Calculus×Computing
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⚛ Computational physics

Fields and waves: vector calculus, Fourier analysis and partial differential equations, up to fluids.

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  1. 01

    Derivative

    f′(a)f'(a) is the instantaneous rate of change of ff at aa: the slope of the tangent line to the graph, defined as the limit of slopes of secant lines.

    FundamentalDerivatives
  2. 02

    Ordinary differential equations

    An equation relating an unknown function to its derivatives, y′=f(t,y)y' = f(t, y). Its solutions are trajectories: given where you start, the equation says where you go next — the mathematical model of anything that evolves in time.

    UniversityDifferential equations
  3. 03

    Second-order linear equations (oscillations)

    mx¨+cx˙+kx=F(t)m\ddot x + c\dot x + kx = F(t): springs, pendulums, circuits, suspensions. The roots of the characteristic equation mr2+cr+k=0m r^2 + c r + k = 0 decide whether the system oscillates (complex roots), returns smoothly (real roots) or resonates.

    UniversityDifferential equations
  4. 04

    Gradient

    ∇f=(∂f∂x1,…,∂f∂xn)\nabla f = \left(\frac{\partial f}{\partial x_1}, \dots, \frac{\partial f}{\partial x_n}\right): the vector of all partial derivatives. It points in the direction of steepest ascent, its length is that steepest slope, and it is perpendicular to the level sets. Walk against it and you go downhill fastest.

    UniversityMultivariable calculus
  5. 05

    Divergence

    ∇⋅F=∂xP+∂yQ+∂zR\nabla\cdot F = \partial_x P + \partial_y Q + \partial_z R: the net outflow per unit volume at a point. Positive at sources, negative at sinks, zero for incompressible flow.

    UniversityVector calculus
  6. 06

    Curl

    ∇×F\nabla\times F: a vector measuring the local rotation of a field — its axis is the axis of spin, its length twice the angular speed. Fields that are gradients have zero curl.

    UniversityVector calculus
  7. 07

    Laplacian

    Δf=∇⋅∇f=∑i∂2f/∂xi2\Delta f = \nabla\cdot\nabla f = \sum_i \partial^2 f/\partial x_i^2: how much ff at a point differs from the average of its neighbours. It appears in the heat, wave, Laplace, Poisson and Schrödinger equations, and its discrete version smooths meshes and detects edges.

    UniversityVector calculus
  8. 08

    Divergence theorem (Gauss)

    The total divergence inside a region equals the flux out through its boundary: what is produced inside must leave through the surface. It is the foundation of conservation laws and finite-volume methods.

    AdvancedVector calculus
  9. 09

    Fourier series

    Any reasonable periodic function is a sum of sines and cosines of multiples of a base frequency: f(t)=∑ncne2πint/Tf(t) = \sum_n c_n e^{2\pi i n t/T}. The coefficients cnc_n — the spectrum — say how much of each harmonic it contains.

    AdvancedTransforms
  10. 10

    Fourier transform

    The continuous version for non-periodic signals: f^(ξ)=∫f(t) e−2πiξt dt\hat f(\xi) = \int f(t)\,e^{-2\pi i\xi t}\,\dd t gives the amount of each frequency ξ\xi. It turns convolution into multiplication and differentiation into multiplication by 2πiξ2\pi i\xi — which is why filtering, compression and solving linear PDEs are easier in frequency space.

    AdvancedTransforms
  11. 11

    Partial differential equations

    Equations for fields that depend on space and time, involving partial derivatives: heat diffusion, waves, fluid flow, electromagnetism, quantum mechanics. Solved numerically by discretizing space (finite differences, finite volumes, finite elements, spectral methods).

    AdvancedDifferential equations
  12. 12

    Heat equation and diffusion

    ut=αΔuu_t = \alpha\Delta u: temperature (or concentration, or probability) flows from high to low and smooths out. Solved by finite differences or Fourier series; Gaussian blur of an image is exactly running this equation.

    AdvancedPhysics and simulation
  13. 13

    Wave equation

    utt=c2Δuu_{tt} = c^2\Delta u: disturbances travel at speed cc. Sound, vibrating strings, seismic waves and light obey it; games use discretized versions for water ripples and room acoustics.

    AdvancedPhysics and simulation
  14. 14

    Fluid dynamics and CFD

    The Navier–Stokes equations for a velocity field uu and pressure pp. Engineering CFD solves them on meshes with finite volumes; film and games use "stable fluids" (Stam, 1999): advect, add forces, and project to zero divergence by solving a Poisson equation.

    SpecializationPhysics and simulation
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