r} _{N}\Psi |\mathbf {r} ,\mathbf {s_{z}} \rangle$ for non-interacting particles: $\Psi =\prod _{n=1}^{N}\Psi \left(\mathbf {r} _{n},s_{zn},t\right)$
Position-momentum Fourier transform (1 particle in 3d)	Φ = momentum-space wavefunction Ψ = position-space wavefunction	${\begin{aligned}\Phi (\mathbf {p} ,s_{z},t)&={\frac {1}{{\sqrt {2\pi \hbar }}^{3}}}\int \limits _{\mathrm {all\,space} }e^{-i\mathbf {p} \cdot \mathbf {r} /\hbar }\Psi (\mathbf {r} ,s_{z},t)\mathrm {d} ^{3}\mathbf {r} \\&\upharpoonleft \downharpoonright \\\Psi (\mathbf {r} ,s_{z},t)&={\frac {1}{{\sqrt {2\pi \hbar }}^{3}}}\int \limits _{\mathrm {all\,space} }e^{+i\mathbf {p} \cdot \mathbf {r} /\hbar }\Phi (\mathbf {p} ,s_{z},t)\mathrm {d} ^{3}\mathbf {p} _{n}\\\end{aligned}}$
General probability distribution	Vj = volume (3d region) particle may occupy, P = Probability that particle 1 has position r1 in volume V1 with spin sz1 and particle 2 has position r2 in volume V2 with spin sz2, etc.	$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int _{V_{N}}\cdots \int _{V_{2}}\int _{V_{1}}\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}\,\!$
General normalization condition		$P=\sum _{s_{zN}}\cdots \sum _{s_{z2}}\sum _{s_{z1}}\int \limits _{\mathrm {all\,space} }\cdots \int \limits _{\mathrm {all\,space} }\;\int \limits _{\mathrm {all\,space} }\left|\Psi \right|^{2}\mathrm {d} ^{3}\mathbf {r} _{1}\mathrm {d} ^{3}\mathbf {r} _{2}\cdots \mathrm {d} ^{3}\mathbf {r} _{N}=1\,\!$

Equations

Wave-particle duality and time evolution

Property or effect	Nomenclature	Equation
Planck–Einstein equation and de Broglie wavelength relations	P = (E/c, p) is the four-momentum, K = (ω/c, k) is the four-wavevector, E = energy of particle ω = 2πf is the angular frequency and frequency of the particle ħ = h/2π are the Planck constants c = speed of light	$\mathbf {P} =(E/c,\mathbf {p} )=\hbar (\omega /c,\mathbf {k} )=\hbar \mathbf {K}$
Schrödinger equation	Ψ = wavefunction of the system Ĥ = Hamiltonian operator, E = energy eigenvalue of system i is the imaginary unit t = time	General time-dependent case: $i\hbar {\frac {\partial }{\partial t}}\Psi ={\hat {H}}\Psi$ Time-independent case: ${\hat {H}}\Psi =E\Psi$
Heisenberg equation	 = operator of an observable property [ ] is the commutator $\langle \,\rangle$ denotes the average	${\frac {d}{dt}}{\hat {A}}(t)={\frac {i}{\hbar }}[{\hat {H}},{\hat {A}}(t)]+{\frac {\partial {\hat {A}}(t)}{\partial t}}$
Time evolution in Heisenberg picture (Ehrenfest theorem)	m = mass, V = potential energy, r = position, p = momentum, of a particle.	${\frac {d}{dt}}\langle {\hat {A}}\rangle ={\frac {1}{i\hbar }}\langle [{\hat {A}},{\hat {H}}]\rangle +\left\langle {\frac {\partial {\hat {A}}}{\partial t}}\right\rangle$For momentum and position; $m{\frac {d}{dt}}\langle \mathbf {r} \rangle =\langle \mathbf {p} \rangle$ ${\frac {d}{dt}}\langle \mathbf {p} \rangle =-\langle \nabla V\rangle$

Non-relativistic time-independent Schrödinger equation

Summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of wavefunction solutions. Notice in the case of one spatial dimension, for one particle, the partial derivative reduces to an ordinary derivative.

	One particle	N particles
One dimension	${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x)=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)$	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {p}}_{n}^{2}}{2m_{n}}}+V(x_{1},x_{2},\cdots x_{N})\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}+V(x_{1},x_{2},\cdots x_{N})\end{aligned}}$where the position of particle n is xn.
	$E\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\Psi +V\Psi$	$E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.$
	$\Psi (x,t)=\psi (x)e^{-iEt/\hbar }\,.$There is a further restriction — the solution must not grow at infinity, so that it has either a finite L2-norm (if it is a bound state) or a slowly diverging norm (if it is part of a continuum): $\|\psi \|^{2}=\int |\psi (x)|^{2}\,dx.\,$	$\Psi =e^{-iEt/\hbar }\psi (x_{1},x_{2}\cdots x_{N})$for non-interacting particles $\Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (x_{n})\,,\quad V(x_{1},x_{2},\cdots x_{N})=\sum _{n=1}^{N}V(x_{n})\,.$
Three dimensions	${\begin{aligned}{\hat {H}}&={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} )\\&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\end{aligned}}$where the position of the particle is r = (x, y, z).	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {\mathbf {p} }}_{n}\cdot {\hat {\mathbf {p} }}_{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})\end{aligned}}$where the position of particle n is r n = (xn, yn, zn), and the Laplacian for particle n using the corresponding position coordinates is $\nabla _{n}^{2}={\frac {\partial ^{2}}{{\partial x_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial y_{n}}^{2}}}+{\frac {\partial ^{2}}{{\partial z_{n}}^{2}}}$
	$E\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi$	$E\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}\Psi +V\Psi$
	$\Psi =\psi (\mathbf {r} )e^{-iEt/\hbar }$	$\Psi =e^{-iEt/\hbar }\psi (\mathbf {r} _{1},\mathbf {r} _{2}\cdots \mathbf {r} _{N})$for non-interacting particles $\Psi =e^{-i{Et/\hbar }}\prod _{n=1}^{N}\psi (\mathbf {r} _{n})\,,\quad V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})=\sum _{n=1}^{N}V(\mathbf {r} _{n})$

Non-relativistic time-dependent Schrödinger equation

Again, summarized below are the various forms the Hamiltonian takes, with the corresponding Schrödinger equations and forms of solutions.

	One particle	N particles
One dimension	${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(x,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}+V(x,t)$	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {p}}_{n}^{2}}{2m_{n}}}+V(x_{1},x_{2},\cdots x_{N},t)\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}+V(x_{1},x_{2},\cdots x_{N},t)\end{aligned}}$where the position of particle n is xn.
	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\Psi +V\Psi$	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}{\frac {\partial ^{2}}{\partial x_{n}^{2}}}\Psi +V\Psi \,.$
	$\Psi =\Psi (x,t)$	$\Psi =\Psi (x_{1},x_{2}\cdots x_{N},t)$
Three dimensions	${\begin{aligned}{\hat {H}}&={\frac {{\hat {\mathbf {p} }}\cdot {\hat {\mathbf {p} }}}{2m}}+V(\mathbf {r} ,t)\\&=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} ,t)\\\end{aligned}}$	${\begin{aligned}{\hat {H}}&=\sum _{n=1}^{N}{\frac {{\hat {\mathbf {p} }}_{n}\cdot {\hat {\mathbf {p} }}_{n}}{2m_{n}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)\\&=-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)\end{aligned}}$
	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi +V\Psi$	$i\hbar {\frac {\partial }{\partial t}}\Psi =-{\frac {\hbar ^{2}}{2}}\sum _{n=1}^{N}{\frac {1}{m_{n}}}\nabla _{n}^{2}\Psi +V\Psi$This last equation is in a very high dimension, so the solutions are not easy to visualize.
	$\Psi =\Psi (\mathbf {r} ,t)$	$\Psi =\Psi (\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N},t)$

Photoemission

Property/Effect	Nomenclature	Equation
Photoelectric equation	Kmax = Maximum kinetic energy of ejected electron (J) h = Planck's constant f = frequency of incident photons (Hz = s−1) φ, Φ = Work function of the material the photons are incident on (J)	$K_{\mathrm {max} }=hf-\Phi \,\!$
Threshold frequency and Work function	φ, Φ = Work function of the material the photons are incident on (J) f0, ν0 = Threshold frequency (Hz = s−1)	Can only be found by experiment. The De Broglie relations give the relation between them: $\phi =hf_{0}\,\!$
Photon momentum	p = momentum of photon (kg m s−1) f = frequency of photon (Hz = s−1) λ = wavelength of photon (m)	The De Broglie relations give: $p=hf/c=h/\lambda \,\!$

Quantum uncertainty

Property or effect	Nomenclature	Equation
Heisenberg's uncertainty principles	n = number of photons φ = wave phase [, ] = commutator	Position-momentum $\sigma (x)\sigma (p)\geq {\frac {\hbar }{2}}\,\!$ Energy-time $\sigma (E)\sigma (t)\geq {\frac {\hbar }{2}}\,\!$ Number-phase $\sigma (n)\sigma (\phi )\geq {\frac {\hbar }{2}}\,\!$
Dispersion of observable	A = observables (eigenvalues of operator)	${\begin{aligned}\sigma (A)^{2}&=\langle (A-\langle A\rangle )^{2}\rangle \\&=\langle A^{2}\rangle -\langle A\rangle ^{2}\end{aligned}}$
General uncertainty relation	A, B = observables (eigenvalues of operator)	$\sigma (A)\sigma (B)\geq {\frac {1}{2}}\langle i[{\hat {A}},{\hat {B}}]\rangle$

Probability Distributions

Property or effect	Equation
Density of states	$N(E)=8{\sqrt {2}}\pi m^{3/2}E^{1/2}/h^{3}\,\!$
Fermi–Dirac distribution (fermions)	$P(E_{i})={\frac {g(E_{i})}{e^{(E-\mu )/kT}+1}}$where P(Ei) = probability of energy Ei g(Ei) = degeneracy of energy Ei (no of states with same energy) μ = chemical potential
Bose–Einstein distribution (bosons)	$P(E_{i})={\frac {g(E_{i})}{e^{(E_{i}-\mu )/kT}-1}}$

Angular momentum

Property or effect	Nomenclature	Equation
Angular momentum quantum numbers	s = spin quantum number ms = spin magnetic quantum number ℓ = Azimuthal quantum number mℓ = azimuthal magnetic quantum number j = total angular momentum quantum number mj = total angular momentum magnetic quantum number	Spin: ${\begin{aligned}&\Vert \mathbf {s} \Vert ={\sqrt {s\,(s+1)}}\,\hbar \\&m_{s}\in \{-s,-s+1\cdots s-1,s\}\\\end{aligned}}\,\!$ Orbital: ${\begin{aligned}&\ell \in \{0\cdots n-1\}\\&m_{\ell }\in \{-\ell ,-\ell +1\cdots \ell -1,\ell \}\\\end{aligned}}\,\!$ Total: ${\begin{aligned}&j=\ell +s\\&m_{j}\in \{|\ell -s|,|\ell -s|+1\cdots |\ell +s|-1,|\ell +s|\}\\\end{aligned}}\,\!$
Angular momentum magnitudes	angular momementa: S = Spin, L = orbital, J = total	Spin magnitude: $|\mathbf {S} |=\hbar {\sqrt {s(s+1)}}\,\!$ Orbital magnitude: $|\mathbf {L} |=\hbar {\sqrt {\ell (\ell +1)}}\,\!$ Total magnitude: $\mathbf {J} =\mathbf {L} +\mathbf {S} \,\!$ $|\mathbf {J} |=\hbar {\sqrt {j(j+1)}}\,\!$
Angular momentum components		Spin: $S_{z}=m_{s}\hbar \,\!$ Orbital: $L_{z}=m_{\ell }\hbar \,\!$

Magnetic moments

In what follows, B is an applied external magnetic field and the quantum numbers above are used.

Property or effect	Nomenclature	Equation
orbital magnetic dipole moment	e = electron charge me = electron rest mass L = electron orbital angular momentum gℓ = orbital Landé g-factor μB = Bohr magneton	${\boldsymbol {\mu }}_{\ell }=-e\mathbf {L} /2m_{e}=g_{\ell }{\frac {\mu _{B}}{\hbar }}\mathbf {L} \,\!$z-component: $\mu _{\ell ,z}=-m_{\ell }\mu _{B}\,\!$
spin magnetic dipole moment	S = electron spin angular momentum gs = spin Landé g-factor	${\boldsymbol {\mu }}_{s}=-e\mathbf {S} /m_{e}=g_{s}{\frac {\mu _{B}}{\hbar }}\mathbf {S} \,\!$z-component: $\mu _{s,z}=-eS_{z}/m_{e}=g_{s}eS_{z}/2m_{e}\,\!$
dipole moment potential	U = potential energy of dipole in field	$U=-{\boldsymbol {\mu }}\cdot \mathbf {B} =-\mu _{z}B\,\!$

The Hydrogen atom

Property or effect	Nomenclature	Equation
Energy level	En = energy eigenvalue n = principal quantum number e = electron charge me = electron rest mass ε0 = permittivity of free space h = Planck's constant	$E_{n}=-me^{4}/8\varepsilon _{0}^{2}h^{2}n^{2}=-13.61\,\mathrm {eV} /n^{2}$
Spectrum	λ = wavelength of emitted photon, during electronic transition from Ei to Ej	${\frac {1}{\lambda }}=R\left({\frac {1}{n_{j}^{2}}}-{\frac {1}{n_{i}^{2}}}\right),\,n_{j}<n_{i}\,\!$

Thermodynamics

Definitions

Many of the definitions below are also used in the thermodynamics of chemical reactions.

General basic quantities

Quantity (common name/s)	(Common) symbol/s	SI unit	Dimension
Number of molecules	N	1	1
Amount of substance	n	mol	N
Temperature	T	K	Θ
Heat Energy	Q, q	J	ML2T−2
Latent heat	QL	J	ML2T−2

General derived quantities

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
Thermodynamic beta, inverse temperature	β	$\beta =1/k_{\text{B}}T$	J−1	T2M−1L−2
Thermodynamic temperature	τ	$\tau =k_{\text{B}}T$$\tau =k_{\text{B}}\left(\partial U/\partial S\right)_{N}$ $1/\tau =1/k_{\text{B}}\left(\partial S/\partial U\right)_{N}$	J	ML2T−2
Entropy	S	$S=-k_{\text{B}}\sum _{i}p_{i}\ln p_{i}$$S=-\left(\partial F/\partial T\right)_{V}$ , $S=-\left(\partial G/\partial T\right)_{N,P}$	J⋅K−1	ML2T−2Θ−1
Pressure	P	$P=-\left(\partial F/\partial V\right)_{T,N}$$P=-\left(\partial U/\partial V\right)_{S,N}$	Pa	ML−1T−2
Internal Energy	U	$U=\sum _{i}E_{i}$	J	ML2T−2
Enthalpy	H	$H=U+pV$	J	ML2T−2
Partition Function	Z		1	1
Gibbs free energy	G	$G=H-TS$	J	ML2T−2
Chemical potential (of component i in a mixture)	μi	$\mu _{i}=\left(\partial U/\partial N_{i}\right)_{N_{j\neq i},S,V}$$\mu _{i}=\left(\partial F/\partial N_{i}\right)_{T,V}$, where $F$ is not proportional to $N$ because $\mu _{i}$ depends on pressure. $\mu _{i}=\left(\partial G/\partial N_{i}\right)_{T,P}$, where $G$ is proportional to $N$ (as long as the molar ratio composition of the system remains the same) because $\mu _{i}$ depends only on temperature and pressure and composition. $\mu _{i}/\tau =-1/k_{\text{B}}\left(\partial S/\partial N_{i}\right)_{U,V}$	J	ML2T−2
Helmholtz free energy	A, F	$F=U-TS$	J	ML2T−2
Landau potential, Landau free energy, Grand potential	Ω, ΦG	$\Omega =U-TS-\mu N\$	J	ML2T−2
Massieu potential, Helmholtz free entropy	Φ	$\Phi =S-U/T$	J⋅K−1	ML2T−2Θ−1
Planck potential, Gibbs free entropy	Ξ	$\Xi =\Phi -pV/T$	J⋅K−1	ML2T−2Θ−1

Thermal properties of matter

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
General heat/thermal capacity	C	$C=\partial Q/\partial T$	J⋅K−1	ML2T−2Θ−1
Heat capacity (isobaric)	Cp	$C_{p}=\partial H/\partial T$	J⋅K−1	ML2T−2Θ−1
Specific heat capacity (isobaric)	Cmp	$C_{mp}=\partial ^{2}Q/\partial m\partial T$	J⋅kg−1⋅K−1	L2T−2Θ−1
Molar specific heat capacity (isobaric)	Cnp	$C_{np}=\partial ^{2}Q/\partial n\partial T$	J⋅K−1⋅mol−1	ML2T−2Θ−1N−1
Heat capacity (isochoric/volumetric)	CV	$C_{V}=\partial U/\partial T$	J⋅K−1	ML2T−2Θ−1
Specific heat capacity (isochoric)	CmV	$C_{mV}=\partial ^{2}Q/\partial m\partial T$	J⋅kg−1⋅K−1	L2T−2Θ−1
Molar specific heat capacity (isochoric)	CnV	$C_{nV}=\partial ^{2}Q/\partial n\partial T$	J⋅K⋅−1 mol−1	ML2T−2Θ−1N−1
Specific latent heat	L	$L=\partial Q/\partial m$	J⋅kg−1	L2T−2
Ratio of isobaric to isochoric heat capacity, heat capacity ratio, adiabatic index, Laplace coefficient	γ	$\gamma =C_{p}/C_{V}=c_{p}/c_{V}=C_{mp}/C_{mV}$	1	1

Thermal transfer

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI unit	Dimension
Temperature gradient	No standard symbol	$\nabla T$	K⋅m−1	ΘL−1
Thermal conduction rate, thermal current, thermal/heat flux, thermal power transfer	P	$P=\mathrm {d} Q/\mathrm {d} t$	W	ML2T−3
Thermal intensity	I	$I=\mathrm {d} P/\mathrm {d} A$	W⋅m−2	MT−3
Thermal/heat flux density (vector analogue of thermal intensity above)	q	$Q=\iint \mathbf {q} \cdot \mathrm {d} \mathbf {S} \mathrm {d} t$	W⋅m−2	MT−3

Equations

Thermodynamic processes

Physical situation	Equations
Isentropic process (adiabatic and reversible)	$Q=0,\quad \Delta U=-W$For an ideal gas $p_{1}V_{1}^{\gamma }=p_{2}V_{2}^{\gamma }$ $T_{1}V_{1}^{\gamma -1}=T_{2}V_{2}^{\gamma -1}$ $p_{1}^{1-\gamma }T_{1}^{\gamma }=p_{2}^{1-\gamma }T_{2}^{\gamma }$
Isothermal process	$\Delta U=0,\quad W=Q$For an ideal gas $W=kTN\ln(V_{2}/V_{1})$ $W=nRT\ln(V_{2}/V_{1})$
Isobaric process	p1 = p2, p = constant $W=p\Delta V,\quad Q=\Delta U+p\delta V$
Isochoric process	V1 = V2, V = constant $W=0,\quad Q=\Delta U$
Free expansion	$\Delta U=0$
Work done by an expanding gas	Process $W=\int _{V_{1}}^{V_{2}}p\mathrm {d} V$ Net work done in cyclic processes $W=\oint _{\mathrm {cycle} }p\mathrm {d} V=\oint _{\mathrm {cycle} }\Delta Q$

Kinetic theory

Ideal gas equations

Physical situation	Nomenclature	Equations
Ideal gas law	p = pressure V = volume of container T = temperature n = amount of substance R = gas constant N = number of molecules k = Boltzmann constant	$pV=nRT=kTN$ ${\frac {p_{1}V_{1}}{p_{2}V_{2}}}={\frac {n_{1}T_{1}}{n_{2}T_{2}}}={\frac {N_{1}T_{1}}{N_{2}T_{2}}}$
Pressure of an ideal gas	m = mass of one molecule Mm = molar mass	$p={\frac {Nm\langle v^{2}\rangle }{3V}}={\frac {nM_{m}\langle v^{2}\rangle }{3V}}={\frac {1}{3}}\rho \langle v^{2}\rangle$

Ideal gas

Quantity	General Equation	Isobaric Δp = 0	Isochoric ΔV = 0	Isothermal ΔT = 0	Adiabatic $Q=0$
Work W	$\delta W=-pdV\;$	$-p\Delta V\;$	$0\;$	$-nRT\ln {\frac {V_{2}}{V_{1}}}\;$$-nRT\ln {\frac {P_{1}}{P_{2}}}\;$	${\frac {PV^{\gamma }(V_{f}^{1-\gamma }-V_{i}^{1-\gamma })}{1-\gamma }}=C_{V}\left(T_{2}-T_{1}\right)$
Heat Capacity C	(as for real gas)	$C_{p}={\frac {5}{2}}nR$ (for monatomic ideal gas) $C_{p}={\frac {7}{2}}nR$ (for diatomic ideal gas)	$C_{V}={\frac {3}{2}}nR$ (for monatomic ideal gas) $C_{V}={\frac {5}{2}}nR\;$ (for diatomic ideal gas)
Internal Energy ΔU	$\Delta U=C_{V}\Delta T\;$	$Q+W\;$ $Q_{p}-p\Delta V$	$Q\;$ $C_{V}\left(T_{2}-T_{1}\right)$	$0\;$ $Q=-W$	$W\;$ $C_{V}\left(T_{2}-T_{1}\right)$
Enthalpy ΔH	$H=U+pV\;$	$C_{p}\left(T_{2}-T_{1}\right)\;$	$Q_{V}+V\Delta p\;$	$0\;$	$C_{p}\left(T_{2}-T_{1}\right)\;$
Entropy Δs	$\Delta S=C_{V}\ln {T_{2} \over T_{1}}+nR\ln {V_{2} \over V_{1}}$ $\Delta S=C_{p}\ln {T_{2} \over T_{1}}-nR\ln {p_{2} \over p_{1}}$	$C_{p}\ln {\frac {T_{2}}{T_{1}}}\;$	$C_{V}\ln {\frac {T_{2}}{T_{1}}}\;$	$nR\ln {\frac {V_{2}}{V_{1}}}\;$ ${\frac {Q}{T}}\;$	$C_{p}\ln {\frac {V_{2}}{V_{1}}}+C_{V}\ln {\frac {p_{2}}{p_{1}}}=0\;$
Constant	$\;$	${\frac {V}{T}}\;$	${\frac {p}{T}}\;$	$pV\;$	$pV^{\gamma }\;$

Entropy

• $S=k_{\mathrm {B} }\ln \Omega$, where k_B is the Boltzmann constant, and Ω denotes the volume of macrostate in the phase space or otherwise called thermodynamic probability.
• $dS={\frac {\delta Q}{T}}$, for reversible processes only

Statistical physics

Below are useful results from the Maxwell–Boltzmann distribution for an ideal gas, and the implications of the Entropy quantity. The distribution is valid for atoms or molecules constituting ideal gases.

Physical situation	Nomenclature	Equations
Maxwell–Boltzmann distribution	v = velocity of atom/molecule, m = mass of each molecule (all molecules are identical in kinetic theory), γ(p) = Lorentz factor as function of momentum (see below) Ratio of thermal to rest mass-energy of each molecule: $\theta =k_{\text{B}}T/mc^{2}$ K2 is the modified Bessel function of the second kind.	Non-relativistic speeds $P\left(v\right)=4\pi \left({\frac {m}{2\pi k_{\text{B}}T}}\right)^{3/2}v^{2}e^{-mv^{2}/2k_{\text{B}}T}$ Relativistic speeds (Maxwell–Jüttner distribution) $f(p)={\frac {1}{4\pi m^{3}c^{3}\theta K_{2}(1/\theta )}}e^{-\gamma (p)/\theta }$
Entropy Logarithm of the density of states	Pi = probability of system in microstate i Ω = total number of microstates	$S=-k_{\text{B}}\sum _{i}P_{i}\ln P_{i}=k_{\mathrm {B} }\ln \Omega$where: $P_{i}=1/\Omega$
Entropy change		$\Delta S=\int _{Q_{1}}^{Q_{2}}{\frac {\mathrm {d} Q}{T}}$ $\Delta S=k_{\text{B}}N\ln {\frac {V_{2}}{V_{1}}}+NC_{V}\ln {\frac {T_{2}}{T_{1}}}$
Entropic force		$\mathbf {F} _{\mathrm {S} }=-T\nabla S$
Equipartition theorem	df = degree of freedom	Average kinetic energy per degree of freedom $\langle E_{\mathrm {k} }\rangle ={\frac {1}{2}}kT$ Internal energy $U=d_{\text{f}}\langle E_{\mathrm {k} }\rangle ={\frac {d_{\text{f}}}{2}}kT$

Corollaries of the non-relativistic Maxwell–Boltzmann distribution are below.

Physical situation	Nomenclature	Equations
Mean speed		$\langle v\rangle ={\sqrt {\frac {8k_{\text{B}}T}{\pi m}}}$
Root mean square speed		$v_{\mathrm {rms} }={\sqrt {\langle v^{2}\rangle }}={\sqrt {\frac {3k_{\text{B}}T}{m}}}$
Modal speed		$v_{\mathrm {mode} }={\sqrt {\frac {2k_{\text{B}}T}{m}}}$
Mean free path	σ = effective cross-section n = volume density of number of target particles ℓ = mean free path	$\ell =1/{\sqrt {2}}n\sigma$

Quasi-static and reversible processes

For quasi-static and reversible processes, the first law of thermodynamics is:

$dU=\delta Q-\delta W$

where δQ is the heat supplied to the system and δW is the work done by the system.

Thermodynamic potentials

The following energies are called the thermodynamic potentials.

Name	Symbol	Formula	Natural variables
Internal energy	$U$	$\int \left(T\,\mathrm {d} S-p\,\mathrm {d} V+\sum _{i}\mu _{i}\mathrm {d} N_{i}\right)$	$S,V,\{N_{i}\}$
Helmholtz free energy	$F$	$U-TS$	$T,V,\{N_{i}\}$
Enthalpy	$H$	$U+pV$	$S,p,\{N_{i}\}$
Gibbs free energy	$G$	$U+pV-TS$	$T,p,\{N_{i}\}$
Landau potential, or grand potential	$\Omega$, $\Phi _{\text{G}}$	$U-TS-$$\sum _{i}\,$$\mu _{i}N_{i}$	$T,V,\{\mu _{i}\}$

and the corresponding fundamental thermodynamic relations or "master equations" are:

Potential	Differential
Internal energy	$dU\left(S,V,{N_{i}}\right)=TdS-pdV+\sum _{i}\mu _{i}dN_{i}$
Enthalpy	$dH\left(S,p,{N_{i}}\right)=TdS+Vdp+\sum _{i}\mu _{i}dN_{i}$
Helmholtz free energy	$dF\left(T,V,{N_{i}}\right)=-SdT-pdV+\sum _{i}\mu _{i}dN_{i}$
Gibbs free energy	$dG\left(T,p,{N_{i}}\right)=-SdT+Vdp+\sum _{i}\mu _{i}dN_{i}$

Maxwell's relations

The four most common Maxwell's relations are:

Physical situation	Nomenclature	Equations
Thermodynamic potentials as functions of their natural variables	$U(S,V)\,$ = Internal energy $H(S,P)\,$ = Enthalpy $F(T,V)\,$ = Helmholtz free energy $G(T,P)\,$ = Gibbs free energy	$\left({\frac {\partial T}{\partial V}}\right)_{S}=-\left({\frac {\partial P}{\partial S}}\right)_{V}={\frac {\partial ^{2}U}{\partial S\partial V}}$$\left({\frac {\partial T}{\partial P}}\right)_{S}=+\left({\frac {\partial V}{\partial S}}\right)_{P}={\frac {\partial ^{2}H}{\partial S\partial P}}$ $+\left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial P}{\partial T}}\right)_{V}=-{\frac {\partial ^{2}F}{\partial T\partial V}}$ $-\left({\frac {\partial S}{\partial P}}\right)_{T}=\left({\frac {\partial V}{\partial T}}\right)_{P}={\frac {\partial ^{2}G}{\partial T\partial P}}$

More relations include the following.

$\left({\partial S \over \partial U}\right)_{V,N}={1 \over T}$	$\left({\partial S \over \partial V}\right)_{N,U}={p \over T}$	$\left({\partial S \over \partial N}\right)_{V,U}=-{\mu \over T}$
$\left({\partial T \over \partial S}\right)_{V}={T \over C_{V}}$	$\left({\partial T \over \partial S}\right)_{P}={T \over C_{P}}$
$-\left({\partial p \over \partial V}\right)_{T}={1 \over {VK_{T}}}$

Other differential equations are:

Name	H	U	G
Gibbs–Helmholtz equation	$H=-T^{2}\left({\frac {\partial \left(G/T\right)}{\partial T}}\right)_{p}$	$U=-T^{2}\left({\frac {\partial \left(F/T\right)}{\partial T}}\right)_{V}$	$G=-V^{2}\left({\frac {\partial \left(F/V\right)}{\partial V}}\right)_{T}$
	$\left({\frac {\partial H}{\partial p}}\right)_{T}=V-T\left({\frac {\partial V}{\partial T}}\right)_{P}$	$\left({\frac {\partial U}{\partial V}}\right)_{T}=T\left({\frac {\partial P}{\partial T}}\right)_{V}-P$

Quantum properties

• $U=Nk_{\text{B}}T^{2}\left({\frac {\partial \ln Z}{\partial T}}\right)_{V}$
• $S={\frac {U}{T}}+Nk_{\text{B}}\ln Z-Nk\ln N+Nk$ Indistinguishable Particles

where N is number of particles, h is that Planck constant, I is moment of inertia, and Z is the partition function, in various forms:

Degree of freedom	Partition function
Translation	$Z_{t}={\frac {(2\pi mk_{\text{B}}T)^{\frac {3}{2}}V}{h^{3}}}$
Vibration	$Z_{v}={\frac {1}{1-e^{\frac {-h\omega }{2\pi k_{\text{B}}T}}}}$
Rotation	$Z_{r}={\frac {2Ik_{\text{B}}T}{\sigma ({\frac {h}{2\pi }})^{2}}}$ where: σ = 1 (heteronuclear molecules) σ = 2 (homonuclear)

Thermal properties of matter

Coefficients	Equation
Joule-Thomson coefficient	$\mu _{JT}=\left({\frac {\partial T}{\partial p}}\right)_{H}$
Compressibility (constant temperature)	$K_{T}=-{1 \over V}\left({\partial V \over \partial p}\right)_{T,N}$
Coefficient of thermal expansion (constant pressure)	$\alpha _{p}={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}$
Heat capacity (constant pressure)	$C_{p}=\left({\partial Q_{rev} \over \partial T}\right)_{p}=\left({\partial U \over \partial T}\right)_{p}+p\left({\partial V \over \partial T}\right)_{p}=\left({\partial H \over \partial T}\right)_{p}=T\left({\partial S \over \partial T}\right)_{p}$
Heat capacity (constant volume)	$C_{V}=\left({\partial Q_{rev} \over \partial T}\right)_{V}=\left({\partial U \over \partial T}\right)_{V}=T\left({\partial S \over \partial T}\right)_{V}$

Thermal transfer

Physical situation	Nomenclature	Equations
Net intensity emission/absorption	Texternal = external temperature (outside of system) Tsystem = internal temperature (inside system) ε = emmisivity	$I=\sigma \epsilon \left(T_{\mathrm {external} }^{4}-T_{\mathrm {system} }^{4}\right)$
Internal energy of a substance	CV = isovolumetric heat capacity of substance ΔT = temperature change of substance	$\Delta U=NC_{V}\Delta T$
Meyer's equation	Cp = isobaric heat capacity CV = isovolumetric heat capacity n = amount of substance	$C_{p}-C_{V}=nR$
Effective thermal conductivities	λi = thermal conductivity of substance i λnet = equivalent thermal conductivity	Series $\lambda _{\mathrm {net} }=\sum _{j}\lambda _{j}$ Parallel ${\frac {1}{\lambda }}_{\mathrm {net} }=\sum _{j}\left({\frac {1}{\lambda }}_{j}\right)$

Thermal efficiencies

Physical situation	Nomenclature	Equations
Thermodynamic engines	η = efficiency W = work done by engine QH = heat energy in higher temperature reservoir QL = heat energy in lower temperature reservoir TH = temperature of higher temp. reservoir TL = temperature of lower temp. reservoir	Thermodynamic engine: $\eta =\left|{\frac {W}{Q_{\text{H}}}}\right|$ Carnot engine efficiency: $\eta _{\text{c}}=1-\left|{\frac {Q_{\text{L}}}{Q_{\text{H}}}}\right|=1-{\frac {T_{\text{L}}}{T_{\text{H}}}}$
Refrigeration	K = coefficient of refrigeration performance	Refrigeration performance $K=\left|{\frac {Q_{\text{L}}}{W}}\right|$ Carnot refrigeration performance $K_{\text{C}}={\frac {|Q_{\text{L}}|}{|Q_{\text{H}}|-|Q_{\text{L}}|}}={\frac {T_{\text{L}}}{T_{\text{H}}-T_{\text{L}}}}$

Classical mechanics

Classical mechanics

Mass and inertia

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Linear, surface, volumetric mass density	λ or μ (especially in acoustics, see below) for Linear, σ for surface, ρ for volume.	$m=\int \lambda \,\mathrm {d} \ell$$m=\iint \sigma \,\mathrm {d} S$ $m=\iiint \rho \,\mathrm {d} V$	kg m−n, n = 1, 2, 3	M L−n
Moment of mass	m (No common symbol)	Point mass: $$\mathbf {m} =\mathbf {r} m$$ Discrete masses about an axis $x_{i}$: $$\mathbf {m} =\sum _{i=1}^{N}\mathbf {r} _{i}m_{i}$$ Continuum of mass about an axis $x_{i}$: $$\mathbf {m} =\int \rho \left(\mathbf {r} \right)x_{i}\mathrm {d} \mathbf {r}$$	kg m	M L
Center of mass	rcom (Symbols vary)	i-th moment of mass $\mathbf {m} _{i}=\mathbf {r} _{i}m_{i}$Discrete masses: $$\mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\sum _{i}\mathbf {r} _{i}m_{i}={\frac {1}{M}}\sum _{i}\mathbf {m} _{i}$$ Mass continuum: $$\mathbf {r} _{\mathrm {com} }={\frac {1}{M}}\int \mathrm {d} \mathbf {m} ={\frac {1}{M}}\int \mathbf {r} \,\mathrm {d} m={\frac {1}{M}}\int \mathbf {r} \rho \,\mathrm {d} V$$	m	L
2-Body reduced mass	m12, μ Pair of masses = m1 and m2	$\mu ={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}$	kg	M
Moment of inertia (MOI)	I	Discrete Masses: $$I=\sum _{i}\mathbf {m} _{i}\cdot \mathbf {r} _{i}=\sum _{i}\left|\mathbf {r} _{i}\right|^{2}m$$ Mass continuum: $$I=\int \left|\mathbf {r} \right|^{2}\mathrm {d} m=\int \mathbf {r} \cdot \mathrm {d} \mathbf {m} =\int \left|\mathbf {r} \right|^{2}\rho \,\mathrm {d} V$$	kg m2	M L2

Derived kinematic quantities

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Velocity	v	$\mathbf {v} ={\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}$	m s−1	L T−1
Acceleration	a	$\mathbf {a} ={\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}\mathbf {r} }{\mathrm {d} t^{2}}}$	m s−2	L T−2
Jerk	j	$\mathbf {j} ={\frac {\mathrm {d} \mathbf {a} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{3}\mathbf {r} }{\mathrm {d} t^{3}}}$	m s−3	L T−3
Jounce	s	$\mathbf {s} ={\frac {\mathrm {d} \mathbf {j} }{\mathrm {d} t}}={\frac {\mathrm {d} ^{4}\mathbf {r} }{\mathrm {d} t^{4}}}$	m s−4	L T−4
Angular velocity	ω	${\boldsymbol {\omega }}=\mathbf {\hat {n}} {\frac {\mathrm {d} \theta }{\mathrm {d} t}}$	rad s−1	T−1
Angular Acceleration	α	${\boldsymbol {\alpha }}={\frac {\mathrm {d} {\boldsymbol {\omega }}}{\mathrm {d} t}}=\mathbf {\hat {n}} {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}$	rad s−2	T−2
Angular jerk	ζ	${\boldsymbol {\zeta }}={\frac {\mathrm {d} {\boldsymbol {\alpha }}}{\mathrm {d} t}}=\mathbf {\hat {n}} {\frac {\mathrm {d} ^{3}\theta }{\mathrm {d} t^{3}}}$	rad s−3	T−3

Derived dynamic quantities

Angular momenta of a classical object.Left: intrinsic "spin" angular momentum S is really orbital angular momentum of the object at every point, right: extrinsic orbital angular momentum L about an axis, top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω) bottom: momentum p and its radial position r from the axis. The total angular momentum (spin + orbital) is J.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Momentum	p	$\mathbf {p} =m\mathbf {v}$	kg m s−1	M L T−1
Force	F	$\mathbf {F} =\mathrm {d} \mathbf {p} /\mathrm {d} t$	N = kg m s−2	M L T−2
Impulse	J, Δp, I	$\mathbf {J} =\Delta \mathbf {p} =\int _{t_{1}}^{t_{2}}\mathbf {F} \,\mathrm {d} t$	kg m s−1	M L T−1
Angular momentum about a position point r0,	L, J, S	$\mathbf {L} =\left(\mathbf {r} -\mathbf {r} _{0}\right)\times \mathbf {p}$Most of the time we can set r0 = 0 if particles are orbiting about axes intersecting at a common point.	kg m2 s−1	M L2 T−1
Moment of a force about a position point r0, Torque	τ, M	${\boldsymbol {\tau }}=\left(\mathbf {r} -\mathbf {r} _{0}\right)\times \mathbf {F} ={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}$	N m = kg m2 s−2	M L2 T−2
Angular impulse	ΔL (no common symbol)	$\Delta \mathbf {L} =\int _{t_{1}}^{t_{2}}{\boldsymbol {\tau }}\,\mathrm {d} t$	kg m2 s−1	M L2 T−1

General energy definitions

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Mechanical work due to a Resultant Force	W	$W=\int _{C}\mathbf {F} \cdot \mathrm {d} \mathbf {r}$	J = N m = kg m2 s−2	M L2 T−2
Work done ON mechanical system, Work done BY	WON, WBY	$\Delta W_{\mathrm {ON} }=-\Delta W_{\mathrm {BY} }$	J = N m = kg m2 s−2	M L2 T−2
Potential energy	φ, Φ, U, V, Ep	$\Delta W=-\Delta V$	J = N m = kg m2 s−2	M L2 T−2
Mechanical power	P	$P={\frac {\mathrm {d} E}{\mathrm {d} t}}$	W = J s−1	M L2 T−3

Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:

• Wherever the force is zero, its potential energy is defined to be zero as well.
• Whenever the force does work, potential energy is lost.

Generalized mechanics

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Generalized coordinates	q, Q		varies with choice	varies with choice
Generalized velocities	${\dot {q}},{\dot {Q}}$	${\dot {q}}\equiv \mathrm {d} q/\mathrm {d} t$	varies with choice	varies with choice
Generalized momenta	p, P	$p=\partial L/\partial {\dot {q}}$	varies with choice	varies with choice
Lagrangian	L	$L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {\dot {q}} )-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)$where $\mathbf {q} =\mathbf {q} (t)$ and p = p(t) are vectors of the generalized coords and momenta, as functions of time	J	M L2 T−2
Hamiltonian	H	$H(\mathbf {p} ,\mathbf {q} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {\dot {q}} ,t)$	J	M L2 T−2
Action, Hamilton's principal function	S, $\scriptstyle {\mathcal {S}}$	${\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\mathrm {d} t$	J s	M L2 T−1

Kinematics

In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ, but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector

$$\mathbf {\hat {n}} =\mathbf {\hat {e}} _{r}\times \mathbf {\hat {e}} _{\theta }$$

defines the axis of rotation, $\scriptstyle \mathbf {\hat {e}} _{r}$ = unit vector in direction of r, $\scriptstyle \mathbf {\hat {e}} _{\theta }$ = unit vector tangential to the angle.

	Translation	Rotation
Velocity	Average: $$\mathbf {v} _{\mathrm {average} }={\Delta \mathbf {r} \over \Delta t}$$ Instantaneous: $$\mathbf {v} ={d\mathbf {r} \over dt}$$	Angular velocity$${\boldsymbol {\omega }}=\mathbf {\hat {n}} {\frac {{\rm {d}}\theta }{{\rm {d}}t}}$$Rotating rigid body:$$\mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r}$$
Acceleration	Average: $$\mathbf {a} _{\mathrm {average} }={\frac {\Delta \mathbf {v} }{\Delta t}}$$ Instantaneous: $$\mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}$$	Angular acceleration $${\boldsymbol {\alpha }}={\frac {{\rm {d}}{\boldsymbol {\omega }}}{{\rm {d}}t}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{2}\theta }{{\rm {d}}t^{2}}}$$ Rotating rigid body: $$\mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} +{\boldsymbol {\omega }}\times \mathbf {v}$$
Jerk	Average: $$\mathbf {j} _{\mathrm {average} }={\frac {\Delta \mathbf {a} }{\Delta t}}$$ Instantaneous: $$\mathbf {j} ={\frac {d\mathbf {a} }{dt}}={\frac {d^{2}\mathbf {v} }{dt^{2}}}={\frac {d^{3}\mathbf {r} }{dt^{3}}}$$	Angular jerk $${\boldsymbol {\zeta }}={\frac {{\rm {d}}{\boldsymbol {\alpha }}}{{\rm {d}}t}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{2}\omega }{{\rm {d}}t^{2}}}=\mathbf {\hat {n}} {\frac {{\rm {d}}^{3}\theta }{{\rm {d}}t^{3}}}$$ Rotating rigid body: $$\mathbf {j} ={\boldsymbol {\zeta }}\times \mathbf {r} +{\boldsymbol {\alpha }}\times \mathbf {a}$$

Dynamics

	Translation	Rotation
Momentum	Momentum is the "amount of translation" $$\mathbf {p} =m\mathbf {v}$$ For a rotating rigid body: $$\mathbf {p} ={\boldsymbol {\omega }}\times \mathbf {m}$$	Angular momentum Angular momentum is the "amount of rotation": $$\mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {I} \cdot {\boldsymbol {\omega }}$$ and the cross-product is a pseudovector i.e. if r and p are reversed in direction (negative), L is not. In general I is an order-2 tensor, see above for its components. The dot · indicates tensor contraction.
Force and Newton's 2nd law	Resultant force acts on a system at the center of mass, equal to the rate of change of momentum: $${\begin{aligned}\mathbf {F} &={\frac {d\mathbf {p} }{dt}}={\frac {d(m\mathbf {v} )}{dt}}\\&=m\mathbf {a} +\mathbf {v} {\frac {{\rm {d}}m}{{\rm {d}}t}}\\\end{aligned}}$$ For a number of particles, the equation of motion for one particle i is: $${\frac {\mathrm {d} \mathbf {p} _{i}}{\mathrm {d} t}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}$$ where pi = momentum of particle i, Fij = force on particle i by particle j, and FE = resultant external force (due to any agent not part of system). Particle i does not exert a force on itself.	Torque Torque τ is also called moment of a force, because it is the rotational analogue to force: $${\boldsymbol {\tau }}={\frac {{\rm {d}}\mathbf {L} }{{\rm {d}}t}}=\mathbf {r} \times \mathbf {F} ={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\omega }})}{{\rm {d}}t}}$$ For rigid bodies, Newton's 2nd law for rotation takes the same form as for translation: $${\begin{aligned}{\boldsymbol {\tau }}&={\frac {{\rm {d}}\mathbf {L} }{{\rm {d}}t}}={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\omega }})}{{\rm {d}}t}}\\&={\frac {{\rm {d}}\mathbf {I} }{{\rm {d}}t}}\cdot {\boldsymbol {\omega }}+\mathbf {I} \cdot {\boldsymbol {\alpha }}\\\end{aligned}}$$ Likewise, for a number of particles, the equation of motion for one particle i is: $${\frac {\mathrm {d} \mathbf {L} _{i}}{\mathrm {d} t}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}$$
Yank	Yank is rate of change of force: $${\begin{aligned}\mathbf {Y} &={\frac {d\mathbf {F} }{dt}}={\frac {d^{2}\mathbf {p} }{dt^{2}}}={\frac {d^{2}(m\mathbf {v} )}{dt^{2}}}\\[1ex]&=m\mathbf {j} +\mathbf {2a} {\frac {{\rm {d}}m}{{\rm {d}}t}}+\mathbf {v} {\frac {{\rm {d^{2}}}m}{{\rm {d}}t^{2}}}\end{aligned}}$$ For constant mass, it becomes; $$\mathbf {Y} =m\mathbf {j}$$	Rotatum Rotatum Ρ is also called moment of a Yank, because it is the rotational analogue to yank: $${\boldsymbol {\mathrm {P} }}={\frac {{\rm {d}}{\boldsymbol {\tau }}}{{\rm {d}}t}}=\mathbf {r} \times \mathbf {Y} ={\frac {{\rm {d}}(\mathbf {I} \cdot {\boldsymbol {\alpha }})}{{\rm {d}}t}}$$
Impulse	Impulse is the change in momentum: $$\Delta \mathbf {p} =\int \mathbf {F} \,dt$$ For constant force F: $$\Delta \mathbf {p} =\mathbf {F} \Delta t$$	Twirl/angular impulse is the change in angular momentum: $$\Delta \mathbf {L} =\int {\boldsymbol {\tau }}\,dt$$ For constant torque τ: $$\Delta \mathbf {L} ={\boldsymbol {\tau }}\Delta t$$

Precession

The precession angular speed of a spinning top is given by:

$${\boldsymbol {\Omega }}={\frac {wr}{I{\boldsymbol {\omega }}}}$$

where w is the weight of the spinning flywheel.

Energy

The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system:

General work-energy theorem (translation and rotation)

The work done W by an external agent which exerts a force F (at r) and torque τ on an object along a curved path C is:

$$W=\Delta T=\int _{C}\left(\mathbf {F} \cdot \mathrm {d} \mathbf {r} +{\boldsymbol {\tau }}\cdot \mathbf {n} \,{\mathrm {d} \theta }\right)$$

where θ is the angle of rotation about an axis defined by a unit vector n.

Kinetic energy

The change in kinetic energy for an object initially traveling at speed $v_{0}$ and later at speed $v$ is: $$\Delta E_{k}=W={\frac {1}{2}}m(v^{2}-{v_{0}}^{2})$$

Elastic potential energy

For a stretched spring fixed at one end obeying Hooke's law, the elastic potential energy is

$$\Delta E_{p}={\frac {1}{2}}k(r_{2}-r_{1})^{2}$$

where r₂ and r₁ are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.

Euler's equations for rigid body dynamics

Euler also worked out analogous laws of motion to those of Newton, see Euler's laws of motion. These extend the scope of Newton's laws to rigid bodies, but are essentially the same as above. A new equation Euler formulated is:

$$\mathbf {I} \cdot {\boldsymbol {\alpha }}+{\boldsymbol {\omega }}\times \left(\mathbf {I} \cdot {\boldsymbol {\omega }}\right)={\boldsymbol {\tau }}$$

where I is the moment of inertia tensor.

General planar motion

The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane,

$$\mathbf {r} =\mathbf {r} (t)=r{\hat {\mathbf {r} }}$$

the following general results apply to the particle.

Kinematics	Dynamics
Position $$\mathbf {r} =\mathbf {r} \left(r,\theta ,t\right)=r{\hat {\mathbf {r} }}$$
Velocity $$\mathbf {v} ={\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}$$	Momentum $$\mathbf {p} =m\left({\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}\right)$$ Angular momenta $$\mathbf {L} =m\mathbf {r} \times \left({\hat {\mathbf {r} }}{\frac {\mathrm {d} r}{\mathrm {d} t}}+r\omega {\hat {\mathbf {\theta } }}\right)$$
Acceleration $$\mathbf {a} =\left({\frac {\mathrm {d} ^{2}r}{\mathrm {d} t^{2}}}-r\omega ^{2}\right){\hat {\mathbf {r} }}+\left(r\alpha +2\omega {\frac {\mathrm {d} r}{{\rm {d}}t}}\right){\hat {\mathbf {\theta } }}$$	The centripetal force is $$\mathbf {F} _{\bot }=-m\omega ^{2}R{\hat {\mathbf {r} }}=-\omega ^{2}\mathbf {m}$$ where again m is the mass moment, and the Coriolis force is $$\mathbf {F} _{c}=2\omega m{\frac {{\rm {d}}r}{{\rm {d}}t}}{\hat {\mathbf {\theta } }}=2\omega mv{\hat {\mathbf {\theta } }}$$ The Coriolis acceleration and force can also be written: $$\mathbf {F} _{c}=m\mathbf {a} _{c}=-2m{\boldsymbol {\omega \times v}}$$

Central force motion

For a massive body moving in a central potential due to another object, which depends only on the radial separation between the centers of masses of the two objects, the equation of motion is:

$${\frac {d^{2}}{d\theta ^{2}}}\left({\frac {1}{\mathbf {r} }}\right)+{\frac {1}{\mathbf {r} }}=-{\frac {\mu \mathbf {r} ^{2}}{\mathbf {l} ^{2}}}\mathbf {F} (\mathbf {r} )$$

Equations of motion (constant acceleration)

These equations can be used only when acceleration is constant. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).

Linear motion	Angular motion
$\mathbf {v-v_{0}} =\mathbf {a} t$	${\boldsymbol {\omega -\omega _{0}}}={\boldsymbol {\alpha }}t$
$\mathbf {x-x_{0}} ={\tfrac {1}{2}}(\mathbf {v_{0}+v} )t$	${\boldsymbol {\theta -\theta _{0}}}={\tfrac {1}{2}}({\boldsymbol {\omega _{0}+\omega }})t$
$\mathbf {x-x_{0}} =\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}$	${\boldsymbol {\theta -\theta _{0}}}={\boldsymbol {\omega }}_{0}t+{\tfrac {1}{2}}{\boldsymbol {\alpha }}t^{2}$
$\mathbf {x} _{n^{th}}=\mathbf {v} _{0}+\mathbf {a} (n-{\tfrac {1}{2}})$	${\boldsymbol {\theta }}_{n^{th}}={\boldsymbol {\omega }}_{0}+{\boldsymbol {\alpha }}(n-{\tfrac {1}{2}})$
$v^{2}-v_{0}^{2}=2\mathbf {a(x-x_{0})}$	$\omega ^{2}-\omega _{0}^{2}=2{\boldsymbol {\alpha (\theta -\theta _{0})}}$

Galilean frame transforms

For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame traveling at constant velocity - including zero) to another is the Galilean transform.

Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.

Motion of entities	Inertial frames	Accelerating frames
Translation V = Constant relative velocity between two inertial frames F and F'. A = (Variable) relative acceleration between two accelerating frames F and F'.	Relative position $$\mathbf {r} '=\mathbf {r} +\mathbf {V} t$$ Relative velocity $$\mathbf {v} '=\mathbf {v} +\mathbf {V}$$ Equivalent accelerations $$\mathbf {a} '=\mathbf {a}$$	Relative accelerations $$\mathbf {a} '=\mathbf {a} +\mathbf {A}$$ Apparent/fictitious forces $$\mathbf {F} '=\mathbf {F} -\mathbf {F} _{\mathrm {app} }$$
Rotation Ω = Constant relative angular velocity between two frames F and F'. Λ = (Variable) relative angular acceleration between two accelerating frames F and F'.	Relative angular position $$\theta '=\theta +\Omega t$$ Relative velocity $${\boldsymbol {\omega }}'={\boldsymbol {\omega }}+{\boldsymbol {\Omega }}$$ Equivalent accelerations $${\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}$$	Relative accelerations $${\boldsymbol {\alpha }}'={\boldsymbol {\alpha }}+{\boldsymbol {\Lambda }}$$ Apparent/fictitious torques $${\boldsymbol {\tau }}'={\boldsymbol {\tau }}-{\boldsymbol {\tau }}_{\mathrm {app} }$$
	Transformation of any vector T to a rotating frame $${\frac {{\rm {d}}\mathbf {T} '}{{\rm {d}}t}}={\frac {{\rm {d}}\mathbf {T} }{{\rm {d}}t}}-{\boldsymbol {\Omega }}\times \mathbf {T}$$

Mechanical oscillators

SHM, DHM, SHO, and DHO refer to simple harmonic motion, damped harmonic motion, simple harmonic oscillator and damped harmonic oscillator respectively.

Equations of motion

Physical situation	Nomenclature	Translational equations	Angular equations
SHM	x = Transverse displacement θ = Angular displacement A = Transverse amplitude Θ = Angular amplitude	$${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}=-\omega ^{2}x$$Solution: $$x=A\sin \left(\omega t+\phi \right)$$	$${\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}=-\omega ^{2}\theta$$Solution: $$\theta =\Theta \sin \left(\omega t+\phi \right)$$
Unforced DHM	b = damping constant κ = torsion constant	$${\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} x}{\mathrm {d} t}}+\omega ^{2}x=0$$Solution (see below for ω'): $$x=Ae^{-bt/2m}\cos \left(\omega '\right)$$ Resonant frequency: $$\omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {b}{4m}}\right)^{2}}}$$ Damping rate: $$\gamma =b/m$$ Expected lifetime of excitation: $$\tau =1/\gamma$$	$${\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}+b{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+\omega ^{2}\theta =0$$Solution: $$\theta =\Theta e^{-\kappa t/2m}\cos \left(\omega \right)$$ Resonant frequency: $$\omega _{\mathrm {res} }={\sqrt {\omega ^{2}-\left({\frac {\kappa }{4m}}\right)^{2}}}$$ Damping rate: $$\gamma =\kappa /m$$ Expected lifetime of excitation: $$\tau =1/\gamma$$

Angular frequencies

Physical situation	Nomenclature	Equations
Linear undamped unforced SHO	k = spring constant m = mass of oscillating bob	$\omega ={\sqrt {\frac {k}{m}}}$
Linear unforced DHO	k = spring constant b = Damping coefficient	$\omega '={\sqrt {{\frac {k}{m}}-\left({\frac {b}{2m}}\right)^{2}}}$
Low amplitude angular SHO	I = Moment of inertia about oscillating axis κ = torsion constant	$\omega ={\sqrt {\frac {\kappa }{I}}}$
Low amplitude simple pendulum	L = Length of pendulum g = Gravitational acceleration Θ = Angular amplitude	Approximate value $$\omega ={\sqrt {\frac {g}{L}}}$$ Exact value can be shown to be: $$\omega ={\sqrt {\frac {g}{L}}}\left[1+\sum _{k=1}^{\infty }{\frac {\prod _{n=1}^{k}\left(2n-1\right)}{\prod _{n=1}^{m}\left(2n\right)}}\sin ^{2n}\Theta \right]$$

Energy in mechanical oscillations

Physical situation	Nomenclature	Equations
SHM energy	T = kinetic energy U = potential energy E = total energy	Potential energy $$U={\frac {m}{2}}\left(x\right)^{2}={\frac {m\left(\omega A\right)^{2}}{2}}\cos ^{2}(\omega t+\phi )$$ Maximum value at x = A: $$U_{\mathrm {max} }{\frac {m}{2}}\left(\omega A\right)^{2}$$ Kinetic energy $$T={\frac {\omega ^{2}m}{2}}\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}={\frac {m\left(\omega A\right)^{2}}{2}}\sin ^{2}\left(\omega t+\phi \right)$$ Total energy $$E=T+U$$
DHM energy		$E={\frac {m\left(\omega A\right)^{2}}{2}}e^{-bt/m}$

Wave theory

Definitions

General fundamental quantities

A wave can be longitudinal where the oscillations are parallel (or antiparallel) to the propagation direction, or transverse where the oscillations are perpendicular to the propagation direction. These oscillations are characterized by a periodically time-varying displacement in the parallel or perpendicular direction, and so the instantaneous velocity and acceleration are also periodic and time varying in these directions. (the apparent motion of the wave due to the successive oscillations of particles or fields about their equilibrium positions) propagates at the phase and group velocities parallel or antiparallel to the propagation direction, which is common to longitudinal and transverse waves. Below oscillatory displacement, velocity and acceleration refer to the kinematics in the oscillating directions of the wave - transverse or longitudinal (mathematical description is identical), the group and phase velocities are separate.

Quantity (common name/s)	(Common) symbol/s	SI units	Dimension
Number of wave cycles	N	dimensionless	dimensionless
(Oscillatory) displacement	Symbol of any quantity which varies periodically, such as h, x, y (mechanical waves), x, s, η (longitudinal waves) I, V, E, B, H, D (electromagnetism), u, U (luminal waves), ψ, Ψ, Φ (quantum mechanics). Most general purposes use y, ψ, Ψ. For generality here, A is used and can be replaced by any other symbol, since others have specific, common uses. $\mathbf {A} =A\mathbf {\hat {e}} _{\parallel }\,\!$ for longitudinal waves, $\mathbf {A} =A\mathbf {\hat {e}} _{\bot }\,\!$ for transverse waves.	m	[L]
(Oscillatory) displacement amplitude	Any quantity symbol typically subscripted with 0, m or max, or the capitalized letter (if displacement was in lower case). Here for generality A0 is used and can be replaced.	m	[L]
(Oscillatory) velocity amplitude	V, v0, vm. Here v0 is used.	m s−1	[L][T]−1
(Oscillatory) acceleration amplitude	A, a0, am. Here a0 is used.	m s−2	[L][T]−2
Spatial position Position of a point in space, not necessarily a point on the wave profile or any line of propagation	d, r	m	[L]
Wave profile displacement Along propagation direction, distance travelled (path length) by one wave from the source point r0 to any point in space d (for longitudinal or transverse waves)	L, d, r $\mathbf {r} \equiv r\mathbf {\hat {e}} _{\parallel }\equiv \mathbf {d} -\mathbf {r} _{0}\,\!$	m	[L]
Phase angle	δ, ε, φ	rad	dimensionless

General derived quantities

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Wavelength	λ	General definition (allows for FM): $\lambda =\mathrm {d} r/\mathrm {d} N\,\!$ For non-FM waves this reduces to: $\lambda =\Delta r/\Delta N\,\!$	m	[L]
Wavenumber, k-vector, Wave vector	k, σ	Two definitions are in use: $\mathbf {k} =\left(2\pi /\lambda \right)\mathbf {\hat {e}} _{\angle }\,\!$ $\mathbf {k} =\left(1/\lambda \right)\mathbf {\hat {e}} _{\angle }\,\!$	m−1	[L]−1
Frequency	f, ν	General definition (allows for FM): $f=\mathrm {d} N/\mathrm {d} t\,\!$ For non-FM waves this reduces to: $f=\Delta N/\Delta t\,\!$ In practice N is set to 1 cycle and t = T = time period for 1 cycle, to obtain the more useful relation: $f=1/T\,\!$	Hz = s−1	[T]−1
Angular frequency/ pulsatance	ω	$\omega =2\pi f=2\pi /T\,\!$	Hz = s−1	[T]−1
Oscillatory velocity	v, vt, v	Longitudinal waves: $\mathbf {v} =\mathbf {\hat {e}} _{\parallel }\left(\partial A/\partial t\right)\,\!$ Transverse waves: $\mathbf {v} =\mathbf {\hat {e}} _{\bot }\left(\partial A/\partial t\right)\,\!$	m s−1	[L][T]−1
Oscillatory acceleration	a, at	Longitudinal waves: $\mathbf {a} =\mathbf {\hat {e}} _{\parallel }\left(\partial ^{2}A/\partial t^{2}\right)\,\!$ Transverse waves: $\mathbf {a} =\mathbf {\hat {e}} _{\bot }\left(\partial ^{2}A/\partial t^{2}\right)\,\!$	m s−2	[L][T]−2
Path length difference between two waves	L, ΔL, Δx, Δr	$\mathbf {r} =\mathbf {r} _{2}-\mathbf {r} _{1}\,\!$	m	[L]
Phase velocity	vp	General definition: $\mathbf {v} _{\mathrm {p} }=\mathbf {\hat {e}} _{\parallel }\left(\Delta r/\Delta t\right)\,\!$ In practice reduces to the useful form: $\mathbf {v} _{\mathrm {p} }=\lambda f\mathbf {\hat {e}} _{\parallel }=\left(\omega /k\right)\mathbf {\hat {e}} _{\parallel }\,\!$	m s−1	[L][T]−1
(Longitudinal) group velocity	vg	$\mathbf {v} _{\mathrm {g} }=\mathbf {\hat {e}} _{\parallel }\left(\partial \omega /\partial k\right)\,\!$	m s−1	[L][T]−1
Time delay, time lag/lead	Δt	$\Delta t=t_{2}-t_{1}\,\!$	s	[T]
Phase difference	δ, Δε, Δϕ	$\Delta \phi =\phi _{2}-\phi _{1}\,\!$	rad	dimensionless
Phase	No standard symbol	$\mathbf {k} \cdot \mathbf {r} \mp \omega t+\phi =2\pi N\,\!$Physically; upper sign: wave propagation in +r direction lower sign: wave propagation in −r direction Phase angle can lag if: ϕ > 0 or lead if: ϕ < 0.	rad	dimensionless

Relation between space, time, angle analogues used to describe the phase:

${\frac {\Delta r}{\lambda }}={\frac {\Delta t}{T}}={\frac {\phi }{2\pi }}\,\!$

Modulation indices

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
AM index:	h, hAM	$h_{AM}=A/A_{m}\,\!$A = carrier amplitude Am = peak amplitude of a component in the modulating signal	dimensionless	dimensionless
FM index:	hFM	$h_{FM}=\Delta f/f_{m}\,\!$Δf = max. deviation of the instantaneous frequency from the carrier frequency fm = peak frequency of a component in the modulating signal	dimensionless	dimensionless
PM index:	hPM	$h_{PM}=\Delta \phi \,\!$Δϕ = peak phase deviation	dimensionless	dimensionless

Acoustics

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Acoustic impedance	Z	$Z=\rho v\,\!$v = speed of sound, ρ = volume density of medium	kg m−2 s−1	[M] [L]−2 [T]−1
Specific acoustic impedance	z	$z=ZS\,\!$S = surface area	kg s−1	[M] [T]−1
Sound Level	β	$\beta =\left(\mathrm {dB} \right)10\log \left|{\frac {I}{I_{0}}}\right|\,\!$	dimensionless	dimensionless

Equations

In what follows n, m are any integers (Z = set of integers); $n,m\in \mathbf {Z} \,\!$.

Standing waves

Physical situation	Nomenclature	Equations
Harmonic frequencies	fn = nth mode of vibration, nth harmonic, (n-1)th overtone	$f_{n}={\frac {v}{\lambda _{n}}}={\frac {nv}{2L}}=nf_{1}\,\!$

Propagating waves

Sound waves

Physical situation	Nomenclature	Equations
Average wave power	P0 = Sound power due to source	$\langle P\rangle =\mu v\omega ^{2}x_{m}^{2}/2\,\!$
Sound intensity	Ω = Solid angle	$I=P_{0}/(\Omega r^{2})\,\!$$I=P/A=\rho v\omega ^{2}s_{m}^{2}/2\,\!$
Acoustic beat frequency	f1, f2 = frequencies of two waves (nearly equal amplitudes)	$f_{\mathrm {beat} }=\left|f_{2}-f_{1}\right|\,\!$
Doppler effect for mechanical waves	V = speed of sound wave in medium f0 = Source frequency fr = Receiver frequency v0 = Source velocity vr = Receiver velocity	$f_{r}=f_{0}{\frac {V\pm v_{r}}{V\mp v_{0}}}\,\!$upper signs indicate relative approach, lower signs indicate relative recession.
Mach cone angle (Supersonic shockwave, sonic boom)	v = speed of body vs = local speed of sound θ = angle between direction of travel and conic envelope of superimposed wavefronts	$\sin \theta ={\frac {v}{v_{s}}}\,\!$
Acoustic pressure and displacement amplitudes	p0 = pressure amplitude s0 = displacement amplitude v = speed of sound ρ = local density of medium	$p_{0}=\left(v\rho \omega \right)s_{0}\,\!$
Wave functions for sound		Acoustic beats $s=\left[2s_{0}\cos \left(\omega 't\right)\right]\cos \left(\omega t\right)\,\!$ Sound displacement function $s=s_{0}\cos(kr-\omega t)\,\!$ Sound pressure-variation $p=p_{0}\sin(kr-\omega t)\,\!$

Gravitational waves

Gravitational radiation for two orbiting bodies in the low-speed limit.

Physical situation	Nomenclature	Equations
Radiated power	P = Radiated power from system, t = time, r = separation between centres-of-mass m1, m2 = masses of the orbiting bodies	$P={\frac {\mathrm {d} E}{\mathrm {d} t}}=-{\frac {32}{5}}\,{\frac {G^{4}}{c^{5}}}\,{\frac {(m_{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}}$
Orbital radius decay		${\frac {\mathrm {d} r}{\mathrm {d} t}}=-{\frac {64}{5}}{\frac {G^{3}}{c^{5}}}{\frac {(m_{1}m_{2})(m_{1}+m_{2})}{r^{3}}}\$
Orbital lifetime	r0 = initial distance between the orbiting bodies	$t={\frac {5}{256}}{\frac {c^{5}}{G^{3}}}{\frac {r_{0}^{4}}{(m_{1}m_{2})(m_{1}+m_{2})}}\$

Superposition, interference, and diffraction

Physical situation	Nomenclature	Equations
Principle of superposition	N = number of waves	$y_{\mathrm {net} }=\sum _{i=1}^{N}y_{i}\,\!$
Resonance	ωd = driving angular frequency (external agent) ωnat = natural angular frequency (oscillator)	$\omega _{d}=\omega _{\mathrm {nat} }\,\!$
Phase and interference	Δr = path length difference φ = phase difference between any two successive wave cycles	${\frac {\Delta r}{\lambda }}={\frac {\Delta t}{T}}={\frac {\phi }{2\pi }}\,\!$Constructive interference $n={\frac {\lambda }{\Delta x}}\,\!$ Destructive interference $n+{\frac {1}{2}}={\frac {\lambda }{\Delta x}}\,\!$

Wave propagation

A common misconception occurs between phase velocity and group velocity (analogous to centres of mass and gravity). They happen to be equal in non-dispersive media. In dispersive media the phase velocity is not necessarily the same as the group velocity. The phase velocity varies with frequency.

The phase velocity is the rate at which the phase of the wave propagates in space.

The group velocity is the rate at which the wave envelope, i.e. the changes in amplitude, propagates. The wave envelope is the profile of the wave amplitudes; all transverse displacements are bound by the envelope profile.

Intuitively the wave envelope is the "global profile" of the wave, which "contains" changing "local profiles inside the global profile". Each propagates at generally different speeds determined by the important function called the dispersion relation. The use of the explicit form ω(k) is standard, since the phase velocity ω/k and the group velocity dω/dk usually have convenient representations by this function.

Physical situation	Nomenclature	Equations
Idealized non-dispersive media	p = (any type of) Stress or Pressure, ρ = Volume Mass Density, F = Tension Force, μ = Linear Mass Density of medium	$v={\sqrt {\frac {p}{\rho }}}={\sqrt {\frac {F}{\mu }}}\,\!$
Dispersion relation		Implicit form $D\left(\omega ,k\right)=0$ Explicit form $\omega =\omega \left(k\right)$
Amplitude modulation, AM		$A=A\left(t\right)$
Frequency modulation, FM		$f=f\left(t\right)$

General wave functions

Wave equations

Physical situation	Nomenclature	Wave equation	General solution/s
Non-dispersive Wave Equation in 3d	A = amplitude as function of position and time	$\nabla ^{2}A={\frac {1}{v_{\parallel }^{2}}}{\frac {\partial ^{2}A}{\partial t^{2}}}\,\!$	$A\left(\mathbf {r} ,t\right)=A\left(x-v_{\parallel }t\right)\,\!$
Exponentially damped waveform	A0 = Initial amplitude at time t = 0 b = damping parameter		$A=A_{0}e^{-bt}\sin \left(kx-\omega t+\phi \right)\,\!$
Korteweg–de Vries equation	α = constant	${\frac {\partial y}{\partial t}}+\alpha y{\frac {\partial y}{\partial x}}+{\frac {\partial ^{3}y}{\partial x^{3}}}=0\,\!$	$A(x,t)={\frac {3v_{\parallel }}{\alpha }}\mathrm {sech} ^{2}\left[{\frac {\sqrt {v_{\parallel }}}{2}}\left(x-v_{\parallel }t\right)\right]\,\!$

Sinusoidal solutions to the 3d wave equation

N different sinusoidal waves

Complex amplitude of wave n

$A_{n}=\left|A_{n}\right|e^{i\left(\mathbf {k} _{\mathrm {n} }\cdot \mathbf {r} -\omega _{n}t+\phi _{n}\right)}\,\!$

Resultant complex amplitude of all N waves

$A=\sum _{n=1}^{N}A_{n}\,\!$

Modulus of amplitude

$A={\sqrt {AA^{*}}}={\sqrt {\sum _{n=1}^{N}\sum _{m=1}^{N}\left|A_{n}\right|\left|A_{m}\right|\cos \left[\left(\mathbf {k} _{n}-\mathbf {k} _{m}\right)\cdot \mathbf {r} +\left(\omega _{n}-\omega _{m}\right)t+\left(\phi _{n}-\phi _{m}\right)\right]}}\,\!$

The transverse displacements are simply the real parts of the complex amplitudes.

1-dimensional corollaries for two sinusoidal waves

The following may be deduced by applying the principle of superposition to two sinusoidal waves, using trigonometric identities. The angle addition and sum-to-product trigonometric formulae are useful; in more advanced work complex numbers and fourier series and transforms are used.

Wavefunction	Nomenclature	Superposition	Resultant
Standing wave		${\begin{aligned}y_{1}+y_{2}&=A\sin \left(kx-\omega t\right)\\&+A\sin \left(kx+\omega t\right)\end{aligned}}\,\!$	$y=2A\sin \left(kx\right)\cos \left(\omega t\right)\,\!$
Beats	$\langle \omega \rangle ={\frac {\omega _{1}+\omega _{2}}{2}}\,\!$ $\langle k\rangle ={\frac {k_{1}+k_{2}}{2}}\,\!$ $\Delta \omega =\omega _{1}-\omega _{2}\,\!$ $\Delta k=k_{1}-k_{2}\,\!$	${\begin{aligned}y_{1}+y_{2}&=A\sin \left(k_{1}x-\omega _{1}t\right)\\&+A\sin \left(k_{2}x+\omega _{2}t\right)\end{aligned}}\,\!$	$y=2A\sin \left(\langle k\rangle x-\langle \omega \rangle t\right)\cos \left({\frac {\Delta k}{2}}x-{\frac {\Delta \omega }{2}}t\right)\,\!$
Coherent interference		${\begin{aligned}y_{1}+y_{2}&=2A\sin \left(kx-\omega t\right)\\&+A\sin \left(kx+\omega t+\phi \right)\end{aligned}}\,\!$	$y=2A\cos \left({\frac {\phi }{2}}\right)\sin \left(kx-\omega t+{\frac {\phi }{2}}\right)\,\!$

Electromagnetism

Definitions

Lorentz force on a charged particle (of charge q) in motion (velocity v), used as the definition of the E field and B field.

Here subscripts e and m are used to differ between electric and magnetic charges. The definitions for monopoles are of theoretical interest, although real magnetic dipoles can be described using pole strengths. There are two possible units for monopole strength, Wb (Weber) and A m (Ampere metre). Dimensional analysis shows that magnetic charges relate by q_m(Wb) = μ₀ q_m(Am).

Initial quantities

Quantity (common name/s)	(Common) symbol/s	SI units	Dimension
Electric charge	qe, q, Q	C = As	[I][T]
Monopole strength, magnetic charge	qm, g, p	Wb or Am	[L]2[M][T]−2 [I]−1 (Wb) [I][L] (Am)

Electric quantities

Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal n̂, d is the dipole moment between two point charges, the volume density of these is the polarization density P. Position vector r is a point to calculate the electric field; r′ is a point in the charged object.

Contrary to the strong analogy between (classical) gravitation and electrostatics, there are no "centre of charge" or "centre of electrostatic attraction" analogues.

Electric transport

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Linear, surface, volumetric charge density	λe for Linear, σe for surface, ρe for volume.	$q_{e}=\int \lambda _{e}\mathrm {d} \ell$$q_{e}=\iint \sigma _{e}\mathrm {d} S$ $q_{e}=\iiint \rho _{e}\mathrm {d} V$	C m−n, n = 1, 2, 3	[I][T][L]−n
Capacitance	C	$C=\mathrm {d} q/\mathrm {d} V\,\!$V = voltage, not volume.	F = C V−1	[I]2[T]4[L]−2[M]−1
Electric current	I	$I=\mathrm {d} q/\mathrm {d} t\,\!$	A	[I]
Electric current density	J	$I=\mathbf {J} \cdot \mathrm {d} \mathbf {S}$	A m−2	[I][L]−2
Displacement current density	Jd	$\mathbf {J} _{\mathrm {d} }=\epsilon _{0}\left(\partial \mathbf {E} /\partial t\right)=\partial \mathbf {D} /\partial t\,\!$	A m−2	[I][L]−2
Convection current density	Jc	$\mathbf {J} _{\mathrm {c} }=\rho \mathbf {v} \,\!$	A m−2	[I][L]−2

Electric fields

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Electric field, field strength, flux density, potential gradient	E	$\mathbf {E} =\mathbf {F} /q\,\!$	N C−1 = V m−1	[M][L][T]−3[I]−1
Electric flux	ΦE	$\Phi _{E}=\int _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} \,\!$	N m2 C−1	[M][L]3[T]−3[I]−1
Absolute permittivity;	ε	$\epsilon =\epsilon _{r}\epsilon _{0}\,\!$	F m−1	[I]2 [T]4 [M]−1 [L]−3
Electric dipole moment	p	$\mathbf {p} =q\mathbf {a} \,\!$a = charge separation directed from -ve to +ve charge	C m	[I][T][L]
Electric Polarization, polarization density	P	$\mathbf {P} =\mathrm {d} \langle \mathbf {p} \rangle /\mathrm {d} V\,\!$	C m−2	[I][T][L]−2
Electric displacement field, flux density	D	$\mathbf {D} =\epsilon \mathbf {E} =\epsilon _{0}\mathbf {E} +\mathbf {P} \,$	C m−2	[I][T][L]−2
Electric displacement flux	ΦD	$\Phi _{D}=\int _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} \,\!$	C	[I][T]
Absolute electric potential, EM scalar potential relative to point $r_{0}\,\!$Theoretical: $r_{0}=\infty \,\!$ Practical: $r_{0}=R_{\mathrm {earth} }\,\!$ (Earth's radius)	φ ,V	$V=-{\frac {W_{\infty r}}{q}}=-{\frac {1}{q}}\int _{\infty }^{r}\mathbf {F} \cdot \mathrm {d} \mathbf {r} =-\int _{r_{1}}^{r_{2}}\mathbf {E} \cdot \mathrm {d} \mathbf {r} \,\!$	V = J C−1	[M] [L]2 [T]−3 [I]−1
Voltage, Electric potential difference	Δφ,ΔV	$\Delta V=-{\frac {\Delta W}{q}}=-{\frac {1}{q}}\int _{r_{1}}^{r_{2}}\mathbf {F} \cdot \mathrm {d} \mathbf {r} =-\int _{r_{1}}^{r_{2}}\mathbf {E} \cdot \mathrm {d} \mathbf {r} \,\!$	V = J C−1	[M] [L]2 [T]−3 [I]−1

Magnetic quantities

Magnetic transport

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Linear, surface, volumetric pole density	λm for Linear, σm for surface, ρm for volume.	$q_{m}=\int \lambda _{m}\mathrm {d} \ell$$q_{m}=\iint \sigma _{m}\mathrm {d} S$ $q_{m}=\iiint \rho _{m}\mathrm {d} V$	Wb m−n A m(−n + 1), n = 1, 2, 3	[L]2[M][T]−2 [I]−1 (Wb) [I][L] (Am)
Monopole current	Im	$I_{m}=\mathrm {d} q_{m}/\mathrm {d} t\,\!$	Wb s−1 A m s−1	[L]2[M][T]−3 [I]−1 (Wb) [I][L][T]−1 (Am)
Monopole current density	Jm	$I=\iint \mathbf {J} _{\mathrm {m} }\cdot \mathrm {d} \mathbf {A}$	Wb s−1 m−2 A m−1 s−1	[M][T]−3 [I]−1 (Wb) [I][L]−1[T]−1 (Am)

Magnetic fields

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Magnetic field, field strength, flux density, induction field	B	$\mathbf {F} =q_{e}\left(\mathbf {v} \times \mathbf {B} \right)\,\!$	T = N A−1 m−1 = Wb m−2	[M][T]−2[I]−1
Magnetic potential, EM vector potential	A	$\mathbf {B} =\nabla \times \mathbf {A}$	T m = N A−1 = Wb m3	[M][L][T]−2[I]−1
Magnetic flux	ΦB	$\Phi _{B}=\int _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} \,\!$	Wb = T m2	[L]2[M][T]−2[I]−1
Magnetic permeability	$\mu \,\!$	$\mu \ =\mu _{r}\,\mu _{0}\,\!$	V·s·A−1·m−1 = N·A−2 = T·m·A−1 = Wb·A−1·m−1	[M][L][T]−2[I]−2
Magnetic moment, magnetic dipole moment	m, μB, Π	Two definitions are possible: using pole strengths, $\mathbf {m} =q_{m}\mathbf {a} \,\!$ using currents: $\mathbf {m} =NIA\mathbf {\hat {n}} \,\!$ a = pole separation N is the number of turns of conductor	A m2	[I][L]2
Magnetization	M	$\mathbf {M} =\mathrm {d} \langle \mathbf {m} \rangle /\mathrm {d} V\,\!$	A m−1	[I] [L]−1
Magnetic field intensity, (AKA field strength)	H	Two definitions are possible: most common: $\mathbf {B} =\mu \mathbf {H} =\mu _{0}\left(\mathbf {H} +\mathbf {M} \right)\,$ using pole strengths, $\mathbf {H} =\mathbf {F} /q_{m}\,$	A m−1	[I] [L]−1
Intensity of magnetization, magnetic polarization	I, J	$\mathbf {I} =\mu _{0}\mathbf {M} \,\!$	T = N A−1 m−1 = Wb m−2	[M][T]−2[I]−1
Self Inductance	L	Two equivalent definitions are possible: $L=N\left(\mathrm {d} \Phi /\mathrm {d} I\right)\,\!$ $L\left(\mathrm {d} I/\mathrm {d} t\right)=-NV\,\!$	H = Wb A−1	[L]2 [M] [T]−2 [I]−2
Mutual inductance	M	Again two equivalent definitions are possible: $M_{1}=N\left(\mathrm {d} \Phi _{2}/\mathrm {d} I_{1}\right)\,\!$ $M\left(\mathrm {d} I_{2}/\mathrm {d} t\right)=-NV_{1}\,\!$ 1,2 subscripts refer to two conductors/inductors mutually inducing voltage/ linking magnetic flux through each other. They can be interchanged for the required conductor/inductor; $M_{2}=N\left(\mathrm {d} \Phi _{1}/\mathrm {d} I_{2}\right)\,\!$ $M\left(\mathrm {d} I_{1}/\mathrm {d} t\right)=-NV_{2}\,\!$	H = Wb A−1	[L]2 [M] [T]−2 [I]−2
Gyromagnetic ratio (for charged particles in a magnetic field)	γ	$\omega =\gamma B\,\!$	Hz T−1	[M]−1[T][I]

Electric circuits

DC circuits, general definitions

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Terminal Voltage for Power Supply	Vter		V = J C−1	[M] [L]2 [T]−3 [I]−1
Load Voltage for Circuit	Vload		V = J C−1	[M] [L]2 [T]−3 [I]−1
Internal resistance of power supply	Rint	$R_{\mathrm {int} }=V_{\mathrm {ter} }/I\,\!$	Ω = V A−1 = J s C−2	[M][L]2 [T]−3 [I]−2
Load resistance of circuit	Rext	$R_{\mathrm {ext} }=V_{\mathrm {load} }/I\,\!$	Ω = V A−1 = J s C−2	[M][L]2 [T]−3 [I]−2
Electromotive force (emf), voltage across entire circuit including power supply, external components and conductors	E	${\mathcal {E}}=V_{\mathrm {ter} }+V_{\mathrm {load} }\,\!$	V = J C−1	[M] [L]2 [T]−3 [I]−1

AC circuits

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Resistive load voltage	VR	$V_{R}=I_{R}R\,\!$	V = J C−1	[M] [L]2 [T]−3 [I]−1
Capacitive load voltage	VC	$V_{C}=I_{C}X_{C}\,\!$	V = J C−1	[M] [L]2 [T]−3 [I]−1
Inductive load voltage	VL	$V_{L}=I_{L}X_{L}\,\!$	V = J C−1	[M] [L]2 [T]−3 [I]−1
Capacitive reactance	XC	$X_{C}={\frac {1}{\omega _{\mathrm {d} }C}}\,\!$	Ω−1 m−1	[I]2 [T]3 [M]−2 [L]−2
Inductive reactance	XL	$X_{L}=\omega _{d}L\,\!$	Ω−1 m−1	[I]2 [T]3 [M]−2 [L]−2
AC electrical impedance	Z	$V=IZ\,\!$$Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}\,\!$	Ω−1 m−1	[I]2 [T]3 [M]−2 [L]−2
Phase constant	δ, φ	$\tan \phi ={\frac {X_{L}-X_{C}}{R}}\,\!$	dimensionless	dimensionless
AC peak current	I0	$I_{0}=I_{\mathrm {rms} }{\sqrt {2}}\,\!$	A	[I]
AC root mean square current	Irms	$I_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[I\left(t\right)\right]^{2}\mathrm {d} t}}\,\!$	A	[I]
AC peak voltage	V0	$V_{0}=V_{\mathrm {rms} }{\sqrt {2}}\,\!$	V = J C−1	[M] [L]2 [T]−3 [I]−1
AC root mean square voltage	Vrms	$V_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}\left[V\left(t\right)\right]^{2}\mathrm {d} t}}\,\!$	V = J C−1	[M] [L]2 [T]−3 [I]−1
AC emf, root mean square	${\mathcal {E}}_{\mathrm {rms} },{\sqrt {\langle {\mathcal {E}}\rangle }}\,\!$	${\mathcal {E}}_{\mathrm {rms} }={\mathcal {E}}_{\mathrm {m} }/{\sqrt {2}}\,\!$	V = J C−1	[M] [L]2 [T]−3 [I]−1
AC average power	$\langle P\rangle \,\!$	$\langle P\rangle ={\mathcal {E}}I_{\mathrm {rms} }\cos \phi \,\!$	W = J s−1	[M] [L]2 [T]−3
Capacitive time constant	τC	$\tau _{C}=RC\,\!$	s	[T]
Inductive time constant	τL	$\tau _{L}=L/R\,\!$	s	[T]

Magnetic circuits

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Magnetomotive force, mmf	F, ${\mathcal {F}},{\mathcal {M}}$	${\mathcal {M}}=NI$N = number of turns of conductor	A	[I]

Electromagnetism

Electric fields

General Classical Equations

Physical situation	Equations
Electric potential gradient and field	$\mathbf {E} =-\nabla V$$\Delta V=-\int _{r_{1}}^{r_{2}}\mathbf {E} \cdot d\mathbf {r} \,\!$
Point charge	$\mathbf {E} ={\frac {q}{4\pi \epsilon _{0}\left|\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} \,\!$
At a point in a local array of point charges	$\mathbf {E} =\sum \mathbf {E} _{i}={\frac {1}{4\pi \epsilon _{0}}}\sum _{i}{\frac {q_{i}}{\left|\mathbf {r} _{i}-\mathbf {r} \right|^{2}}}\mathbf {\hat {r}} _{i}\,\!$
At a point due to a continuum of charge	$\mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}\int _{V}{\frac {\mathbf {r} \rho \mathrm {d} V}{\left|\mathbf {r} \right|^{3}}}\,\!$
Electrostatic torque and potential energy due to non-uniform fields and dipole moments	${\boldsymbol {\tau }}=\int _{V}\mathrm {d} \mathbf {p} \times \mathbf {E}$$U=-\int _{V}\mathrm {d} \mathbf {p} \cdot \mathbf {E}$

Magnetic fields and moments

General classical equations

Physical situation	Equations
Magnetic potential, EM vector potential	$\mathbf {B} =\nabla \times \mathbf {A}$
Due to a magnetic moment	$\mathbf {A} ={\frac {\mu _{0}}{4\pi }}{\frac {\mathbf {m} \times \mathbf {r} }{\left|\mathbf {r} \right|^{3}}}$$\mathbf {B} ({\mathbf {r} })=\nabla \times {\mathbf {A} }={\frac {\mu _{0}}{4\pi }}\left({\frac {3\mathbf {r} (\mathbf {m} \cdot \mathbf {r} )}{\left|\mathbf {r} \right|^{5}}}-{\frac {\mathbf {m} }{\left|\mathbf {r} \right|^{3}}}\right)$
Magnetic moment due to a current distribution	$\mathbf {m} ={\frac {1}{2}}\int _{V}\mathbf {r} \times \mathbf {J} \mathrm {d} V$
Magnetostatic torque and potential energy due to non-uniform fields and dipole moments	${\boldsymbol {\tau }}=\int _{V}\mathrm {d} \mathbf {m} \times \mathbf {B}$$U=-\int _{V}\mathrm {d} \mathbf {m} \cdot \mathbf {B}$

Electric circuits and electronics

Below N = number of conductors or circuit components. Subscript net refers to the equivalent and resultant property value.

Physical situation	Nomenclature	Series	Parallel
Resistors and conductors	Ri = resistance of resistor or conductor i Gi = conductance of resistor or conductor i	$R_{\mathrm {net} }=\sum _{i=1}^{N}R_{i}\,\!$${1 \over G_{\mathrm {net} }}=\sum _{i=1}^{N}{1 \over G_{i}}\,\!$	${1 \over R_{\mathrm {net} }}=\sum _{i=1}^{N}{1 \over R_{i}}\,\!$$G_{\mathrm {net} }=\sum _{i=1}^{N}G_{i}\,\!$
Charge, capacitors, currents	Ci = capacitance of capacitor i qi = charge of charge carrier i	$q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!$${1 \over C_{\mathrm {net} }}=\sum _{i=1}^{N}{1 \over C_{i}}\,\!$ $I_{\mathrm {net} }=I_{i}\,\!$	$q_{\mathrm {net} }=\sum _{i=1}^{N}q_{i}\,\!$$C_{\mathrm {net} }=\sum _{i=1}^{N}C_{i}\,\!$ $I_{\mathrm {net} }=\sum _{i=1}^{N}I_{i}\,\!$
Inductors	Li = self-inductance of inductor i Lij = self-inductance element ij of L matrix Mij = mutual inductance between inductors i and j	$L_{\mathrm {net} }=\sum _{i=1}^{N}L_{i}\,\!$	${1 \over L_{\mathrm {net} }}=\sum _{i=1}^{N}{1 \over L_{i}}\,\!$$V_{i}=\sum _{j=1}^{N}L_{ij}{\frac {\mathrm {d} I_{j}}{\mathrm {d} t}}\,\!$

Series circuit equations

Circuit	DC Circuit equations	AC Circuit equations
RC circuits	Circuit equation $R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\,\!$ Capacitor charge $q=C{\mathcal {E}}\left(1-e^{-t/RC}\right)\,\!$ Capacitor discharge $q=C{\mathcal {E}}e^{-t/RC}\,\!$
RL circuits	Circuit equation $L{\frac {\mathrm {d} I}{\mathrm {d} t}}+RI={\mathcal {E}}\,\!$ Inductor current rise $I={\frac {\mathcal {E}}{R}}\left(1-e^{-Rt/L}\right)\,\!$ Inductor current fall $I={\frac {\mathcal {E}}{R}}e^{-t/\tau _{L}}=I_{0}e^{-Rt/L}\,\!$
LC circuits	Circuit equation $L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+q/C={\mathcal {E}}\,\!$	Circuit equation $L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+q/C={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!$ Circuit resonant frequency $\omega _{\mathrm {res} }=1/{\sqrt {LC}}\,\!$ Circuit charge $q=q_{0}\cos(\omega t+\phi )\,\!$ Circuit current $I=-\omega q_{0}\sin(\omega t+\phi )\,\!$ Circuit electrical potential energy $U_{E}=q^{2}/2C=Q^{2}\cos ^{2}(\omega t+\phi )/2C\,\!$ Circuit magnetic potential energy $U_{B}=Q^{2}\sin ^{2}(\omega t+\phi )/2C\,\!$
RLC Circuits	Circuit equation $L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\,\!$	Circuit equation $L{\frac {\mathrm {d} ^{2}q}{\mathrm {d} t^{2}}}+R{\frac {\mathrm {d} q}{\mathrm {d} t}}+{\frac {q}{C}}={\mathcal {E}}\sin \left(\omega _{0}t+\phi \right)\,\!$ Circuit charge $q=q_{0}eT^{-Rt/2L}\cos(\omega 't+\phi )\,\!$