Fields

Type any formula for a vector field F = (Fx, Fy, Fz) or a scalar field f. Choose an operator to visualize its gradient, divergence, curl (rotation) or Laplacian, computed live.

To read off circles or geometric structure, raise Arrow grid N, or set Arrow layout → plane for a dense 2-D slice of arrows (pick the plane axis & position); streamlines trace the field lines directly.

Values at a point: move the point P (magenta marker) and the readout shows the actual numbers behind the graph: ∇·F, ∇×F, ∇²F (or ∇f, ∇²f for a scalar) at P, with ◀ in graph marking whichever operator you're currently displaying.

Line integral ∮F·dr: enable it, give a curve r(t) and a t-range (or use a preset), and the app computes the work of your field along the path, colours the curve by F·T̂, and animates a trace point. For a closed loop it flags whether the circulation is ≈ 0, a hint that F is conservative (has a potential φ).

Drop bodies into the field: position P, drop a die-faced cube, and release it. The same arrows mean different physics depending on the reading of F:

• Velocity field (flow): the body is advected, ẋ = F(x), and (Cauchy–Stokes) it spins with ω = ½∇×F, its volume changes at the rate ∇·F, and with Deform on it carries the full deformation gradient Ȧ = (∇F)A (shear + stretch; det A is the exact volume factor). Try Rigid rotation (orbits and spins), Simple shear (tumbles at rate ½ while sliding), and the Irrotational vortex: bodies orbit the axis but never rotate, the classic paddle-wheel surprise (∇×F = 0 there).
• Force field (Newton): m·ẍ = F(x) with a chosen mass and launch velocity v₀: orbits, oscillations, escape. A point mass does not spin here: the curl of a force field measures non-conservativity, not rotation. For a scalar field the body feels F = −∇f (f as potential energy) and the readout shows E = ½m|v|² + f staying constant.

If the field animates in t, the bodies' clock locks to the field clock: one shared time. Bodies that leave 3× the domain fade out and stop.

Field designer: pick a property and hit Generate; the app constructs a random field for which the property holds as a mathematical identity, and shows the construction. Conservative fields come as F = ∇φ (so ∇×F ≡ 0 because curl∘grad ≡ 0), incompressible ones as F = ∇×A (div∘curl ≡ 0), Laplace fields from a harmonic potential (both at once), uniform source / vorticity fields carry a prescribed constant ∇·F or ∇×F, and the confined eddy is built from a stream function ψ whose level curves are the field lines. For scalar fields: harmonic functions, Laplacian eigenfunctions (∇²f = −k²f), and localized bumps. Each generated field arrives with a “why it works” note and a suggested experiment: the recipe is the lesson.

Variables & syntax

• Coordinates: x, y, z, and time t.
• Handy shortcuts: r=√(x²+y²+z²), rho=√(x²+y²), phi, theta, r2=x²+y²+z².
• Constants: pi, tau, e. Power is ^. Implicit products work: 2x, 3sin(x).
• Also: absolute value |x| − |y|, superscripts x² = x^2, and juxtaposed coordinates xyz = x·y·z (an exponent binds to the last letter: xy² = x·y²).
• Functions: sin cos tan asin acos atan atan2 sinh cosh tanh exp ln log10 log2 sqrt cbrt abs sign floor ceil round min max mod clamp step smoothstep hypot gauss sinc.

Comparisons & regions

Comparisons < <= > >= == != can be part of a formula: each evaluates to 1 (true) or 0 (false). This lets you build indicator/characteristic functions, piecewise definitions and region masks:

• Indicator of a set: (x^2+y^2 < 1), 1 inside the unit disk, 0 outside.
• Chained (a mathematician's interval): 0 <= x <= 1, or an annulus 1 < x^2+y^2 < 4.
• Piecewise: (x<0)*(-x) + (x>=0)*x = |x|;   mask a field: -y*(rho<2).
• Combine regions: multiply for AND, max for OR, 1- for NOT, or use and(a,b), or(a,b), not(a).
• Proper piecewise: if(cond, a, b) or cases(c₁, v₁, c₂, v₂, …, else); only the active branch is evaluated, so if(||x,y|| != 0, x*y/||x,y||, 0) is exactly 0 at the origin (no 0/0). Derivatives are taken branch-wise.
• Norms: ||a, b, …|| is the current norm (Euclidean by default; switch it in the Functions lab: 1-norm, max-norm, p-norm). Fixed variants: norm(…), norm1(…), norminf(…), normp(p, …).

Comparisons are piecewise-constant, so a gradient/Taylor of a masked function is that of the smooth part inside the region and 0 outside; 1-D plots break the line cleanly across a jump.

Operators

• ∇f gradient: steepest-ascent vector field of a scalar.
• ∇·F divergence: scalar source/sink density.
• ∇×F curl: local rotation axis & strength.
• ∇² Laplacian: of a scalar or (component-wise) a vector.

Matrix Lab

A 3×3 matrix A acts on space. See the deformed unit cube, the images of the basis vectors (the columns of A), the eigenvectors, and the animated flow x(t) = exp(tA)·x₀. The readout shows determinant, trace, rank, eigen-decomposition and the matrix exponential.

Functions

Pick a type in the dropdown; the panel shows only what's relevant:

• Scalar f(x)/f(x,y)/f(x,y,z): curve / surface / volume, with a Taylor approximation (drag the expansion point), plus gradient, Jacobian (=∇fᵀ) and Hessian.
• Vector F(x,y,z): a vector field whose Jacobian is a full 3×3 matrix; the readout gives det J, ∇·F (= trace) and ∇×F (antisymmetric part), and the Jacobian's local action is drawn as a deformed cube.

(Line integrals ∮F·dr over a curve live in the Fields tab, where the field is defined.)

Total derivative. The section below Taylor treats df = ∇f·h as what it is: a linear map. Show its graph as the tangent plane at P, and switch on the analysis to test total differentiability numerically: the remainder f(P+hv) − f(P) − h∇f·v divided by h is measured over shrinking h in every direction; it must vanish, in all directions at once. The direction probe slices the surface along a chosen v and compares the true one-sided slope D_v f with the prediction ∇f·v. Try the preset xy/‖(x,y)‖ at 0 with P at the origin: all partial derivatives exist, yet the remainder sticks at ½ along the diagonal: partials existing does not imply differentiability. For a vector map the total derivative is the Jacobian, and the same remainder test applies to F.

Constraints & constrained extrema (Extrema unter Nebenbedingungen). Each row of the constraint list is one relation; several rows hold simultaneously (AND). An inequality like x^2+y^2 <= 4 or 2 < x^2+y^2 < 9 clips the graph cleanly to the region where it holds (boundary cells are cut exactly at the edge); an equality like x^2+y^2 = 4 draws the constraint curve on the surface. Every boundary is drawn in red with the Lagrange candidates marked in yellow: the points where ∇f = λ·∇g (sign changes of ∇f × ∇g along the curve). The readout lists each candidate with f, λ and a max/min tag. Try saddle + circle g = 4 (maxima f = 4 at (±2, 0) with λ = 1, minima f = −4 at (0, ±2) with λ = −1) and paraboloid on disk for the inequality case. For f(x,y,z) the constraint set is a surface: it is drawn (marching tetrahedra) coloured by f, so the constrained extrema appear as its hottest and coldest spots; candidates come from Newton on ∇f = λ·∇g, g = c and carry ∇f ∥ ∇g arrows. Try plane on sphere r = 3: max f = 9 at (2,−1,−2) with λ = ½.

Continuity (Stetigkeit). The continuity section classifies f on the plotted domain into Lipschitz ⊂ uniformly continuous ⊂ continuous, each with a reason: suspected discontinuities are refined by bisection (a gap that survives every refinement is a jump; exploding values mark a pole), and the Lipschitz bound L = sup ‖∇f‖ is verified by zooming in on the steepest point: if it stabilises, the mean value theorem gives |f(x)−f(y)| ≤ L|x−y|; if it keeps growing (try sqrt(|x|)), no L exists, yet Heine–Cantor still gives uniform continuity on the compact domain. The class depends on the domain: 1/x is not uniformly continuous on (0, R] but becomes Lipschitz with the constraint x >= 1, and for x² an outlook warns that the slope keeps growing beyond the box: on all of ℝ it would not be uniformly continuous.

All derivatives and Taylor terms come from automatic differentiation. Axis scale x/y/z stretches the view to reveal detail along an axis.

Manifolds

Explore submanifolds of ℝ³ and their geometry. Pick a kind:

• Parametric surface φ(u,v): coloured by Gaussian curvature K (red = elliptic, blue = hyperbolic). The readout gives the tangent vectors, normal, first & second fundamental forms, K, mean H, principal curvatures, area, and the Euler characteristic χ via Gauss–Bonnet (sphere → 2, torus → 0).
• Level set g(x,y,z)=c: a real isosurface (marching tetrahedra) with the ∇g normal field and critical points. The readout tells you whether c is a regular value: if so, the level set is a smooth 2-submanifold (regular value theorem); at a critical value it can be singular (try the cone). The tangent plane is drawn at the footpoint Q (the point P projected onto the surface by Newton steps along ∇g) where T_Q M = ker dg(Q) = ∇g(Q)⊥: exactly the directions the total derivative sends to 0.
• Curve r(t): the Frenet frame T (red), N (green), B (blue), with curvature κ, torsion τ, and arc length. Press Play to run the frame along the curve.

Surfaces use u, v as parameters; move the point to see how the local geometry changes.

Minkowski (spacetime)

A flat 1+1 spacetime diagram of special relativity: the pseudo-Euclidean sibling of the Matrix lab. Time runs up (ct), space across (x), and light travels at 45°. A Lorentz boost is a hyperbolic rotation Λ = exp(φK): same exp-of-a-generator as R = exp(θK), but K is symmetric and cos/sin → cosh/sinh, so it preserves the interval s² = (ct)² − x² instead of length.

• Frame velocity β: drag it (or press Space to animate) and watch the moving frame’s ct′, x′ axes scissor toward the light cone. Rapidity φ = artanh β is the “angle”; it adds under composition, which is exactly relativistic velocity addition.
• Calibration hyperbolae s² = ±k² mark the true unit lengths of the tilted axes. Read time dilation and length contraction off them (Euclidean distance on the page is misleading here).
• Lines of simultaneity (ct′ = const) are parallel to x′: two events on one are simultaneous in the moving frame but generally not in the lab.
• Worldlines are constant-velocity particles (0 = at rest/vertical, ±1 = light). Events show their coordinates in both frames and the invariant s² (timelike / spacelike / lightlike).
• Scenarios set up simultaneity, time dilation, length contraction, the light cone, velocity addition and the twin paradox.

This tab uses a fixed, undistorted 2-D view (no orbit) so 45° really means 45°. R resets the range.

Quantum (Schrödinger 1-D)

The time-independent Schrödinger equation −½ψ″ + V(x)ψ = Eψ (units ℏ = m = 1), solved by diagonalising the finite-difference Hamiltonian. Type any potential V(x):

• Eigenstates: the energy levels Eₙ (horizontal lines) with each wavefunction ψₙ drawn on its level; toggle |ψ|² for the probability density. Nodes increase with n. Try harmonic (equally spaced), infinite/finite well, double well (tunnelling pairs), Morse.
• Wave packet: launch a Gaussian packet (start x₀, width σ, momentum k₀) and press ▶/Space. It is expanded in the eigenbasis, so ψ(x,t)=Σ cₙψₙe^{−iEₙt} is exact. See it spread, bounce, or tunnel through a barrier when ⟨E⟩ is below the top: small grey dots mark the classical turning points V = ⟨E⟩, beyond which a classical particle could never go.
• Superposition and beats: an equal mix of two eigenstates ψₙ₁ + ψₙ₂. One eigenstate alone is stationary; the pair sloshes at the beat frequency ω = E₂ − E₁ (the readout gives the period 2π/ΔE): quantum motion comes from energy differences.

Phase space (classical mechanics)

A 1-degree-of-freedom system ẍ = a(x, v, t) is a flow on the phase plane (x, v): ẋ = v, v̇ = a. Type the acceleration a using x, the velocity v, and time t.

• The direction field, trajectories (RK4) and fixed points (classified as centre / saddle / node / spiral from the Jacobian) are drawn live; for a conservative force, energy contours ½v²+U(x)=E are the exact orbits.
• Presets: pendulum (separatrix, libration vs rotation), damped & driven pendulum, the double well (Duffing), the van der Pol limit cycle. Drag the start point and press ▶/Space to launch a trajectory.

Fourier

Any periodic f is a sum of sines and cosines: S_N = a₀/2 + Σ aₙcos(nπx/L) + bₙsin(nπx/L).

• Series: watch the partial sum S_N converge as you raise N (or press ▶/Space); near a jump it overshoots by ~9 % forever: the Gibbs phenomenon. The bar chart is the amplitude spectrum √(aₙ²+bₙ²). Try square, sawtooth, triangle.
• Transform: F(k)=∫f e^{−ikx}dx. A narrow signal has a wide spectrum: the readout shows Δx·Δk, minimal for a Gaussian: the mathematical seed of the uncertainty relation.

Waves (1-D PDE evolution)

Separation of variables, animated. The initial profile u₀(x) on [0, L] (ends held at zero) is projected once onto the sine modes, aₙ = (2/L)∫u₀ sin(nπx/L)dx, and then every mode evolves exactly, no time-stepping, no numerical error:

• Wave u_tt = c²u_xx: mode n oscillates at ωₙ = cnπ/L (all multiples of ω₁, why a string plays a note). Initial velocity v₀ supported. Try the piano-hammer preset. Energy is conserved exactly.
• Heat u_t = D·u_xx: mode n decays as e^(−Dkₙ²t); high harmonics die ~n² faster, which is why diffusion smooths sharp edges first. Watch the spectrum bars vanish top-down.
• Schrödinger iψ_t = −½ψ_xx: the same u₀, but nothing decays; phases rotate at Eₙ ∝ n², the shape scrambles (dispersion) while ∫|ψ|² stays exactly 1, and in a box the packet even revives.

Feeding the same bump to all three equations is the whole lesson: identical modes, different physics.

Modes (kleine Schwingungen)

N masses and springs written in matrix form M·ü = −K·u (mass matrix M, stiffness matrix K; both shown for small N). The ansatz u = φe^(iωt) gives the generalized eigenvalue problem (K − ω²M)φ = 0, symmetrised as K̃ = M^(−1/2)K M^(−1/2) and diagonalised with the same QL eigensolver as the Schrödinger lab. Modes are M-orthonormal, so any initial condition splits as cₙ = φₙᵀMu(0) and evolves exactly.

• Two pendula: displace one; energy sloshes between them with beat period 2π/Δω.
• Uniform chain: mode shapes are sampled sine waves; the dispersion view plots ωₙ against q and matches ω = 2√(k/m)·sin(q/2): the doorway to phonons.
• Diatomic chain (m, 3m): acoustic + optical branches with a band gap.
• Free ends: translation symmetry ⇒ a zero mode ω₁ = 0 (the chain drifts as a whole).
• Pendulum term g/ℓ: adds gravity restoring to every mass; pure chain physics becomes coupled pendula.

Scatter (Streuung & Wirkungsquerschnitt)

A parallel beam of particles (energy E = ½v∞², impact parameter b) hits a central potential V(r). The deflection function is computed from the exact classical scattering integral Θ(b) = π − 2∫ b·du/√(1 − b²u² − V/E) (u = 1/r, turning-point singularity removed analytically), and the beam view integrates real RK4 trajectories with true timestamps: particles visibly slow near the turning point. The two methods agree, and the self-tests prove both against closed forms.

• dσ/dΩ = (b/sin θ)·|db/dθ| is the Jacobian of the map b → θ: it counts how much beam area lands in each solid angle. No single trajectory has a cross-section: only the family does. Where Θ(b) has a flat spot (dθ/db = 0) the cross-section spikes: rainbow scattering (try Lennard-Jones).
• Rutherford V = k/r: Θ = 2·atan(k/2Eb), dσ/dΩ = (k/4E)²/sin⁴(θ/2) (green analytic overlay). Head-on rays stop at r₀ = k/E: how the nucleus was sized. The attractive −1/r gives the identical cross-section: scattering can't tell + from −.
• Hard sphere: isotropic dσ/dΩ = R²/4, total σ = πR², the only case where the cross-section is just the geometric shadow. Long-range potentials (Coulomb) have divergent classical σ_tot.
• Attractive potentials can capture rays (spirals into the centre, gaps in Θ(b)).
• Annotate a ray: pick one impact parameter and the graph labels the textbook triple: the Stoßparameter b (bracket between beam axis and incoming asymptote), r_min (segment from the centre to the closest approach), and the Streuwinkel θ (arc between the undeflected line and the outgoing asymptote). Slide b through the beam and watch θ and r_min respond: the deflection function traced by hand.
• Annotated Wirkungsquerschnitt: the beam view also draws the tube b ± db (blue companion rays with a db bracket) opening into the dθ wedge: the beam-annulus area 2πb·db landing in the solid-angle ring 2π·sinθ·dθ, whose ratio is dσ/dΩ. The readout computes the per-ray value from a local derivative of the exact integral (with the analytic comparison for Rutherford / hard sphere), and the same ray appears as a labelled marker on the Θ(b) and dσ/dΩ plots: one slider, three views, one ray.

The newest labs

The tab bar is grouped into Math and Physics. In brief:

• Charges: point-charge electrostatics with field lines, equipotentials and a movable Gauss sphere: its numeric flux ∮E·dA is compared live against 4π·Q_enc.
• Spin: a qubit on the Bloch sphere: exact Larmor precession about Ω, probabilities, and measurement buttons that collapse the state.
• Atom: the real hydrogen orbitals (n ≤ 3): probability cloud or sign-coloured |ψ|² isosurface, with Eₙ, ⟨r⟩ and node counts.
• Rigid body: build a body from parts (boxes, spheres, cylinders, rings; the T-handle and tennis racket are presets): the app assembles and diagonalises the inertia tensor, draws the principal axes, and you choose the spin axis, then watch the Dzhanibekov flip about the unstable middle one. Place the rotation centre where you like: the COM at the origin, the body left at its built position, or pinned at a point you pick: the tensor is Steiner-shifted there and the COM orbits the pin.
• Kepler: central-force orbits with the Laplace–Runge–Lenz vector, all three Kepler laws, the V_eff view, and perihelion precession for p ≠ 2.
• Chaos: the double pendulum: diverging twin trajectories and a Poincaré section.
• Sequences: the ε–N game, series as partial sums, and the ε-tube that separates pointwise from uniform convergence.
• Complex: domain colouring and grid images of complex maps, with a numeric Cauchy–Riemann check for conformality.

Theme

Use the ☀ / ☾ button (top bar) to switch between dark and light mode. Your choice is remembered.

Saving a plot

Click ⤓ Save (top bar) to download the current 3-D view as a high-resolution PNG image, captioned with the expression you're plotting, ready to drop into notes or a report.

Custom points

The Custom points panel (bottom of the sidebar, always available) lets you drop colour-coded markers at exact coordinates to gauge where structures sit. Points persist across every tab; click a point's colour swatch to change its colour, delete points individually, or clear all.

Plotting several functions

In the Functions tab (types f(x) and f(x,y)), use + Add function to overlay extra graphs in one view, each in its own colour and sharing one vertical scale for comparison. The first (primary) function keeps the full Taylor / gradient / Hessian analysis.

Controls

Left-drag orbit · right-drag (or Shift+drag) pan · wheel zoom · Space play/pause · R reset view.

Every slider's value box is editable. Click it and type an exact number (then Enter) instead of dragging to the right spot. Number boxes accept constant expressions too: pi, 2pi, pi/4, sqrt(2), 1/3. This works in matrix cells and t/u/v ranges too.