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Lorentz Factor
γ=11−v2c2\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}
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Heisenberg Uncertainty Principle
ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}
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Physics

Newtonian Mechanics

Newton's Second Law
F=maF = ma
Kinetic Energy
KE=12mv2KE = \frac{1}{2}mv^2
Work
W=Fdcos⁡θW = Fd\cos\theta
Newton's Third Law
F12=−F21F_{12} = -F_{21}
Linear Momentum
p=mvp = mv
Impulse
J=Δp=FΔtJ = \Delta p = F\Delta t
Centripetal Force
Fc=mv2rF_c = \frac{mv^2}{r}
Power
P=FvP = Fv

Electricity & Magnetism

Coulomb's Constant
k=14πε0≈9.0×109 N⋅m2/C2k = \frac{1}{4\pi\varepsilon_0} \approx 9.0\times10^9\,N\cdot m^2/C^2
Electric Field of a Point Charge
E(r)=kqr2r^E(\mathbf{r}) = k \frac{q}{r^2} \hat{r}
Linear Charge Density
λ=dqdl=QL\lambda = \frac{dq}{dl} = \frac{Q}{L}
Surface Charge Density
σ=dqdA=QA\sigma = \frac{dq}{dA} = \frac{Q}{A}
Volume Charge Density
ρ=dqdV=QV\rho = \frac{dq}{dV} = \frac{Q}{V}
Electric Flux (Gauss's Law)
ΦE=∮E⋅dA=Qencε0\Phi_E = \oint \mathbf{E}\cdot d\mathbf{A} = \frac{Q_{enc}}{\varepsilon_0}
Capacitance of a Parallel Plate Capacitor
C=ε0Ad=QVC = \frac{\varepsilon_0 A}{d} = \frac{Q}{V}
Electric Field Between Infinite Sheets
E=σε0E = \frac{\sigma}{\varepsilon_0}
Electric Potential Outside a Spherical Conductor
V(r)=kqrV(r) = k \frac{q}{r}
Electric Potential on a Conductor's Surface
V(P)=kqRV(P) = k \frac{q}{R}
Ohm's Law
V=IRV = IR
Magnetic Flux
ΦB=∫SB⋅dA\Phi_B = \int_S \mathbf{B} \cdot d\mathbf{A}
Equivalent Resistance (Series)
Req=R1+R2+⋯+RnR_{eq} = R_1 + R_2 + \dots + R_n
Kirchhoff’s Loop Rule
∑V=0\sum V = 0
Equivalent Resistance (Parallel)
Req=(1R1+1R2+⋯+1Rn)−1R_{eq} = \left( \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n} \right)^{-1}
Magnetic Torque on a Coil
τ=NIABsin⁡(θ)\tau = N I A B \sin(\theta)
Magnetic Force on a Wire
F=Il×B\mathbf{F} = I \mathbf{l} \times \mathbf{B}
RC Circuit Discharge
Q=Q0e−t/(RC)Q = Q_0 e^{-t/(RC)}
Current Density
J=IAJ = \frac{I}{A}
Magnetic Field of a Long Straight Conductor
B=μ02πIrB = \frac{\mu_0}{2\pi} \frac{I}{r}
Electromagnetic Wave Relation
νλ=c\nu \lambda = c
Speed of Light
c=3×108 m/sc = 3 \times 10^8\,m/s
Ampère's Law with Maxwell's Correction
∮CB⋅dl=μ0Ienc+μ0ε0dΦEdt\oint_C \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc} + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}
Self-Inductance
L=ΦBIL = \frac{\Phi_B}{I}
Power Dissipated in a Resistor
Ploss=I2RP_{loss} = I^2 R
Electrical Power
P=VIP = V I
Phase Angle in AC Circuits
ϕ=arctan⁡(XL−XCR)\phi = \arctan\left(\frac{X_L - X_C}{R}\right)
Inductive Reactance
XL=ωLX_L = \omega L
Capacitive Reactance
XC=1ωCX_C = \frac{1}{\omega C}
Impedance of an AC Circuit
Z=R2+(XL−XC)2Z = \sqrt{R^2 + (X_L - X_C)^2}
Average AC Power
Pav=12VIcos⁡ϕP_{av} = \frac{1}{2} V I \cos\phi
Coulomb's Law
F=kq1q2r2F = k \frac{q_1 q_2}{r^2}
Lorentz Force
F=q(E+v×B)F = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})
Biot–Savart Law
dB=μ04πIdl×r^r2d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \hat{\mathbf{r}}}{r^2}
Electric Potential
V=−∫CE⋅dlV = -\int_C \mathbf{E} \cdot d\mathbf{l}
Capacitance
C=QVC = \frac{Q}{V}
Energy Stored in a Capacitor
U=12CV2U = \frac{1}{2} C V^2
Inductance
L=NΦBIL = \frac{N\Phi_B}{I}
Energy Stored in an Inductor
U=12LI2U = \frac{1}{2} L I^2
EMF (Battery with Internal Resistance)
ε=V+Ir\varepsilon = V + I r
Faraday's Law of Induction
ε=−dΦdt\varepsilon = -\frac{d\Phi}{dt}
Faraday's Law for a Coil
ε=−NdΦdt\varepsilon = -N\frac{d\Phi}{dt}

Waves & Fluid Mechanics

Maximum Power (Harmonic Wave)
Pmax=μFω2A2P_{max} = \sqrt{\mu F \omega^2 A^2}
Average Power (Harmonic Wave)
Pav=12μFω2A2P_{av} = \frac{1}{2}\sqrt{\mu F \omega^2 A^2}
Intensity of a Spherical Wave
I1=P4πr2I_1 = \frac{P}{4\pi r^2}
Young's Modulus
Y=F⊥/AΔl/l0=F⊥l0AΔlY = \frac{F_{\perp}/A}{\Delta l/l_0} = \frac{F_{\perp}l_0}{A\Delta l}
Bulk Modulus
B=−ΔpΔV/V0B = -\frac{\Delta p}{\Delta V/V_0}
Shear Modulus
S=F∥/Ax/h=F∥hAxS = \frac{F_{\parallel}/A}{x/h} = \frac{F_{\parallel} h}{A x}
Strain Tensor (Small Deformations)
εij=12(∂sj∂xi+∂si∂xj)\varepsilon_{ij} = \frac{1}{2}\left(\frac{\partial s_j}{\partial x_i} + \frac{\partial s_i}{\partial x_j}\right)
Strain in Isotropic Materials
εxx=1Y(σxx−ν(σyy+σzz))\varepsilon_{xx} = \frac{1}{Y}\left(\sigma_{xx} - \nu(\sigma_{yy} + \sigma_{zz})\right)
Energy Density in a Mechanical Wave
ε=μv2(∂ξ∂x)2\varepsilon = \mu v^2 \left(\frac{\partial \xi}{\partial x}\right)^2
Reflection Coefficient
yr0=Z1−Z2Z1+Z2y0y_{r0} = \frac{Z_1 - Z_2}{Z_1 + Z_2} y_0
Transmission Coefficient
yt0=2Z1Z1+Z2y0y_{t0} = \frac{2Z_1}{Z_1 + Z_2} y_0
Wave Impedance (String)
Z=μSZ = \sqrt{\mu S}
Wave Impedance (Mass-Spring System)
Z=KμZ = \sqrt{K\mu}
Superposition of Two Waves (Beat Phenomenon)
ξ=2ξ0cos⁡(Δk x−Δω t)sin⁡(kx−ωt),k=k1+k22,  ω=ω1+ω22,  Δk=k1−k22,  Δω=ω1−ω22\xi = 2\xi_0 \cos\left(\Delta k\,x - \Delta \omega\,t\right)\sin\left(kx - \omega t\right), \quad k = \frac{k_1+k_2}{2}, \; \omega = \frac{\omega_1+\omega_2}{2}, \; \Delta k = \frac{k_1-k_2}{2}, \; \Delta \omega = \frac{\omega_1-\omega_2}{2}
Fourier Series Expansion
f(x)=a02+∑n=1∞[ancos⁡(2πnxP)+bnsin⁡(2πnxP)]f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n \cos\left(\frac{2\pi n x}{P}\right) + b_n \sin\left(\frac{2\pi n x}{P}\right)\right]
Fourier Cosine Coefficients
an=2P∫Pf(x)cos⁡(2πnxP)dxa_n = \frac{2}{P} \int_P f(x) \cos\left(\frac{2\pi n x}{P}\right) dx
Fourier Sine Coefficients
bn=2P∫Pf(x)sin⁡(2πnxP)dxb_n = \frac{2}{P} \int_P f(x) \sin\left(\frac{2\pi n x}{P}\right) dx
Surface Energy
E=αΔAE = \alpha \Delta A
Laplace Pressure
Δp=ΔFΔA=α(1R1+1R2)\Delta p = \frac{\Delta F}{\Delta A} = \alpha\left(\frac{1}{R_1} + \frac{1}{R_2}\right)
Hydrostatic Equilibrium
−∇p+ρg=0-\nabla p + \rho g = 0
Continuity Equation
∂ρ∂t+∇⋅(ρv)=0\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho v) = 0
Volumetric Flow Rate
Φv=∫v⋅dA\Phi_v = \int v \cdot dA
Material Derivative of Velocity
Dtv=(∂∂t+v⋅∇)vD_t v = \left(\frac{\partial}{\partial t} + v \cdot \nabla\right)v
Vorticity
ω=∇×v\omega = \nabla\times v
Vorticity Transport Equation
Dtω=(ω⋅∇)v+ν∇2ωD_t \omega = (\omega \cdot \nabla)v + \nu\nabla^2 \omega
Kinematic Viscosity
ν=μρ\nu = \frac{\mu}{\rho}
Reynolds Number
Re=ρvLμRe = \frac{\rho v L}{\mu}
Velocity from Stream Function
v=(∂ψ∂y,−∂ψ∂x)v = \left(\frac{\partial \psi}{\partial y}, -\frac{\partial \psi}{\partial x}\right)
Bernoulli's Equation
p+ρgz+12ρv2=kp + \rho g z + \frac{1}{2}\rho v^2 = k
Transport Theorem
dBdt=ddt∫V(t)ρβ dV+∫∂V(t)ρβ (v⋅n^) dA\frac{dB}{dt} = \frac{d}{dt}\int_{V(t)} \rho\beta\,dV + \int_{\partial V(t)} \rho\beta\,(v \cdot \hat{n})\,dA
Harmonic Plane Wave (1D)
ξ(x,t)=ξ0sin⁡(kx−ωt+φ)\xi(x,t) = \xi_0 \sin\left(kx - \omega t + \varphi\right)
Harmonic Plane Wave (3D)
ξ(r,t)=ξ0sin⁡(k⋅r−ωt+φ)\xi(\mathbf{r},t) = \xi_0 \sin\left(\mathbf{k}\cdot\mathbf{r} - \omega t + \varphi\right)
Wave Equation (1D)
∂2ξ(x,t)∂x2=1v2∂2ξ(x,t)∂t2\frac{\partial^2 \xi(x,t)}{\partial x^2} = \frac{1}{v^2}\frac{\partial^2 \xi(x,t)}{\partial t^2}
Wave Equation (3D)
∇2ξ(r,t)=1v2∂2ξ(r,t)∂t2\nabla^2 \xi(\mathbf{r},t) = \frac{1}{v^2}\frac{\partial^2 \xi(\mathbf{r},t)}{\partial t^2}
Phase Velocity
v=ωkv = \frac{\omega}{k}
Group Velocity
vg=dωdkv_{g} = \frac{d\omega}{dk}
Non-dispersive Wave Speed in Elastic Media
v=Eρv = \sqrt{\frac{E}{\rho}}
Mechanical Stress in Elastic Media
stress=E×strain\text{stress} = E \times \text{strain}
Transverse Wave on a String
v=Sμv = \sqrt{\frac{S}{\mu}}
Longitudinal Wave in Fluids
v=Bρv = \sqrt{\frac{B}{\rho}}
Longitudinal Wave in Solids
v=Yρv = \sqrt{\frac{Y}{\rho}}
Average Value over Wavelength
A=1λ∫0λA(x,t) dxA = \frac{1}{\lambda}\int_0^{\lambda}A(x,t)\,dx
Average Value over Period
⟨A⟩=1T∫0TA(x,t) dt\langle A \rangle = \frac{1}{T}\int_0^T A(x,t)\,dt
Average Energy per Unit Length (String)
ε=12μω2ξ02\varepsilon = \frac{1}{2}\mu\omega^2\xi_0^2
Average Energy per Unit Volume (Plane Wave)
ε=12ρω2ξ02\varepsilon = \frac{1}{2}\rho\omega^2\xi_0^2
Average Power (String Wave)
P=v ε=12vμω2ξ02P = v\,\varepsilon = \frac{1}{2}v\mu\omega^2\xi_0^2
Intensity of a Plane Wave
I=v ε=12vρω2ξ02I = v\,\varepsilon = \frac{1}{2}v\rho\omega^2\xi_0^2
Average Momentum Density
π=εv\pi = \frac{\varepsilon}{v}
Ideal Gas Law
pV=NkBTpV = Nk_BT
Heat Capacity at Constant Pressure
Cp=(dQdT)pC_p = \left(\frac{dQ}{dT}\right)_p
Heat Capacity at Constant Volume
CV=(dQdT)VC_V = \left(\frac{dQ}{dT}\right)_V
Adiabatic Process (Ideal Gas)
pVγ=constantpV^\gamma = \text{constant}
Adiabatic Constant
γ=CpCV\gamma = \frac{C_p}{C_V}
Bulk Modulus (Ideal Gas, Adiabatic)
B=γpB = \gamma p
Speed of Sound in Gas
v=γpρ=γkBTmv = \sqrt{\frac{\gamma p}{\rho}} = \sqrt{\frac{\gamma k_BT}{m}}
Sound Pressure
Δp=−B∂ξ∂x\Delta p = -B \frac{\partial \xi}{\partial x}
Sound Level
β=10log⁡(II0)\beta = 10\log\left(\frac{I}{I_0}\right)
Doppler Effect (Sound Waves)
νO=(1−vOv1−vSv)νS\nu_O = \left(\frac{1-\frac{v_O}{v}}{1-\frac{v_S}{v}}\right)\nu_S
Shock Waves Relation
sin⁡α=vvS\sin \alpha = \frac{v}{v_S}
Maxwell's Equations (Integral Form)
∮E⋅dA=qε0,∮B⋅dA=0,∮E⋅dl=−ddt∫B⋅dA,∮B⋅dl=μ0I+μ0ε0ddt∫E⋅dA\oint \mathbf{E}\cdot d\mathbf{A} = \frac{q}{\varepsilon_0}, \quad \oint \mathbf{B}\cdot d\mathbf{A} = 0, \quad \oint \mathbf{E}\cdot d\mathbf{l} = -\frac{d}{dt}\int \mathbf{B}\cdot d\mathbf{A}, \quad \oint \mathbf{B}\cdot d\mathbf{l} = \mu_0 I + \mu_0\varepsilon_0\frac{d}{dt}\int \mathbf{E}\cdot d\mathbf{A}
Maxwell's Equations (Differential Form)
∇⋅E=ρε0,∇⋅B=0,∇×E=−∂B∂t,∇×B=μ0j+μ0ε0∂E∂t\nabla\cdot\mathbf{E} = \frac{\rho}{\varepsilon_0}, \quad \nabla\cdot\mathbf{B} = 0, \quad \nabla\times\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}, \quad \nabla\times\mathbf{B} = \mu_0\mathbf{j} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}
Lorentz Force
F=q(E+v×B)F = q\left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)
Wave Equation in Vacuum for EM Fields
∇2E=1c2∂2E∂t2,∇2B=1c2∂2B∂t2,c=1ε0μ0\nabla^2 \mathbf{E} = \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2}, \quad \nabla^2 \mathbf{B} = \frac{1}{c^2}\frac{\partial^2 \mathbf{B}}{\partial t^2}, \quad c = \frac{1}{\sqrt{\varepsilon_0\mu_0}}
Energy Density in EM Field
u=12ε0E2+12μ0B2u = \frac{1}{2}\varepsilon_0E^2 + \frac{1}{2}\mu_0B^2
Intensity of an EM Wave
I=cε0E2=cε0⟨E2⟩I = c\varepsilon_0E^2 = c\varepsilon_0\langle E^2\rangle
Poynting Vector
S=1μ0(E×B)S = \frac{1}{\mu_0}(\mathbf{E}\times\mathbf{B})
Momentum in an EM Wave
π=μ0ε0S\pi = \mu_0\varepsilon_0S
Electric Dipole Moment
p=qdp = qd
Magnetic Dipole Moment
m=IAm = IA
Radiated Power (Electric Dipole)
⟨P⟩=p02ω412πε0c3\langle P \rangle = \frac{p_0^2\omega^4}{12\pi\varepsilon_0c^3}
Radiated Power (Magnetic Dipole)
⟨P⟩=μ0m02ω412πc3\langle P \rangle = \frac{\mu_0m_0^2\omega^4}{12\pi c^3}
Malus' Law
I(θ)=I0cos⁡2θI(\theta) = I_0\cos^2\theta
Linear Media Relations
P=ε0χeE,D=ε0E+P=ε0(1+χe)E=ε0εrE,M=χmH,B=μ0H+μ0M=μ0(1+χm)H=μ0μrHP = \varepsilon_0\chi_eE, \quad D = \varepsilon_0E+P=\varepsilon_0(1+\chi_e)E=\varepsilon_0\varepsilon_rE, \quad M = \chi_mH, \quad B = \mu_0H+\mu_0M=\mu_0(1+\chi_m)H=\mu_0\mu_rH
Gauss's Law for Electric Displacement
∮D⋅dA=qfree\oint \mathbf{D}\cdot d\mathbf{A} = q_{\text{free}}
Gauss's Law for Magnetic Fields
∮B⋅dA=0\oint \mathbf{B}\cdot d\mathbf{A} = 0
Faraday's Law of Induction
ε=−dΦdt\varepsilon = -\frac{d\Phi}{dt}
Lorentz Transformation
x′=γ(x−vt),t′=γ(t−vxc2)x' = \gamma(x - vt), \quad t' = \gamma\left(t - \frac{vx}{c^2}\right)
Time Dilation
Δt=γΔt0\Delta t = \gamma\Delta t_0
Length Contraction
Δx=γΔx0\Delta x = \gamma\Delta x_0
Relativistic Velocity Addition
vAC=vAB+vBC1+vABvBCc2v_{AC} = \frac{v_{AB}+v_{BC}}{1+\frac{v_{AB}v_{BC}}{c^2}}

Relativity

Mass-Energy Equivalence
E=mc2E = mc^2
Lorentz Factor
γ=11−v2c2\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

Thermodynamics

Ideal Gas Law
PV=nRTPV = nRT
First Law of Thermodynamics
ΔU=Q−W\Delta U = Q - W

Quantum Mechanics

Time-Dependent Schrödinger Equation
iℏ∂∂tΨ(r,t)=H^Ψ(r,t)i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)
Heisenberg Uncertainty Principle
ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}

Optics

Thin Lens Formula
1f=1do+1di\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}

Calculus

Differentiation

Limit Definition of Derivative
f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}
Tangent Line Equation
y=f(x0)+f′(x0)(x−x0)y = f(x_0) + f'(x_0)(x - x_0)
Product Rule
(fg)′=f′ g+f g′(fg)' = f'\,g + f\,g'
Quotient Rule
ddx(fg)=g f′−f g′g2\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{g\,f' - f\,g'}{g^2}
Chain Rule
ddxf(g(x))=f′(g(x)) g′(x)\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)
Derivatives of Trigonometric Functions
ddxsin⁡x=cos⁡x;ddxcos⁡x=−sin⁡x;ddxtan⁡x=1cos⁡2x\frac{d}{dx}\sin x = \cos x; \quad \frac{d}{dx}\cos x = -\sin x; \quad \frac{d}{dx}\tan x = \frac{1}{\cos^2 x}
Mean Value Theorem
f(b)−f(a)b−a=f′(c) for some c∈(a,b)\frac{f(b)-f(a)}{b-a} = f'(c) \text{ for some } c \in (a,b)
Derivative of Arcsine
ddxarcsin⁡x=11−x2\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}
Derivative of Arctangent
ddxarctan⁡x=11+x2\frac{d}{dx}\arctan x = \frac{1}{1+x^2}

Hyperbolic Functions

Hyperbolic Functions
cosh⁡x=ex+e−x2;sinh⁡x=ex−e−x2;cosh⁡2x−sinh⁡2x=1\cosh x = \frac{e^x + e^{-x}}{2}; \quad \sinh x = \frac{e^x - e^{-x}}{2}; \quad \cosh^2 x - \sinh^2 x = 1
Derivatives of Hyperbolic Functions
ddxcosh⁡x=sinh⁡x;ddxsinh⁡x=cosh⁡x\frac{d}{dx}\cosh x = \sinh x; \quad \frac{d}{dx}\sinh x = \cosh x

Integration

Integral of Power Function
∫xr dx=xr+1r+1+C(r≠−1)\int x^r\,dx = \frac{x^{r+1}}{r+1} + C \quad (r \neq -1)
Integral of Cosine
∫cos⁡(ax) dx=1asin⁡(ax)+C\int \cos(ax)\,dx = \frac{1}{a}\sin(ax) + C
Integral of 1/x
∫dxx=ln⁡∣x∣+C\int \frac{dx}{x} = \ln|x| + C
Integral Leading to Arcsine
∫dxa2−x2=arcsin⁡(xa)+C(a>0)\int \frac{dx}{\sqrt{a^2-x^2}} = \arcsin\left(\frac{x}{a}\right) + C \quad (a>0)
Integral of Exponential Function
∫eax dx=1aeax+C\int e^{ax}\,dx = \frac{1}{a}e^{ax} + C
Integral of 1/(a^2+x^2)
∫dxa2+x2=1aarctan⁡(xa)+C\int \frac{dx}{a^2+x^2} = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C
Integral of Sine
∫sin⁡(ax) dx=−1acos⁡(ax)+C\int \sin(ax)\,dx = -\frac{1}{a}\cos(ax) + C
Integration by Parts
∫u(x)v′(x) dx=u(x)v(x)−∫u′(x)v(x) dx\int u(x)v'(x)\,dx = u(x)v(x) - \int u'(x)v(x)\,dx

Series & Approximations

Taylor Polynomial
Pn(x)=∑k=0nf(k)(a)k!(x−a)kP_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k
Taylor Remainder
En(x)=f(n+1)(s)(n+1)!(x−a)n+1E_n(x) = \frac{f^{(n+1)}(s)}{(n+1)!}(x-a)^{n+1}
Geometric Series
11−x=∑n=0∞xn(−1<x<1)\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \quad (-1<x<1)
Cosine Series
cos⁡x=∑n=0∞(−1)n(2n)!x2n\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}x^{2n}
Natural Logarithm Series
ln⁡(1+x)=∑n=1∞(−1)n−1nxn(−1<x≤1)\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}x^n \quad (-1<x\leq1)
Exponential Series
ex=∑n=0∞xnn!e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}
Sine Series
sin⁡x=∑n=0∞(−1)n(2n+1)!x2n+1\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}x^{2n+1}

Numerical Methods

Newton's Method
xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
Trapezoidal Rule
Tn=h(y02+y1+⋯+yn−1+yn2)T_n = h\left(\frac{y_0}{2} + y_1 + \cdots + y_{n-1} + \frac{y_n}{2}\right)
Simpson's Rule
S2n=h3(y0+4y1+2y2+⋯+4y2n−1+y2n)S_{2n} = \frac{h}{3}\left(y_0 + 4y_1 + 2y_2 + \cdots + 4y_{2n-1} + y_{2n}\right)
Euler's Method
xn+1=xn+h,yn+1=yn+h f(xn,yn)x_{n+1} = x_n + h, \quad y_{n+1} = y_n + h\,f(x_n,y_n)

Linear_Algebra

Matrix Operations

Matrix Multiplication
(AB)ij=∑k=1nAikBkj(AB)_{ij} = \sum_{k=1}^n A_{ik}B_{kj}
Transpose of a Product
(AB)T=BTAT(AB)^T = B^T A^T
Inverse of a Product
(AB)−1=B−1A−1(AB)^{-1} = B^{-1}A^{-1}

Determinants

Determinant of a 2x2 Matrix
det⁡(abcd)=ad−bc\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc
Multiplicative Property of Determinants
det⁡(AB)=det⁡(A)det⁡(B)\det(AB) = \det(A)\det(B)
Cofactor Expansion
det⁡(A)=∑j=1n(−1)i+jaijMij\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{ij} M_{ij}

Eigen Concepts

Characteristic Polynomial
p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I)
Eigenvalue Equation
Av=λvA\mathbf{v} = \lambda\mathbf{v}
Diagonalization
A=PDP−1A = P D P^{-1}

Vector Spaces

Definition of Span
span{v1,…,vk}={∑i=1kcivi:ci∈R}\text{span}\{\mathbf{v}_1, \dots, \mathbf{v}_k\} = \left\{\sum_{i=1}^k c_i \mathbf{v}_i : c_i \in \mathbb{R} \right\}
Basis and Dimension
dim(V)=nifV=span{v1,…,vn}\text{dim}(V) = n \quad \text{if} \quad V = \text{span}\{\mathbf{v}_1, \dots, \mathbf{v}_n\}
Subspace Criteria
0∈W,if u,v∈W then u+v∈W,and c⋅u∈W0 \in W, \quad \text{if } u,v \in W \text{ then } u+v \in W, \quad \text{and } c\cdot u \in W

Linear Transformations

Definition of Linear Transformation
T(cu+v)=cT(u)+T(v)T(c\mathbf{u}+\mathbf{v}) = cT(\mathbf{u}) + T(\mathbf{v})
Matrix Representation of a Linear Transformation
T(x)=AxT(\mathbf{x}) = A\mathbf{x}
Kernel and Image
ker⁡(T)={x:T(x)=0},im(T)={T(x):x∈V}\ker(T) = \{\mathbf{x} : T(\mathbf{x}) = \mathbf{0}\}, \quad \text{im}(T) = \{T(\mathbf{x}) : \mathbf{x} \in V\}

Orthogonality & Projections

Orthogonality Condition
u⋅v=0\mathbf{u} \cdot \mathbf{v} = 0
Projection of a Vector
projv(u)=u⋅v∥v∥2v\text{proj}_{\mathbf{v}}(\mathbf{u}) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}
Gram-Schmidt Process
v1=u1,v2=u2−u2⋅v1∥v1∥2v1,…v_1 = u_1, \quad v_2 = u_2 - \frac{u_2 \cdot v_1}{\|v_1\|^2} v_1, \quad \ldots

Advanced Topics

Rank-Nullity Theorem
rank(T)+nullity(T)=dim⁡(V)\text{rank}(T) + \text{nullity}(T) = \dim(V)
Spectral Theorem (Symmetric Matrices)
A=QΛQTA = Q \Lambda Q^T
Singular Value Decomposition
A=UΣVTA = U \Sigma V^T

Trigonometry

Sin Identities

Basic Sine Identity
sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
Sine Addition Formula
sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta
Sine Subtraction Formula
sin⁡(α−β)=sin⁡αcos⁡β−cos⁡αsin⁡β\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta
Sine Double-Angle Formula
sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta)=2\sin\theta\cos\theta
Sine Half-Angle Formula
sin⁡(θ2)=±1−cos⁡θ2\sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos\theta}{2}}
Sine of Negative Angle
sin⁡(−θ)=−sin⁡θ\sin(-\theta)=-\sin\theta
Sine of a Complementary Angle
sin⁡(π2−θ)=cos⁡θ\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta
Product-to-Sum for Sine
2sin⁡αsin⁡β=cos⁡(α−β)−cos⁡(α+β)2\sin\alpha\sin\beta=\cos(\alpha-\beta)-\cos(\alpha+\beta)

Cos Identities

Cosine Double-Angle
cos⁡(2θ)=2cos⁡2θ−1\cos(2\theta) = 2\cos^2\theta - 1
Cosine Double-Angle Alternative
cos⁡(2θ)=cos⁡2θ−sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta
Cosine Addition Formula
cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta
Cosine of Negative Angle
cos⁡(−θ)=cos⁡θ\cos(-\theta)=\cos\theta
Cosine of a Complementary Angle
cos⁡(π2−θ)=sin⁡θ\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta
Cosine Subtraction Formula
cos⁡(α−β)=cos⁡αcos⁡β+sin⁡αsin⁡β\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta
Product-to-Sum for Cosine
2cos⁡αcos⁡β=cos⁡(α+β)+cos⁡(α−β)2\cos\alpha\cos\beta=\cos(\alpha+\beta)+\cos(\alpha-\beta)

Tan Identities

Tangent Addition Formula
tan⁡(α+β)=tan⁡α+tan⁡β1−tan⁡αtan⁡β\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}
Tangent Double-Angle Formula
tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}
Tangent Half-Angle Formula
tan⁡(θ2)=1−cos⁡θsin⁡θ\tan\left(\frac{\theta}{2}\right)=\frac{1-\cos\theta}{\sin\theta}

Other Identities

Pythagorean Identity (Tangent-Secant)
1+tan⁡2θ=sec⁡2θ1+\tan^2\theta = \sec^2\theta
Pythagorean Identity (Cotangent-Cosecant)
1+cot⁡2θ=csc⁡2θ1+\cot^2\theta = \csc^2\theta
Sine of a Double Angle
sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta)=2\sin\theta\cos\theta
Cosine of a Half Angle
cos⁡(θ2)=±1+cos⁡θ2\cos\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1+\cos\theta}{2}}
Standard Circle Equation
(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2
Sine of a Half Angle
sin⁡(θ2)=±1−cos⁡θ2\sin\left(\frac{\theta}{2}\right)=\pm\sqrt{\frac{1-\cos\theta}{2}}
Sine Triple-Angle Formula
sin⁡(3θ)=3sin⁡θ−4sin⁡3θ\sin(3\theta)=3\sin\theta-4\sin^3\theta
Cosine Triple-Angle Formula
cos⁡(3θ)=4cos⁡3θ−3cos⁡θ\cos(3\theta)=4\cos^3\theta-3\cos\theta

Sum-Product Identities

Sine Sum-to-Product
sin⁡α+sin⁡β=2sin⁡(α+β2)cos⁡(α−β2)\sin\alpha + \sin\beta = 2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)
Cosine Sum-to-Product
cos⁡α+cos⁡β=2cos⁡(α+β2)cos⁡(α−β2)\cos\alpha + \cos\beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)

Cot Identities

Cotangent Addition Formula
cot⁡(α+β)=cot⁡αcot⁡β−1cot⁡α+cot⁡β\cot(\alpha+\beta)=\frac{\cot\alpha\cot\beta-1}{\cot\alpha+\cot\beta}

Reciprocal Identities

Secant Identity
sec⁡θ=1cos⁡θ\sec\theta = \frac{1}{\cos\theta}
Cosecant Identity
csc⁡θ=1sin⁡θ\csc\theta = \frac{1}{\sin\theta}

Algebra

Quadratic Equations

Quadratic Formula
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Completing the Square
ax2+bx=a(x+b2a)2−b24aax^2 + bx = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a}
Discriminant
D=b2−4acD = b^2 - 4ac
Factoring Quadratics
ax2+bx+c=(mx+n)(px+q)ax^2 + bx + c = (mx + n)(px + q)

Polynomials

Binomial Theorem
(a+b)n=∑k=0n(nk)an−kbk(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
Remainder Theorem
f(x)=(x−k)q(x)+f(k)f(x) = (x-k)q(x) + f(k)
Factor Theorem
(x−k) divides f(x)  ⟺  f(k)=0(x-k) \text{ divides } f(x) \iff f(k)=0
Polynomial Long Division
Dividend=Divisor⋅Quotient+Remainder\text{Dividend} = \text{Divisor} \cdot \text{Quotient} + \text{Remainder}

Exponents and Logarithms

Laws of Exponents
am⋅an=am+n,aman=am−n,(am)n=amna^m \cdot a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}
Change of Base Formula
logb(a)=log⁡k(a)log⁡k(b)log_b(a) = \frac{\log_k(a)}{\log_k(b)}
Logarithm Rules
logb(MN)=logb(M)+logb(N),logb(M/N)=logb(M)−logb(N),logb(Mp)=p logb(M)log_b(MN)=log_b(M)+log_b(N), \quad log_b(M/N)=log_b(M)-log_b(N), \quad log_b(M^p)=p\,log_b(M)
Exponential Growth and Decay
y=y0ekty = y_0 e^{kt}

Systems of Equations

Cramer's Rule
xi=det⁡(Ai)det⁡(A)x_i = \frac{\det(A_i)}{\det(A)}

Sequences and Series

Arithmetic Series Sum
Sn=n2(a1+an)=n2[2a+(n−1)d]S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}\left[2a + (n-1)d\right]
Geometric Series Sum
Sn=a1−rn1−r(r≠1)S_n = a \frac{1 - r^n}{1 - r} \quad (r \neq 1)
Infinite Geometric Series
S=a1−r(∣r∣<1)S = \frac{a}{1 - r} \quad (|r| < 1)

Complex Numbers

Euler's Formula
eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta
Complex Conjugate
z=a+bi  ⟹  zˉ=a−biz = a + bi \implies \bar{z} = a - bi

Factoring Techniques

Difference of Squares
a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)
Sum of Cubes
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - ab + b^2)
Difference of Cubes
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Inequalities

Linear Inequality
ax+b>cax + b > c
Quadratic Inequality
ax2+bx+c<0ax^2 + bx + c < 0

Absolute Value Equations

Absolute Value Equation
∣ax+b∣=c|ax + b| = c
Absolute Value Inequality
∣ax+b∣<c|ax + b| < c

Statistics

Descriptive Statistics

Mean
μ=1N∑i=1Nxi\mu = \frac{1}{N}\sum_{i=1}^{N} x_i
Variance
σ2=1N∑i=1N(xi−μ)2\sigma^2 = \frac{1}{N}\sum_{i=1}^{N} (x_i - \mu)^2
Standard Deviation
σ=1N∑i=1N(xi−μ)2\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^{N} (x_i - \mu)^2}

Probability

Bayes' Theorem
P(A∣B)=P(B∣A)P(A)P(B)P(A|B) = \frac{P(B|A)P(A)}{P(B)}
Normal Distribution (PDF)
f(x)=12πσ2 e−(x−μ)22σ2f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Economics

Macroeconomics

GDP (Expenditure Approach)
GDP=C+I+G+(X−M)GDP = C + I + G + (X - M)
Consumer Surplus (Approximation)
CS≈12Q(Pmax−P)CS \approx \frac{1}{2} Q (P_{max} - P)

Microeconomics

Price Elasticity of Demand
Ep=%ΔQ%ΔPE_p = \frac{\%\Delta Q}{\%\Delta P}
Cobb-Douglas Production Function
Y=AKαL1−αY = A K^{\alpha} L^{1-\alpha}
Present Value
PV=FV(1+r)nPV = \frac{FV}{(1 + r)^n}