§F
Math Foundations
§F · Cheat sheet

Cheat sheet

Every key formula, in canonical order, at a glance. Each formula links to the lesson that derives it.

0 — Pre-calculus: trigonometry and logarithms

Unit-circle point

P=(cos⁡θ, sin⁡θ)P=(\cos\theta,\ \sin\theta)

Pythagorean identity

cos⁡2θ+sin⁡2θ=1\cos^2\theta+\sin^2\theta=1

Angle-sum (cosine)

cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha+\beta)=\cos\alpha\cos\beta-\sin\alpha\sin\beta

Angle-sum (sine)

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta

Power reduction

sin⁡2α=12(1−cos⁡2α)\sin^2\alpha=\tfrac12(1-\cos 2\alpha)

Pure oscillation

x(t)=Acos⁡(ωt+φ), T=2π/ωx(t)=A\cos(\omega t+\varphi),\ T=2\pi/\omega

ee is its own derivative

ddxex=ex\tfrac{d}{dx}e^x=e^x

Log of a product

log⁡a(xy)=log⁡ax+log⁡ay\log_a(xy)=\log_a x+\log_a y

Change of base

log⁡ax=log⁡bxlog⁡ba\log_a x=\dfrac{\log_b x}{\log_b a}

Decibel level

L=10log⁡10 ⁣PP0=20log⁡10 ⁣AA0L=10\log_{10}\!\tfrac{P}{P_0}=20\log_{10}\!\tfrac{A}{A_0}

Exponential decay

x(t)=x0 e−t/τx(t)=x_0\,e^{-t/\tau}

1 — Single-variable calculus

Derivative (limit)

f′(t)=lim⁡h→0f(t+h)−f(t)hf'(t)=\lim_{h\to0}\dfrac{f(t+h)-f(t)}{h}

Product rule

(fg)′=f′g+fg′(fg)'=f'g+fg'

Chain rule

(f(g(t)))′=f′(g(t)) g′(t)\big(f(g(t))\big)'=f'(g(t))\,g'(t)

Fundamental theorem

∫abf(t) dt=F(b)−F(a)\int_a^b f(t)\,dt=F(b)-F(a)

Substitution

∫f(g(t)) g′(t) dt=∫f(u) du\int f(g(t))\,g'(t)\,dt=\int f(u)\,du

Integration by parts

∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du

Root-mean-square

frms=1T∫0Tf(t)2 dtf_\text{rms}=\sqrt{\tfrac1T\int_0^T f(t)^2\,dt}

Taylor expansion

f(t0+ε)=f(t0)+εf′(t0)+12ε2f′′(t0)+⋯f(t_0+\varepsilon)=f(t_0)+\varepsilon f'(t_0)+\tfrac12\varepsilon^2 f''(t_0)+\cdots

2 — Partial derivatives and vector calculus

Total differential

df=∂f∂xdx+∂f∂ydy+∂f∂zdzdf=\tfrac{\partial f}{\partial x}dx+\tfrac{\partial f}{\partial y}dy+\tfrac{\partial f}{\partial z}dz

Gradient

∇ϕ=(∂ϕ∂x,∂ϕ∂y,∂ϕ∂z)\nabla\phi=\big(\tfrac{\partial\phi}{\partial x},\tfrac{\partial\phi}{\partial y},\tfrac{\partial\phi}{\partial z}\big)

Directional derivative

Du^ϕ=∇ϕ⋅u^=∣∇ϕ∣cos⁡θD_{\hat u}\phi=\nabla\phi\cdot\hat{\mathbf u}=\lvert\nabla\phi\rvert\cos\theta

Divergence

∇⋅v=∂vx∂x+∂vy∂y+∂vz∂z\nabla\cdot\mathbf v=\tfrac{\partial v_x}{\partial x}+\tfrac{\partial v_y}{\partial y}+\tfrac{\partial v_z}{\partial z}

Curl (zz-component)

(∇×v)z=∂vy∂x−∂vx∂y(\nabla\times\mathbf v)_z=\tfrac{\partial v_y}{\partial x}-\tfrac{\partial v_x}{\partial y}

Laplacian

∇2ϕ=∂2ϕ∂x2+∂2ϕ∂y2+∂2ϕ∂z2\nabla^2\phi=\tfrac{\partial^2\phi}{\partial x^2}+\tfrac{\partial^2\phi}{\partial y^2}+\tfrac{\partial^2\phi}{\partial z^2}

3 — Complex exponentials and phasors

Euler’s formula

eiθ=cos⁡θ+isin⁡θe^{i\theta}=\cos\theta+i\sin\theta

Phasor representation

X~=A eiφ,x(t)=Re⁡[X~ eiωt]\tilde X=A\,e^{i\varphi},\quad x(t)=\operatorname{Re}[\tilde X\,e^{i\omega t}]

Complex-argument decay

e(−γ+iω)t=e−γt(cos⁡ωt+isin⁡ωt)e^{(-\gamma+i\omega)t}=e^{-\gamma t}(\cos\omega t+i\sin\omega t)

Plane wave

p(r,t)=Re⁡[P0 ei(ωt−k⋅r)]p(\mathbf r,t)=\operatorname{Re}[P_0\,e^{i(\omega t-\mathbf k\cdot\mathbf r)}]

Dispersion relation

ω2=c2∣k∣2\omega^2=c^2\lvert\mathbf k\rvert^2

Specific acoustic impedance

Z0=ρcZ_0=\rho c

Series RLC impedance

Z(ω)=R+i ⁣(ωL−1ωC)Z(\omega)=R+i\!\left(\omega L-\tfrac{1}{\omega C}\right)

4 — Linear algebra

Matrix–vector product

(Av)i=∑jaijvj(A\mathbf v)_i=\sum_j a_{ij}v_j

2×22\times2 determinant

det⁡A=a11a22−a12a21\det A=a_{11}a_{22}-a_{12}a_{21}

Eigenvalue equation

Av=λvA\mathbf v=\lambda\mathbf v

Characteristic equation

det⁡(A−λI)=0\det(A-\lambda I)=0

2×22\times2 characteristic poly

λ2−(tr⁡A)λ+det⁡A=0\lambda^2-(\operatorname{tr}A)\lambda+\det A=0

Dot product

u⋅v=∑kukvk\mathbf u\cdot\mathbf v=\sum_k u_k v_k

Norm

∥v∥=v⋅v\lVert\mathbf v\rVert=\sqrt{\mathbf v\cdot\mathbf v}

Angle between vectors

cos⁡θ=u⋅v∥u∥ ∥v∥\cos\theta=\dfrac{\mathbf u\cdot\mathbf v}{\lVert\mathbf u\rVert\,\lVert\mathbf v\rVert}

Orthonormal expansion

v=∑k(v⋅ek) ek\mathbf v=\sum_k(\mathbf v\cdot\mathbf e_k)\,\mathbf e_k

Eigendecomposition

A=Q D QTA=Q\,D\,Q^{T}

5 — Linear ordinary differential equations

First-order linear ODE

x˙+αx=f(t)\dot x+\alpha x=f(t)

Exponential solution

x(t)=x0 e−αt, τ=1/αx(t)=x_0\,e^{-\alpha t},\ \tau=1/\alpha

Second-order linear ODE

ax¨+bx˙+c x=f(t)a\ddot x+b\dot x+c\,x=f(t)

Simple harmonic motion

x¨+ω02x=0, ω0=k/m\ddot x+\omega_0^2 x=0,\ \omega_0=\sqrt{k/m}

Damped roots

λ=−γ±γ2−ω02\lambda=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}

Underdamped solution

x(t)=e−γt[Acos⁡ωdt+Bsin⁡ωdt], ωd=ω02−γ2x(t)=e^{-\gamma t}[A\cos\omega_d t+B\sin\omega_d t],\ \omega_d=\sqrt{\omega_0^2-\gamma^2}

Driven steady-state amplitude

X~(ω)=F0/m(ω02−ω2)+2iγω\tilde X(\omega)=\dfrac{F_0/m}{(\omega_0^2-\omega^2)+2i\gamma\omega}

Phase-plane system

x˙=Ax, x=(x, v)T\dot{\mathbf x}=A\mathbf x,\ \mathbf x=(x,\ v)^{T}

6 — Linear partial differential equations

Wave equation

∂t2u=c2∇2u\partial_t^2 u=c^2\nabla^2 u

Heat equation

∂tu=D∇2u\partial_t u=D\nabla^2 u

Laplace’s equation

∇2u=0\nabla^2 u=0

d’Alembert solution

u(x,t)=12[f(x−ct)+f(x+ct)]+12c∫x−ctx+ctg(s) dsu(x,t)=\tfrac12[f(x-ct)+f(x+ct)]+\tfrac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds

Clamped-string modes

u(x,t)=∑nAnsin⁡ ⁣nπxLcos⁡ ⁣nπctLu(x,t)=\sum_n A_n\sin\!\tfrac{n\pi x}{L}\cos\!\tfrac{n\pi ct}{L}

Mode coefficient

cn=2L∫0Lf(x)sin⁡ ⁣nπxL dxc_n=\tfrac{2}{L}\int_0^L f(x)\sin\!\tfrac{n\pi x}{L}\,dx

Heat-equation solution

u(x,t)=∑nAnsin⁡ ⁣nπxL e−D(nπ/L)2tu(x,t)=\sum_n A_n\sin\!\tfrac{n\pi x}{L}\,e^{-D(n\pi/L)^2 t}

Helmholtz equation

∇2ϕ+k2ϕ=0, k=ω/c\nabla^2\phi+k^2\phi=0,\ k=\omega/c

Rectangular cavity modes

ωmn=cπ(m/Lx)2+(n/Ly)2\omega_{mn}=c\pi\sqrt{(m/L_x)^2+(n/L_y)^2}

Schrödinger equation

iℏ ∂tΨ=−ℏ22m∇2Ψ+VΨi\hbar\,\partial_t\Psi=-\tfrac{\hbar^2}{2m}\nabla^2\Psi+V\Psi

Particle-in-a-box energies

En=n2π2ℏ22mL2E_n=\dfrac{n^2\pi^2\hbar^2}{2mL^2}

7 — Fourier series and the Fourier transform

Fourier series

f(t)=∑ncnei2πnt/T, cn=1T∫0Tf(t)e−i2πnt/Tdtf(t)=\sum_n c_n e^{i2\pi nt/T},\ c_n=\tfrac1T\int_0^T f(t)e^{-i2\pi nt/T}dt

Fourier transform

f~(ω)=∫−∞∞f(t) e−iωt dt\tilde f(\omega)=\int_{-\infty}^{\infty} f(t)\,e^{-i\omega t}\,dt

Inverse transform

f(t)=12π∫−∞∞f~(ω) eiωt dωf(t)=\tfrac{1}{2\pi}\int_{-\infty}^{\infty}\tilde f(\omega)\,e^{i\omega t}\,d\omega

Uncertainty relation

Δt⋅Δω≥12\Delta t\cdot\Delta\omega\geq\tfrac12

Convolution

(f∗g)(t)=∫−∞∞f(τ)g(t−τ) dτ(f*g)(t)=\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau

Convolution theorem

(f∗g)(t) ⟷ f~(ω)⋅g~(ω)(f*g)(t)\ \longleftrightarrow\ \tilde f(\omega)\cdot\tilde g(\omega)

Parseval’s theorem

∫−∞∞∣f(t)∣2dt=12π∫−∞∞∣f~(ω)∣2dω\int_{-\infty}^{\infty}\lvert f(t)\rvert^2 dt=\tfrac{1}{2\pi}\int_{-\infty}^{\infty}\lvert\tilde f(\omega)\rvert^2 d\omega

Nyquist criterion

fs≥2fmax⁡f_s\geq 2f_{\max}

Discrete Fourier transform

Xk=∑n=0N−1xn e−i2πkn/NX_k=\sum_{n=0}^{N-1}x_n\,e^{-i2\pi kn/N}

Aliased frequency

falias=∣f0−nfs∣f_\text{alias}=\lvert f_0-nf_s\rvert

8 — Dimensional analysis

Buckingham π\pi count

groups=n−k\text{groups}=n-k

Sound speed (fluid)

c∼K/ρc\sim\sqrt{K/\rho}

Wave speed (string)

c∼T/μc\sim\sqrt{T/\mu}

Pendulum period

τ∼ℓ/g\tau\sim\sqrt{\ell/g}

Reynolds number

Re=ρvL/μRe=\rho vL/\mu

Mach number

M=v/cM=v/c

Strouhal number

St=fL/vSt=fL/v

Helmholtz number

ka=2πa/λka=2\pi a/\lambda

9 — Numerical methods

Forward difference

f′(x)≈f(x+h)−f(x)hf'(x)\approx\dfrac{f(x+h)-f(x)}{h}

Centred difference

f′(x)≈f(x+h)−f(x−h)2hf'(x)\approx\dfrac{f(x+h)-f(x-h)}{2h}

Second-derivative stencil

f′′(x)≈f(x+h)−2f(x)+f(x−h)h2f''(x)\approx\dfrac{f(x+h)-2f(x)+f(x-h)}{h^2}

Forward Euler

xn+1=xn+h f(tn,xn)x_{n+1}=x_n+h\,f(t_n,x_n)

Runge–Kutta 4

xn+1=xn+h6(k1+2k2+2k3+k4)x_{n+1}=x_n+\tfrac{h}{6}(k_1+2k_2+2k_3+k_4)

Wave-equation leapfrog

ujn+1=2ujn−ujn−1+C2(uj+1n−2ujn+uj−1n)u^{n+1}_j=2u^n_j-u^{n-1}_j+C^2(u^n_{j+1}-2u^n_j+u^n_{j-1})

CFL stability

C=c Δt/Δx≤1C=c\,\Delta t/\Delta x\leq1

Jacobi relaxation

ui,j=14(ui+1,j+ui−1,j+ui,j+1+ui,j−1)u_{i,j}=\tfrac14(u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1})

Newton’s method

xn+1=xn−f(xn)f′(xn)x_{n+1}=x_n-\dfrac{f(x_n)}{f'(x_n)}

10 — Statistics and probability

Expected value

E[X]=∑kk p(k)\mathbb E[X]=\sum_k k\,p(k)

Variance

Var⁡[X]=E[X2]−μ2\operatorname{Var}[X]=\mathbb E[X^2]-\mu^2

Binomial PMF

Pr⁡(X=k)=(nk)pk(1−p)n−k\Pr(X=k)=\binom{n}{k}p^k(1-p)^{n-k}

Gaussian PDF

f(x)=1σ2πexp⁡ ⁣(−(x−μ)22σ2)f(x)=\tfrac{1}{\sigma\sqrt{2\pi}}\exp\!\big(-\tfrac{(x-\mu)^2}{2\sigma^2}\big)

Central limit theorem

Sn−nμσn→N(0,1)\dfrac{S_n-n\mu}{\sigma\sqrt n}\to\mathcal N(0,1)

Standard error

σXˉ=σ/n\sigma_{\bar X}=\sigma/\sqrt n

Random-walk mean square

E[XN2]=N\mathbb E[X_N^2]=N

Brownian variance

E[B(t)2]=2Dt\mathbb E[B(t)^2]=2Dt

Einstein relation

D=kBT/ζD=k_B T/\zeta

Poisson PMF

Pr⁡(N(T)=k)=(λT)ke−λTk!\Pr(N(T)=k)=\dfrac{(\lambda T)^k e^{-\lambda T}}{k!}

Bayes’ rule

Pr⁡(A∣B)=Pr⁡(B∣A) Pr⁡(A)Pr⁡(B)\Pr(A\mid B)=\dfrac{\Pr(B\mid A)\,\Pr(A)}{\Pr(B)}

11 — Chaos and nonlinear dynamics

Logistic map

xn+1=r xn(1−xn)x_{n+1}=r\,x_n(1-x_n)

Feigenbaum constant

δ=lim⁡k→∞rk−rk−1rk+1−rk=4.6692016…\delta=\lim_{k\to\infty}\dfrac{r_k-r_{k-1}}{r_{k+1}-r_k}=4.6692016\ldots

Lorenz system

x˙=σ(y−x), y˙=x(ρ−z)−y, z˙=xy−βz\dot x=\sigma(y-x),\ \dot y=x(\rho-z)-y,\ \dot z=xy-\beta z

Lyapunov separation

∣δ(t)∣≈∣δ0∣ eλt\lvert\boldsymbol\delta(t)\rvert\approx\lvert\boldsymbol\delta_0\rvert\,e^{\lambda t}

Prediction horizon

thorizon=1λln⁡Lδ0t_\text{horizon}=\tfrac{1}{\lambda}\ln\tfrac{L}{\delta_0}

12 — Calculus of variations

Functional

J[y]=∫abF(x,y,y′) dxJ[y]=\int_a^b F(x,y,y')\,dx

First variation

δJ=∫ab ⁣(∂F∂y η+∂F∂y′ η′)dx=0\delta J=\int_a^b\!\left(\tfrac{\partial F}{\partial y}\,\eta+\tfrac{\partial F}{\partial y'}\,\eta'\right)dx=0

Euler–Lagrange equation

∂F∂y−ddx∂F∂y′=0\frac{\partial F}{\partial y}-\frac{d}{dx}\frac{\partial F}{\partial y'}=0

First integral (F independent of y)

∂F∂y′=const\frac{\partial F}{\partial y'}=\text{const}

Beltrami identity (F independent of x)

F−y′∂F∂y′=constF-y'\frac{\partial F}{\partial y'}=\text{const}

Brachistochrone (cycloid)

y(1+y′2)=ky(1+y'^2)=k

Catenary

y=acosh⁡ ⁣((x−b)/a)y=a\cosh\!\big((x-b)/a\big)

Snell from Fermat

n(y)sin⁡θ=constn(y)\sin\theta=\text{const}

Isoperimetric (multiplier)

δ ⁣(J−λK)=0\delta\!\big(J-\lambda K\big)=0

Hamilton’s principle

S[q]=∫t1t2(T−U) dt,δS=0S[q]=\int_{t_1}^{t_2}(T-U)\,dt,\quad \delta S=0

Euler–Lagrange (mechanics)

ddt∂L∂q˙−∂L∂q=0\frac{d}{dt}\frac{\partial L}{\partial\dot q}-\frac{\partial L}{\partial q}=0

Field Euler–Lagrange

∂t∂L∂(∂tϕ)+∇⋅∂L∂(∇ϕ)−∂L∂ϕ=0\partial_t\frac{\partial\mathcal L}{\partial(\partial_t\phi)}+\nabla\cdot\frac{\partial\mathcal L}{\partial(\nabla\phi)}-\frac{\partial\mathcal L}{\partial\phi}=0

13 — Operators and observables

Adjoint

⟨f,A^g⟩=⟨A^†f,g⟩,self-adjoint: A^†=A^\langle f,\hat A g\rangle=\langle \hat A^\dagger f,g\rangle,\quad \text{self-adjoint: }\hat A^\dagger=\hat A

Position and momentum operators

x^ ψ=xψ,p^=−iℏ ddx\hat x\,\psi=x\psi,\qquad \hat p=-i\hbar\,\frac{d}{dx}

Commutator

[A^,B^]=A^B^−B^A^[\hat A,\hat B]=\hat A\hat B-\hat B\hat A

Canonical commutation relation

[x^,p^]=iℏ[\hat x,\hat p]=i\hbar

Uncertainty relation

ΔA ΔB≥12∣⟨[A^,B^]⟩∣,Δx Δp≥ℏ2\Delta A\,\Delta B\ge\tfrac12\lvert\langle[\hat A,\hat B]\rangle\rvert,\qquad \Delta x\,\Delta p\ge\tfrac{\hbar}{2}

Ladder commutator

[a^,a^†]=1,N^=a^†a^[\hat a,\hat a^\dagger]=1,\qquad \hat N=\hat a^\dagger\hat a

Oscillator Hamiltonian and spectrum

H^=ℏω(N^+12),En=ℏω(n+12)\hat H=\hbar\omega\big(\hat N+\tfrac12\big),\qquad E_n=\hbar\omega\big(n+\tfrac12\big)

Ladder actions

a^†∣n⟩=n+1 ∣n+1⟩,a^∣n⟩=n ∣n−1⟩\hat a^\dagger\lvert n\rangle=\sqrt{n+1}\,\lvert n+1\rangle,\quad \hat a\lvert n\rangle=\sqrt{n}\,\lvert n-1\rangle

Time evolution

U^(t)=e−iH^t/ℏ\hat U(t)=e^{-i\hat H t/\hbar}

Heisenberg equation

dA^dt=iℏ[H^,A^]+∂A^∂t\frac{d\hat A}{dt}=\frac{i}{\hbar}[\hat H,\hat A]+\frac{\partial\hat A}{\partial t}

Canonical quantization

{A,B}PB⟶1iℏ[A^,B^]\{A,B\}_\text{PB}\longrightarrow\frac{1}{i\hbar}[\hat A,\hat B]