Complete Interactive Mathematical Functions, Surfaces & Solvers Directory

Models underdamped mechanical springs and RLC circuits where exponential envelope e^(−ax) bounds rapid cosine oscillations.

Normalized probability density function with total area equal to 1, peak at x = μ, and inflection points at x = μ ± σ.

Bounded S-shaped activation and population carrying-capacity curve with inflection symmetry at y = L/2.

Ideal low-pass Nyquist reconstruction kernel in DSP; its square sinc²(x) governs single-slit optical diffraction fringes.

Rational bell curve studied by Maria Gaetana Agnesi (1748), identical in form to spectral Lorentzian line shapes and Cauchy distributions.

Exact equilibrium shape of a uniform flexible cable hanging under gravity, compared side-by-side against its osculating parabola.

Demonstrates exact pole branch splitting at x = ±1, roots at x = ±2, y-intercept at (0, 4), and horizontal asymptote y = 1.

Visualizes how local maximum at x = −√a and minimum at x = +√a align directly with the roots of f′(x) = 3x² − 3a.

Superposition of two close frequencies produces rapid carrier waves modulated by slow beat envelope 2·cos(Δω·x/2).

Minimax polynomial on [−1, 1] exhibiting six alternating ±1 extrema that suppress Runge’s phenomenon in polynomial interpolation.

Balances Pauli short-range r⁻¹² repulsion against London dispersion r⁻⁶ attraction with equilibrium minimum at r = 2^(1/6)σ.

Compares smooth non-monotonic neural network activations that eliminate ReLU dead-neuron gradients.

Illustrates continuous curves with sharp corner points and vertical-tangent cusps where classical derivatives fail to exist.

Plots analytic continuation of factorials Γ(n) = (n−1)! across negative poles alongside bounded integral erf(x).

Generates k petals when k is odd and 2k petals when k is even; tune slider b live to watch petal topology transform.

Transitions smoothly from convex dimpled limaçon (a > b) to cusp cardioid (a = b) and inner-loop limaçon (a < b).

Successive turns maintain constant radial separation Δr = 2πb, used in scroll compressors, watch springs, and antenna arrays.

Famous 1989 transcendental polar curve combining exponential, harmonic 4θ, and subharmonic sin⁵((2θ−π)/24) wing modulation.

Unified polar equation of planetary orbits with focus at the pole: circle (e=0), ellipse (0<e<1), parabola (e=1), hyperbola (e>1).

Orthogonal harmonic oscillations used on laboratory oscilloscopes to measure frequency ratios a:b and phase difference δ.

Locus of a point on a circle of radius a/4 rolling inside a fixed circle of radius a; satisfies x^(2/3) + y^(2/3) = a^(2/3).

Generated by rolling a unit circle around the outside of a circle of radius k, foundational to cycloidal gear tooth geometry.

Bernoulli’s curve of fastest descent under uniform gravity and Huygens’ tautochrone pendulum path.

Jakob Bernoulli’s self-similar spiral cutting all radial rays at constant angle cot⁻¹(b), observed in nautilus shells and spiral galaxies.

Doubly-ruled negative Gaussian curvature surface curving upward along x and downward along y with canonical saddle point at (0,0,0).

Axisymmetric 2D Fourier transform of a circular aperture, displaying central Airy-like peak surrounded by concentric annular ripples.

Strictly convex quadratic surface with positive definite Hessian H = diag(2a, 2b) and nested elliptical contour isolines.

Real part of complex cubic Re(z³) possessing three alternating valleys and ridges where both gradient ∇f and Hessian vanish at (0,0).

Multi-modal optimization test landscape featuring one central local maximum and four distinct global minima equal to zero.

Narrow parabolic valley where finding the valley floor is easy, but converging to the global minimum at (1, 1) challenges first-order optimizers.

Antisymmetric smooth scalar potential exhibiting paired global maximum and minimum separated across the nodal line x = 0.

Eigenfunction solution of the 2D Helmholtz wave equation on a rectangular drumhead with orthogonal nodal grid lines.

Negative normalized Laplacian of a 2D Gaussian kernel used across computer vision and seismology for multi-scale edge detection.

Parabolic bowl superposed with periodic cosine lattice creating dozens of deceptive local minima trapping greedy solvers.

Conical surface smooth everywhere except the apex singularity (0,0), exhibiting perfectly concentric equidistant circular contours.

Upper hemisphere of radius R = 3 demonstrating normal vector alignment along position vector (x, y, z) and steep equatorial rim.

Fundamental Green’s function of the 2D Laplace equation Δu = 0 governing line-charge electrostatics and ideal fluid sources.

Visualizes complex argument arg(x + iy) spiraling smoothly around the origin up to its branch-cut jump along the negative x-axis.

Landau free-energy surface displaying unstable central peak, four saddle passes, and four symmetric degenerate ground-state wells.

Directionally selective localized wave packet achieving the Heisenberg uncertainty bound simultaneously in space and spatial frequency.

Electronic energy band surface of a 2D square crystal lattice showing Van Hove saddle singularities along diamond Fermi contours.

Heavy-tailed rational surface modeling focused laser beam intensity profiles and 2D nuclear magnetic resonance spectral peaks.

Produces nested hyperbolic crest-and-trough fringes seen in astigmatic optical wavefront interferometry.

Smoothly regularized conoid oscillating as sin(2θ) around every circle, illustrating directional limit dependence near the origin.

Transforms any matrix into canonical Reduced Row Echelon Form using exact GCD-simplified rational elementary row operations.

Solves 2x₁ + x₂ − x₃ = 8, −3x₁ − x₂ + 2x₃ = −11, −2x₁ + x₂ + 2x₃ = −3 to exact solution (2, 3, −1) with zero residual.

Solves diagonally dominant 4-node steady-state Laplace finite-difference / resistor mesh equations with exact rational fractions.

Identifies pivot vs. free variables when rank(A) = rank([A|b]) < n and constructs full vector solution x = x_p + t₁v₁ + t₂v₂.

Demonstrates step-by-step how rank(A) < rank([A|b]) produces an explicit contradiction row 0 = 4, proving zero solutions exist.

Computes exact determinant via Gaussian elimination while tracking row-swap parity (−1)^s and parallelepiped volume orientation.

Avoids 24-term Leibniz cofactor expansion by reducing 4×4 matrices cleanly to four exact rational diagonal pivots.

Augments A with the identity matrix I₃ and executes full Gauss–Jordan elimination to extract exact rational A⁻¹.

Recovers exact integer inverse [[9,−36,30],[−36,192,−180],[30,−180,180]] and det(H₃) = 1/2160 without floating-point rounding loss.

Reveals R₃ − 2R₂ + R₁ = 0 during row reduction, proving rank(A) = 2 < 3 and det(A) = 0.

Computes characteristic polynomial λ³ − 12λ² + 45λ − 54 = 0, spectral eigenvalues λ = 6, 3, 3, and real eigenvectors.

Explains why spiral rotation-dilation matrices have negative discriminant Δ < 0 and complex conjugate eigenvalues α ± βi.

Analyzes doubly stochastic transition matrices where dominant eigenvalue λ₁ = 1 governs stationary equilibrium and |λ₂| controls convergence speed.

Extracts pivot columns of A for Col(A) and special free-variable vectors for Null(A), verifying rank(A) + nullity(A) = n.

Decomposes square matrices into permutation P, unit lower-triangular L, and upper-triangular U factors for fast multi-RHS solves.

Orthonormalizes the columns of classic Golub–Van Loan benchmark [[12,−51,4],[6,167,−68],[−4,24,−41]] into orthogonal Q and upper-triangular R.

Determines exact cubic coefficients c₀ + c₁x + c₂x² + c₃x³ passing through (−1,2), (0,1), (1,2), and (2,11).

Binomial-coefficient matrix Pᵢⱼ = C(i+j, i) whose LU triangular factors have unit diagonals, forcing det(Pₙ) = 1 for every dimension n.

Applies symbolic Product and Chain rules step-by-step, locating stationary extrema at x = 0 and x = 4 plus concavity inflections.

Derives exact rational first and second derivatives, curvature κ(x₀), and bounded global extrema without pole singularities.

Finds all stationary maxima, minima, roots, and concavity sign-flips across [−π, π] with live tangent line visualization.

Classic integration-by-parts identity where terms x/(1+x²) cancel algebraically to leave f′(x) = arctan(x) and f″(x) = 1/(1+x²).

Evaluates ∫₋₂² e^(−x²) dx = √π·erf(2) ≈ 1.76416278 across 6 quadrature rules, revealing exponential accuracy gain.

Shades and integrates the famous ±1.96σ two-tailed statistical threshold to 7-digit Gauss–Legendre precision.

Rapidly chirping optical diffraction integral testing midpoint/trapezoid cancellation errors against 192-node Gauss–Legendre quadrature.

Simultaneously verifies area = 2π ≈ 6.283185, arc circumference = 2π, and solid-of-revolution sphere volume V = (4/3)π(2)³ ≈ 33.5103.

Classic benchmark integral evaluating to exactly 1, illustrating concave-down trapezoidal underestimation vs. midpoint overestimation.

Computes Si(π) ≈ 1.85193705, exceeding π/2 ≈ 1.570796 by the exact 8.95% Fourier Gibbs overshoot ratio.

Overlays odd-degree alternating Maclaurin approximations onto sin(x), showing how each added term hugs another half-wavelength.

Every derivative f^(k)(0) = 1 generates clean reciprocal-factorial coefficients converging across all real x.

Visually demonstrates why Taylor polynomials match ln(1+x) tightly inside (−1, 1] yet diverge wildly for x > 1 due to the pole at x = −1.

Canonical geometric expansion with unit coefficients cₖ = 1 bounded strictly by the vertical asymptote at x = 1.

Synthesizes odd sine harmonics 1, 3, 5, … up to N = 25, measuring RMS energy error and persistent 9% jump-discontinuity ringing.

Contains both odd and even integer harmonics decaying as 1/n, foundational to subtractive audio synthesizer waveforms.

Continuity of the triangle wave accelerates Fourier decay to 1/n², reaching near-visual perfection in just 5 harmonics.

Extracts DC average a₀/2 = 2/π ≈ 0.63662 plus even ripple harmonics cos(2nx)/(4n² − 1) for AC-DC filter capacitor design.

All trajectories exponentially approach horizontal equilibrium asymptote y = 1 regardless of initial temperature y(0).

Autonomous phase portrait contrasting unstable extinction equilibrium y = 0 against stable carrying capacity y = 3.

Initial states y(0) > 0 snap to upper attractor y = +2 while y(0) < 0 snap to y = −2, modeling Schmitt triggers and genetic switches.

Shows transient initial offset decaying away until all RK4 trajectories lock onto the same phase-lagged sinusoidal steady state.

Compares Euler’s 1st-order drift against 4th-order Runge–Kutta (RK4) across symmetric Gaussian peak trajectories.

Non-separable quadratic ODE appearing in optimal LQR control theory and WKB quantum tunneling approximations.

Reveals seasonal limit-cycle oscillation above critical biomass vs. sudden population collapse below the saddle-node separatrix.

Unlike symmetric logistic growth (inflection at K/2), Gompertz growth inflects earlier at y = K/e ≈ 0.368K.

Transforms into a linear ODE via substitution u = y^(1−n) = y⁻², providing an exact closed-form check on RK4 numerical precision.

Multiplying by μ(x) = x + 4 yields exact derivative d/dx[(x+4)y] = (x+4)cos(x), solved alongside interactive RK4 trajectories.

Demonstrates Hubbard’s antifunnel/funnel theorem as trajectories converge tightly onto the upper parabolic branch y ≈ √(x/α).

Illustrates periodic potential valleys with viscous damping pulling trajectories into stable phase basins.