"Here is a gigantic detailed map of key equations used in special and general relativity: Special Relativity: 1. Lorentz factor: γ = 1/√(1 - v²/c²) 2. Time dilation: Δt' = γΔt 3. Length contraction: L' = L/γ 4. Relativistic velocity addition: v = (u + v)/(1 + uv/c²) 5. Relativistic momentum: p = γmv 6. Relativistic kinetic energy: E_k = (γ - 1)mc² 7. Mass-energy equivalence: E = mc² 8. Relativistic Doppler effect: - Moving away: f' = f√((1 - v/c)/(1 + v/c)) - Moving towards: f' = f√((1 + v/c)/(1 - v/c)) General Relativity: 1. Einstein field equations: G_μν = 8πG/c⁴ * T_μν - G_μν: Einstein tensor - T_μν: Stress-energy tensor - G: Gravitational constant - c: Speed of light 2. Metric tensor: g_μν - Describes the geometry of spacetime - Used to calculate distances, angles, and time intervals 3. Christoffel symbols (Γ^ρ_μν): - Describe the connection between the metric tensor and the derivatives of the metric tensor - Used to calculate geodesics (paths of particles in curved spacetime) 4. Riemann curvature tensor (R^ρ_σμν): - Describes the curvature of spacetime - Calculated using the Christoffel symbols and their derivatives 5. Ricci tensor (R_μν) and Ricci scalar (R): - Contractions of the Riemann curvature tensor - Used in the Einstein field equations 6. Geodesic equation: d²x^μ/dτ² + Γ^μ_ρσ * dx^ρ/dτ * dx^σ/dτ = 0 - Describes the motion of particles in curved spacetime - τ: Proper time 7. Schwarzschild metric: - Describes the geometry of spacetime around a non-rotating, uncharged, spherically symmetric mass - Used to derive predictions like gravitational time dilation and the existence of black holes 8. Kerr metric: - Describes the geometry of spacetime around a rotating, uncharged, axially symmetric mass - Used to study the properties of rotating black holes 9. Friedmann equations: - Describe the expansion of the universe in the context of general relativity - Derived from the Einstein field equations assuming a homogeneous and isotropic universe These equations form the mathematical foundation of special and general relativity, allowing for the description of phenomena such as time dilation, length contraction, gravitational lensing, black holes, and the expansion of the universe. The interplay between these equations and their physical interpretations has led to a profound understanding of the nature of space, time, and gravity." "Here is a detailed map of many key equations used in classical mechanics: Kinematics - Position: x(t) = x₀ + v₀t + ½at² - Velocity: v(t) = v₀ + at - Acceleration: a = (v - v₀) / t - Projectile motion: y = y₀ + v₀t - ½gt² Dynamics - Newton's 2nd Law: F = ma - Weight: W = mg - Friction: f ≤ μN - Tension: T = F - Hooke's Law: F = -kx Work, Energy, Power - Work: W = Fd (cos θ) - Kinetic Energy: K = ½mv² - Potential Energy: U = mgh (gravitational), U = ½kx² (elastic) - Conservation of Energy: ΔK + ΔU = W - Power: P = W/t Momentum & Collisions - Momentum: p = mv - Impulse: J = FΔt = Δp - Conservation of Momentum: m₁v₁ + m₂v₂ = m₁v'₁ + m₂v'₂ - Elastic Collision: v₁' = (m₁-m₂)/(m₁+m₂)v₁ + 2m₂/(m₁+m₂)v₂, v₂' = 2m₁/(m₁+m₂)v₁ + (m₂-m₁)/(m₁+m₂)v₂ Circular & Rotational Motion - Angular displacement: θ = s/r - Angular velocity: ω = Δθ/Δt = v/r - Angular acceleration: α = Δω/Δt = a/r - Centripetal acceleration: a_c = v²/r = rω² - Centripetal force: F_c = mv²/r = mrω² - Moment of inertia: I = Σmr² (discrete), I = ∫r²dm (continuous) - Torque: τ = rF sin θ = Iα - Angular momentum: L = Iω - Rotational KE: K = ½Iω² Gravitation - Newton's Law of Universal Gravitation: F = Gm₁m₂/r² - Gravitational PE: U = -Gm₁m₂/r - Orbital velocity: v = √(GM/r) - Escape velocity: v_e = √(2GM/r) Simple Harmonic Motion - Displacement: x = A sin(ωt + φ) - Velocity: v = Aω cos(ωt + φ) - Acceleration: a = -Aω² sin(ωt + φ) - Period: T = 2π/ω = 2π√(m/k) (mass-spring), T = 2π√(L/g) (pendulum) Fluid Mechanics - Density: ρ = m/V - Pressure: P = F/A - Pascal's Principle: ΔP = ρgh - Archimedes' Principle: F_b = ρgV - Continuity Equation: A₁v₁ = A₂v₂ - Bernoulli's Principle: P + ½ρv² + ρgy = constant This covers many of the core equations of classical mechanics, organized by major topics. Of course, this is not an exhaustive list, as classical mechanics is a vast field with many other important equations, principles, and applications. But hopefully this provides a helpful overview and reference for much of the key material. Let me know if you need any clarification or have additional questions!" "Here is a gigantic detailed map of key equations used in classical field theory: Maxwell's Equations: - Gauss' Law: ∇ ⋅ E = ρ/ε₀ - Gauss' Law for Magnetism: ∇ ⋅ B = 0 - Faraday's Law: ∇ × E = -∂B/∂t - Ampère's Circuital Law (with Maxwell's correction): ∇ × B = μ₀(J + ε₀ ∂E/∂t) Electromagnetic Potentials: - Electric Potential: E = -∇ϕ - Magnetic Vector Potential: B = ∇ × A - Lorentz Gauge: ∇ ⋅ A + (1/c²) ∂ϕ/∂t = 0 Wave Equations: - Wave Equation for Electric Field: ∇²E - (1/c²) ∂²E/∂t² = 0 - Wave Equation for Magnetic Field: ∇²B - (1/c²) ∂²B/∂t² = 0 - Wave Equation for Electric Potential: ∇²ϕ - (1/c²) ∂²ϕ/∂t² = -ρ/ε₀ - Wave Equation for Magnetic Vector Potential: ∇²A - (1/c²) ∂²A/∂t² = -μ₀J Poynting Vector and Electromagnetic Energy: - Poynting Vector: S = E × H - Electromagnetic Energy Density: u = (1/2)(ε₀E² + (1/μ₀)B²) Lorentz Force Law: - Force on a point charge: F = q(E + v × B) - Force density: f = ρE + J × B Stress-Energy Tensor: - Electromagnetic Stress-Energy Tensor: T^μν = F^μα F_α^ν - (1/4) η^μν F^αβ F_αβ Lagrangian and Hamiltonian Formulations: - Lagrangian Density: L = -(1/4)F^μν F_μν - J^μ A_μ - Hamiltonian Density: H = (1/2)(ε₀E² + (1/μ₀)B²) + ρϕ - J ⋅ A Gauge Transformations: - Scalar Potential Transformation: ϕ → ϕ - ∂Λ/∂t - Vector Potential Transformation: A → A + ∇Λ Charge and Current Conservation: - Continuity Equation: ∂ρ/∂t + ∇ ⋅ J = 0 Biot-Savart Law: - Magnetic Field due to Current Distribution: B(r) = (μ₀/4π) ∫ (J(r') × (r - r'))/|r - r'|³ d³r' These are some of the key equations that form the foundation of classical field theory, encompassing electromagnetism and its relativistic formulation. The equations describe the relationships between electric and magnetic fields, their sources (charges and currents), and the forces they exert on charged particles. The Lagrangian and Hamiltonian formulations provide alternative ways to describe the dynamics of the electromagnetic field." "Here is a detailed map of many key equations used in quantum field theory: Quantum Fields & Field Operators • Scalar field φ(x) • Fermion field ψ(x) • Gauge field Aμ(x) • Field strength tensor Fμν = ∂μAν - ∂νAμ • Conjugate momentum field π(x) = ∂L/∂(∂0φ) • Equal-time commutation relations: [φ(x),π(y)]|x0=y0 = iħδ(x-y) {ψ(x),ψ†(y)}|x0=y0 = ħδ(x-y) • Creation/annihilation operators: a†(p), a(p) for bosons b†(p), b(p), d†(p), d(p) for fermions Lagrangian Formalism • Lagrangian density L determines dynamics • Scalar field Lagrangian: L = ½(∂μφ)(∂μφ) - ½m2φ2 • Dirac Lagrangian: L = iψ̄γμ∂μψ - mψ̄ψ • Electromagnetic Lagrangian: L = -¼FμνFμν • Euler-Lagrange equation: ∂μ(∂L/∂(∂μφ)) = ∂L/∂φ • Noether's theorem relates symmetries to conservation laws Canonical Quantization • Promote fields to operators, impose commutation relations • Free field mode expansions: φ(x) = ∫d³p (a(p)e-ipx + a†(p)e+ipx) ψ(x) = ∫d³p (b(p)u(p)e-ipx + d†(p)v(p)e+ipx) • Momentum operator Pμ = ∫d³x : T0μ : where Tμν is stress-energy tensor • Hamiltonian H = P0 Perturbation Theory & Feynman Diagrams • S-matrix relates initial and final states Sfi = ⟨f|S|i⟩ = ⟨f|T{exp(-i/ħ∫d4xHint(x))}|i⟩ • Dyson series expands S-matrix in powers of interaction • Wick's theorem reduces time-ordered products to normal-ordered • Feynman diagrams represent terms in perturbation series Propagators, vertices given by Feynman rules • LSZ reduction relates S-matrix to correlation functions • Cross sections, decay rates extracted from S-matrix Renormalization • Regularization (cutoffs, dimensional regularization) to handle infinities • Introduce renormalized parameters and counterterms • Compute physical observables in terms of renormalized quantities • Renormalization group describes scale dependence β(gi) = μ∂gi/∂μ for coupling constants gi • Asymptotic freedom: β<0 (e.g. QCD) Gauge Theory • Lagrangian has local gauge invariance under eiα(x) • Covariant derivative Dμ = ∂μ + iqAμ preserves invariance • Gauge fixing (e.g. ∂μAμ=0) to quantize • BRST symmetry in gauge-fixed Lagrangian • Ghost fields to maintain unitarity • Gauge-invariant observables Spontaneous Symmetry Breaking • Minimum of potential breaks a symmetry of Lagrangian • Massless Goldstone bosons for broken global symmetry • Higgs mechanism: gauge bosons acquire mass, eat Goldstone modes • Higgs boson is massive excitation of condensate • Elitzur's theorem: local gauge symmetry can't break spontaneously This map hits many of the key concepts and equations of quantum field theory, from the basic fields and Lagrangians to gauge symmetry and renormalization. The equations describe the dynamics and quantization of fields, the perturbative expansion and Feynman diagrams, and important phenomena like spontaneous symmetry breaking. Of course, many more technical details could be added, but this provides a high-level overview of much of the key mathematical machinery used in the theory." "Here is a gigantic detailed map of equations used in optics, organized by topic: ``` Optics Equations Map ==================== 1. Geometric Optics 1.1 Reflection - Law of Reflection: θ_i = θ_r 1.2 Refraction - Snell's Law: n_1 sin θ_1 = n_2 sin θ_2 - Critical Angle: θ_c = sin^(-1)(n_2 / n_1) 1.3 Thin Lenses - Lens Maker's Equation: 1/f = (n_lens - 1) * (1/R_1 - 1/R_2) - Thin Lens Equation: 1/f = 1/d_o + 1/d_i - Magnification: M = -d_i / d_o = h_i / h_o 1.4 Mirrors - Mirror Equation: 1/f = 1/d_o + 1/d_i - Magnification: M = -d_i / d_o = h_i / h_o - Focal Length (Spherical Mirror): f = R / 2 2. Wave Optics 2.1 Interference - Path Difference: Δ = d_2 - d_1 - Constructive Interference: Δ = mλ (m = 0, ±1, ±2, ...) - Destructive Interference: Δ = (m + 1/2)λ (m = 0, ±1, ±2, ...) 2.2 Diffraction - Single Slit Diffraction: sin θ = mλ / a (m = 0, ±1, ±2, ...) - Double Slit Diffraction: d sin θ = mλ (m = 0, ±1, ±2, ...) - Diffraction Grating: d sin θ = mλ (m = 0, ±1, ±2, ...) - Rayleigh Criterion: sin θ ≈ 1.22 λ / D 2.3 Polarization - Malus' Law: I = I_0 cos^2 θ - Brewster's Angle: tan θ_B = n_2 / n_1 3. Electromagnetic Optics 3.1 Maxwell's Equations - Gauss' Law for Electric Fields: ∇ · E = ρ / ε_0 - Gauss' Law for Magnetic Fields: ∇ · B = 0 - Faraday's Law: ∇ × E = -∂B/∂t - Ampère's Circuital Law (with Maxwell's correction): ∇ × B = μ_0 (J + ε_0 ∂E/∂t) 3.2 Electromagnetic Waves - Wave Equation for Electric Field: ∇^2 E = μ_0 ε_0 ∂^2 E / ∂t^2 - Wave Equation for Magnetic Field: ∇^2 B = μ_0 ε_0 ∂^2 B / ∂t^2 - Speed of Light in Vacuum: c = 1 / √(μ_0 ε_0) - Poynting Vector: S = E × H 4. Fourier Optics 4.1 Fourier Transforms - Fourier Transform: F(ω) = ∫ f(t) e^(-iωt) dt - Inverse Fourier Transform: f(t) = (1/2π) ∫ F(ω) e^(iωt) dω 4.2 Fraunhofer Diffraction - Fraunhofer Diffraction Formula: U(x, y) = (e^(ikz) / iλz) ∬ U(ξ, η) e^(-ik(xξ+yη)/z) dξdη 4.3 Fresnel Diffraction - Fresnel Diffraction Formula: U(x, y) = (e^(ikz) / iλz) ∬ U(ξ, η) e^(ik((x-ξ)^2+(y-η)^2)/2z) dξdη 5. Fiber Optics 5.1 Fiber Modes - Normalized Frequency (V-number): V = (2πa/λ) √(n_core^2 - n_clad^2) - Cutoff Wavelength: λ_c = (2πa/V_c) √(n_core^2 - n_clad^2) 5.2 Dispersion - Chromatic Dispersion: D = (λ/c) (d^2n/dλ^2) - Polarization Mode Dispersion: Δτ = L * Δβ_1 5.3 Nonlinear Effects - Kerr Effect: n = n_0 + n_2 * I - Self-Phase Modulation: ΔΦ = (2π/λ) * n_2 * I * L 6. Quantum Optics 6.1 Photon Statistics - Bose-Einstein Distribution: <n> = 1 / (e^(hν/kT) - 1) - Planck's Law: I(ν, T) = (2hν^3 / c^2) * (1 / (e^(hν/kT) - 1)) 6.2 Quantum States of Light - Coherent States: |α> = e^(-|α|^2/2) ∑ (α^n / √n!) |n> - Squeezed States: |ξ> = e^((ξ* a^2 - ξ a^†2) / 2) |0> 6.3 Quantum Entanglement - Bell States: |Φ±> = (|00> ± |11>) / √2, |Ψ±> = (|01> ± |10>) / √2 - Entanglement Measures: Concurrence, Negativity, Entanglement of Formation 7. Optical Materials 7.1 Dispersion Relations - Sellmeier Equation: n^2(λ) = 1 + ∑ (B_i * λ^2 / (λ^2 - C_i)) - Cauchy's Equation: n(λ) = A + B/λ^2 + C/λ^4 7.2 Nonlinear Optics - Second Harmonic Generation: P^(2) = ε_0 * χ^(2) * E^2 - Third Harmonic Generation: P^(3) = ε_0 * χ^(3) * E^3 - Pockels Effect: Δn = -rE / 2 - Kerr Effect: Δn = λ * K * E^2 ``` This map covers a wide range of optics equations, from geometric optics and wave optics to electromagnetic optics, Fourier optics, fiber optics, quantum optics, and optical materials. Each section includes key equations and concepts related to that area of optics. The equations are written using standard mathematical notation and symbols commonly used in the field of optics." "Here is a detailed map of many key equations used in quantum mechanics: Schrödinger Equation: iℏ ∂Ψ/∂t = ĤΨ - Fundamental equation of quantum mechanics - Ĥ is the Hamiltonian operator - Ψ is the wavefunction Time-Independent Schrödinger Equation: ĤΨ = EΨ - Eigenvalue equation for stationary states - E is the energy eigenvalue Position and Momentum Operators: x̂ = x p̂ = -iℏ ∂/∂x - x̂ is the position operator - p̂ is the momentum operator Commutator: [x̂,p̂] = iℏ - Fundamental commutation relation - Leads to Heisenberg uncertainty principle Uncertainty Principle: ΔxΔp ≥ ℏ/2 - Relationship between uncertainties in position (Δx) and momentum (Δp) Expectation Values: ⟨Â⟩ = ∫Ψ*ÂΨ dx - Average value of an observable A - Ψ* is the complex conjugate of Ψ Ehrenfest Theorem: d⟨x̂⟩/dt = ⟨p̂⟩/m d⟨p̂⟩/dt = -⟨∂V/∂x⟩ - Relates expectation values to classical quantities Spin Operators: Ŝx = (ℏ/2)σx, Ŝy = (ℏ/2)σy, Ŝz = (ℏ/2)σz - Ŝx, Ŝy, Ŝz are spin angular momentum operators - σx, σy, σz are Pauli matrices Spin Eigenvalues: Ŝz|↑⟩ = (ℏ/2)|↑⟩, Ŝz|↓⟩ = -(ℏ/2)|↓⟩ - |↑⟩ and |↓⟩ are spin eigenstates - Eigenvalues are +ℏ/2 and -ℏ/2 Angular Momentum Commutation Relations: [Ĵx,Ĵy] = iℏĴz, [Ĵy,Ĵz] = iℏĴx, [Ĵz,Ĵx] = iℏĴy - Ĵx, Ĵy, Ĵz are angular momentum operators Angular Momentum Eigenvalues: Ĵ2|j,m⟩ = j(j+1)ℏ2|j,m⟩, Ĵz|j,m⟩ = mℏ|j,m⟩ - |j,m⟩ are angular momentum eigenstates - j is the total angular momentum quantum number - m is the magnetic quantum number Perturbation Theory: Ĥ = Ĥ0 + Ĥ' En = E(0)n + E(1)n + E(2)n + ... - Ĥ0 is the unperturbed Hamiltonian - Ĥ' is a small perturbation - En is the nth energy level with corrections Selection Rules: Δl = ±1, Δm = 0,±1, ΔS = 0 - Rules for allowed transitions - Based on conservation of angular momentum Fermi's Golden Rule: Γi→f = (2π/ℏ)|⟨f|Ĥ'|i⟩|2ρ(Ef) - Transition rate from initial state |i⟩ to final states |f⟩ - ρ(Ef) is density of final states Density Matrix: ρ = ∑i pi|Ψi⟩⟨Ψi| - Describes a quantum statistical ensemble - pi is probability of state |Ψi⟩ There are many more equations, but this covers a broad overview of key ones used across quantum mechanics, including the fundamental Schrödinger equation, operators and expectation values, spin and angular momentum, perturbation theory, selection rules, transition rates, and the density matrix formalism. Let me know if you would like me to elaborate on any part of this!" "Here is a detailed map of many key equations used in fluid mechanics: ```markdown # Fluid Mechanics Equations ## Fluid Properties - Density: ρ = m/V - Specific Weight: γ = ρg - Specific Gravity: SG = ρ/ρ_ref - Viscosity: - Dynamic Viscosity: τ = μ(du/dy) - Kinematic Viscosity: ν = μ/ρ - Bulk Modulus: K = -V(dp/dV) ## Fluid Statics - Pressure: - Absolute Pressure: p_abs = p_gauge + p_atm - Hydrostatic Pressure: p = ρgh - Pascal's Law: Δp = ρgΔh - Archimedes' Principle: F_buoyant = ρgV_displaced ## Fluid Kinematics - Velocity Field: V(x, y, z, t) = u(x,y,z,t)i + v(x,y,z,t)j + w(x,y,z,t)k - Acceleration Field: a = ∂V/∂t + (V·∇)V - Continuity Equation: - Differential Form: ∂ρ/∂t + ∇·(ρV) = 0 - Integral Form: dm/dt = ∫∫ρV·dA - Stream Function (2D incompressible): u = ∂ψ/∂y, v = -∂ψ/∂x - Velocity Potential (irrotational flow): V = ∇ϕ ## Fluid Dynamics - Navier-Stokes Equations: - ρ(∂V/∂t + (V·∇)V) = -∇p + ρg + μ∇²V - Euler Equations (inviscid flow): - ρ(∂V/∂t + (V·∇)V) = -∇p + ρg - Bernoulli Equation: - Steady, Incompressible, Inviscid Flow Along a Streamline: p/ρ + V²/2 + gz = constant - Darcy-Weisbach Equation (pipe flow): hf = f(L/D)(V²/2g) - Hagen-Poiseuille Equation (laminar pipe flow): Q = (πΔpR⁴)/(8μL) ## Dimensional Analysis - Reynolds Number: Re = ρVD/μ = VD/ν - Froude Number: Fr = V/√(gL) - Mach Number: M = V/c ## Boundary Layers - Boundary Layer Thickness: δ = 4.91x/√Re_x - Displacement Thickness: δ* = ∫(1 - u/U)dy - Momentum Thickness: θ = ∫(u/U)(1 - u/U)dy - Skin Friction Coefficient: Cf = τ_wall/(0.5ρU²) ## Turbulence - Reynolds-Averaged Navier-Stokes (RANS) Equations: - ρ(∂V̄/∂t + (V̄·∇)V̄) = -∇p̄ + ρḡ + ∇·(μ∇V̄ - ρ(V'V')) - Turbulence Kinetic Energy: k = 0.5(u'² + v'² + w'²) - Turbulence Dissipation Rate: ε = ν(∂u_i'/∂x_j)(∂u_i'/∂x_j) ## Compressible Flow - Speed of Sound: c = √(∂p/∂ρ) - Isentropic Flow Relations: - p/p₀ = (ρ/ρ₀)^γ - T/T₀ = (p/p₀)^((γ-1)/γ) - Normal Shock Relations: - ρ₂/ρ₁ = (γ+1)M₁²/((γ-1)M₁² + 2) - p₂/p₁ = (2γM₁² - (γ-1))/(γ+1) ## Potential Flow - Laplace Equation: ∇²ϕ = 0 - Velocity Potential for Uniform Flow: ϕ = Ux - Velocity Potential for Source/Sink: ϕ = ±(λ/2π)ln(r) - Velocity Potential for Doublet: ϕ = -(κ/2π)(cosθ/r) - Complex Potential: F(z) = ϕ(x,y) + iψ(x,y) ## Computational Fluid Dynamics (CFD) - Finite Volume Method: - ∂/∂t∫∫∫QdV + ∫∫F·dA = ∫∫∫SdV - Finite Difference Method: - ∂u/∂x ≈ (u(i+1,j) - u(i-1,j))/(2Δx) - ∂²u/∂x² ≈ (u(i+1,j) - 2u(i,j) + u(i-1,j))/(Δx²) ``` This map covers key equations from various areas of fluid mechanics, including fluid properties, fluid statics, kinematics, dynamics, dimensional analysis, boundary layers, turbulence, compressible flow, potential flow, and computational fluid dynamics. Each section includes fundamental equations and principles that are essential for analyzing and understanding fluid flow phenomena. This map serves as a comprehensive reference for engineers and researchers working in the field of fluid mechanics."