Interactive Mathematics & Teaching Tools

A Catalog of Browser-Native Resources

Shelvean Kapita · Department of Mathematics, Texas A&M University · kapita@tamu.edu

Every tool below runs entirely in the browser with no installation and no server-side processing: all computation happens client-side. The collection is built in single-file HTML and JavaScript, designed to meet WCAG 2.1 AA accessibility standards, with source code released under the MIT License and educational content under CC BY 4.0. Tool names are hyperlinked — click any name to open it. Each tool is listed exactly once, under a single class. This catalog lists 73 tools.

◆ Mathematical Tools

Differential Equations 10

  1. Direction Field Plots the slope field for any first-order ODE \(dy/dx = f(x, y)\) and overlays solution curves. The field redraws in real time as the function is edited.
  2. Phase Line Plot Analyzes equilibria and stability for autonomous first-order ODEs \(dy/dt = f(y)\). Classifies each fixed point as stable, unstable, or semi-stable on the phase line.
  3. Mass-Spring-Damping Simulator Simulates a single mass on a spring with a damper (dashpot), governed by \(m\ddot{x} + c\dot{x} + kx = 0\), across the underdamped, critically damped, and overdamped regimes. Shows the displacement curve and the phase-plane trajectory as the mass oscillates and settles.
  4. Multiple Degree of Freedom Mass Spring Simulates a chain of masses linked in series by springs and resolves the system’s normal modes of vibration. Reveals how energy transfers back and forth between the coupled masses, producing beats.
  5. Forced Vibrations and Resonance Simulates a mass-spring-damper driven by an external periodic force, \(m\ddot{x} + c\dot{x} + kx = F_0\cos(\omega t)\). Shows the transient and steady-state response and how the driving frequency shapes the amplitude and produces resonance.
  6. Laplace Transform Visualizer Renders the Laplace transform as a signed-area integral whose weighted integrand accumulates toward \(F(s)\). Tracks convergence and illustrates the first and second shifting theorems.
  7. Convolution Visualizer Animates the flip-and-slide construction as \(g(t - \tau)\) moves across \(f(\tau)\). The overlapping area builds up the convolution \(h(t)\) in real time.
  8. Linear Phase Portraits Classifies fixed points of planar linear systems \(x' = Ax\) through eigenvalue analysis. Draws nodes, spirals, saddles, and centers from the matrix you enter.
  9. Nonlinear Phase Portraits Plots vector fields and nullclines for nonlinear autonomous systems. Locates fixed points and assesses their local stability via linearization.
  10. Newton’s Law of Cooling Simulates a warm object losing heat to its cooler surroundings, governed by \(dT/dt = -k(T - T_a)\), as its temperature decays toward the ambient value \(T_a\). Includes realistic object visuals, a temperature colormap, steam effects, and a live numerical solution.

Dynamical Systems 17

  1. Damped Pendulum Simulator Simulates a rigid pendulum — a bob swinging on a rod under gravity with frictional damping — governed by \(\ddot{\theta} + (b/m)\dot{\theta} + (g/L)\sin\theta = 0\). Contrasts small-angle linear behavior with the full large-angle nonlinear dynamics and draws the phase portrait.
  2. Elastic (Spring) Pendulum Simulates a mass on a spring that can both swing and stretch. Demonstrates energy exchange between radial and angular modes, autoparametric resonance, and quasi-periodic orbits.
  3. Coupled Pendula with Spring Models two pendulums joined by a spring. Reveals normal modes, energy transfer, and beat phenomena.
  4. Fourier Series Approximation Builds periodic functions from partial sums of sines and cosines. Lets you vary the number of terms and watch the approximation converge.
  5. Lotka-Volterra Model Simulates the classic predator-prey cycle in phase space. Shows the periodic orbits linking predator and prey densities.
  6. Lotka-Volterra with Carrying Capacity Extends the predator-prey model with logistic growth limits on the prey. Illustrates how carrying capacity alters or stabilizes the dynamics.
  7. Lorenz System Visualizes the Lorenz attractor, a three-variable idealization of atmospheric convection rolls, and its sensitive dependence on initial conditions. Lets you vary \(\sigma\), \(\rho\), and \(\beta\) to reshape the chaotic flow.
  8. Multi-Pendulum Simulator Simulates double and triple pendulums — chains of rigid rods linked end to end and swinging under gravity — with chaotic trajectory traces. Highlights the extreme sensitivity to initial conditions of this Lagrangian system.
  9. Particle in a Cubic Potential Studies motion and phase-space orbits in an asymmetric cubic potential well. Shows bounded and unbounded trajectories depending on initial energy.
  10. Particle in a Double Well Potential Explores bistability in a symmetric double-well potential. Demonstrates how initial energy decides which well the particle settles into.
  11. Bifurcation Diagram Sweeps a parameter to reveal period-doubling cascades and the onset of chaos, with points colored by Lyapunov exponent. Resolves the Feigenbaum structure and supports custom maps.
  12. Bifurcation Family Explorer Walks through the four canonical one-dimensional normal forms — saddle-node, transcritical, and supercritical/subcritical pitchfork — pairing the phase line of \(\dot{x} = f(x; r)\) with its \((r, x^*)\) bifurcation diagram in a coupled view. Drag or animate the parameter \(r\) to watch equilibria appear, collide, exchange stability, and jump with hysteresis in the subcritical case.
  13. Pendulum Wave Machine Tunes a row of pendulums so each completes a set number of swings per cycle, producing snake, butterfly, and wave patterns. Includes a 3D view, a displacement wave strip, live pattern detection, and a preset library.
  14. Van der Pol Oscillator Simulates the Van der Pol oscillator — a nonlinear oscillator with amplitude-dependent damping, first introduced to model triode (vacuum-tube) electronic circuits — governed by \(\ddot{x} + \mu(x^2 - 1)\dot{x} + x = 0\). Explores its limit cycles and relaxation oscillations with a hybrid RK4/SDIRK2 solver for stiff regimes at large \(\mu\).
  15. Duffing Oscillator Simulates a driven mass on a nonlinear hardening/softening spring whose double-well potential allows two stable rest positions, governed by \(\ddot{x} + \delta\dot{x} + \alpha x + \beta x^3 = \gamma\cos(\omega t)\). Plots phase portraits, time series, and potential energy with an adaptive RK45 solver.
  16. Logistic Map Explorer Draws cobweb diagrams for the logistic, tent, and sine maps and traces the full period-doubling route to chaos. Offers preset parameters, a live Lyapunov exponent, a regime indicator, and step-by-step orbit animation.
  17. Cobweb Diagram Visualizes fixed-point iteration for general one-dimensional maps. Shows convergence, periodic orbits, and chaos with interactive map selection and parameter control.

Linear Algebra 29

  1. Geometry of Linear Systems Shows the row picture and column picture for \(2 \times 2\) linear systems side by side. Connects the algebra of a system to its geometric solution.
  2. 2D Matrix Transformation Visualizer Animates rotation, reflection, scaling, shear, projection, and translation acting on objects in the plane. Also displays the SVD of the transformation and lets you compose several maps.
  3. 2D SVD Visualization Breaks the action of any \(2 \times 2\) matrix into three geometric steps: rotate \((V^{\top})\), scale \((\Sigma)\), rotate \((U)\). Shows how rank deficiency collapses shapes onto a line.
  4. Coordinate Transform Visualizer Animates change-of-basis for vectors in the plane. Demonstrates how a vector’s coordinates change between different bases.
  5. Matrix Product Computes matrix multiplication and interprets it as the composition of linear transformations. Visualizes how the product map acts.
  6. Determinant and Trace Computes the determinant and trace of small matrices through elementary operations. Explains what each quantity reveals about the matrix.
  7. RREF Solver Performs step-by-step Gauss-Jordan elimination to reach reduced row echelon form. Highlights the pivot at each stage.
  8. Matrix Inverse Finds the inverse of a square matrix via augmented row reduction. Walks through the elimination steps that produce it.
  9. Elementary Matrices Shows how each elementary row operation corresponds to multiplication by an elementary matrix. Makes the matrix algebra behind row reduction explicit.
  10. REF Solver Performs Gaussian elimination to compute the row echelon form. Displays the sequence of row operations.
  11. Change of Coordinates (Transition Matrix) Converts a vector’s coordinates between two bases using the transition matrix. Computes the change-of-basis matrix for you.
  12. Similarity Transforms Finds the matrix representation of a linear operator in a chosen basis. Supports similarity transforms between representations.
  13. Gram-Schmidt Orthogonalization Applies the Gram-Schmidt process to produce an orthogonal or orthonormal basis. Displays each projection-and-subtraction step.
  14. Matrix Diagonalization Diagonalizes a matrix using its eigenvectors whenever diagonalization is possible. Returns the eigenvector matrix and the diagonal factor.
  15. Cholesky Factorization Computes the Cholesky decomposition \(A = LL^{\top}\) for symmetric positive-definite matrices. Shows the factorization steps.
  16. Four Fundamental Subspaces Computes bases for the column space, null space, row space, and left null space. Ties the four subspaces back to row reduction.
  17. \(A = CR\) Factorization Computes the rank-revealing \(A = CR\) factorization. Exposes the column space and row space directly.
  18. Magic Factorization Computes a rank-revealing factorization of a matrix. Presents the resulting factors.
  19. Linear Equation Solver Solves a linear system with detailed step-by-step row reduction. Reports the full solution set, including any free variables.
  20. Matrix Inverse or Pseudo-Inverse Computes the ordinary inverse when it exists, or the Moore-Penrose pseudo-inverse otherwise. Handles non-square and rank-deficient matrices.
  21. \(PA = LU\) Factorization Computes the LU decomposition with partial and scaled-partial pivoting. Shows the permutation together with the triangular factors.
  22. QR Factorization Computes the orthogonal-triangular decomposition \(A = QR\). Displays the orthogonal and upper-triangular factors.
  23. Least Squares Solver using QR Solves the least-squares problem via QR decomposition. Offers the numerically stable route for well-conditioned problems.
  24. Least Squares Solver using Normal Equations Solves least squares by forming and solving the normal equations \(A^{\top}A\,x = A^{\top}b\). Shows the classical, fast approach.
  25. Least Squares Solver using SVD Solves the least-squares problem through the singular value decomposition. Handles rank-deficient and ill-conditioned systems gracefully.
  26. Eigenvalues and Eigenvectors Computes the eigenvalues and eigenvectors of a real matrix. Reports the spectrum and the corresponding eigenvectors.
  27. Singular Value Decomposition (SVD) Computes the full SVD \(A = U\Sigma V^{\top}\). Returns the singular values and the orthogonal factor matrices.
  28. Determinant Visualizer Depicts the determinant as the oriented area of a 2D parallelogram or the oriented volume of a 3D parallelepiped. Builds the figure from the column vectors of the matrix you enter.
  29. Matrix-Vector Multiplication Computes \(Ax\) interactively and visualizes the result. For \(2 \times 2\) matrices it also displays the eigenvalues and eigenvectors.

Linear Programming 1

  1. Method of Corners (Graphical Method) Constructs the feasible region for a two-variable linear program, identifies its corner points, and finds the optimum. Visualizes how the objective function selects the optimal vertex.

Numerical Methods 7

  1. Bisection Method Animated root-finding by repeated interval halving, with interval zoom and an iteration table. Includes several worked examples.
  2. Newton’s Method Visualizer Animates Newton’s iterative root-finding and the tangent-line steps. Lets you observe convergence behavior and failure cases.
  3. Polynomial Interpolation Constructs and plots the interpolating polynomial through a set of data points. Shows how the curve passes exactly through each node.
  4. Polynomial Least Squares Data Fitting Fits polynomials to data by least squares. Plots the resulting fit and analyzes the residuals.
  5. Rational Function Least Squares Data Fitting Approximates data with rational functions via least-squares optimization. Useful in cases where polynomial fits perform poorly.
  6. Finite Difference Generator Generates finite-difference stencils for derivatives of any order. Reports the stencil weights together with a truncation-error estimate.
  7. Numerical Integration Approximates definite integrals with Left, Right, Midpoint, Trapezoidal, Simpson, and Gauss-Legendre rules, plus adaptive Simpson and Romberg extrapolation. Shades signed areas, compares convergence rates, and finds the subdivisions needed for a target tolerance.

Business Mathematics 1

  1. TVM Solver (FinanceCalc Pro) A Time-Value-of-Money calculator for present value, future value, payment, interest rate, and number of periods. Generates amortization schedules and supports continuous compounding.

◆ Administrative & Classroom Utilities

  1. Exam Timer / Time Display A large-digit classroom clock and countdown timer for exams, presentations, and timed activities. Runs entirely in the browser with simple start, stop, and reset controls.
  2. Grades Statistics Calculator Computes a grade distribution from an uploaded roster (CSV, XLSX, TXT, or ODS) under several letter-grade schemes. Produces summary statistics and a distribution chart for the class.
  3. Seat Allocator Spreads a given number of students evenly across a chosen Texas A&M classroom, avoiding broken seats and assigning left-handed seats as needed. Produces a fresh random arrangement on each run and exports both a roster (TXT) and a seating map (PDF).

◆ PDF & Document Tools

All PDF and document tools process files entirely in the browser; nothing is uploaded to any server.

  1. PDF & Image Merger Combines multiple PDFs and images into a single document. Supports drag-and-drop reordering, page preview, and download of the merged result.
  2. PDF Organizer Reorders, rotates, and deletes individual pages within a PDF. Drag pages into the desired sequence and export the reorganized document.
  3. PDF Extract Pages Pulls selected pages or page ranges out of a PDF into a new file. Pick pages from a thumbnail grid and download the extract.
  4. PDF Darken Darkens faded or lightly scanned PDF pages to improve readability. Adjust contrast and brightness, then export the enhanced document.
  5. Office to PDF Viewer Opens DOCX and XLSX files directly in the browser and converts them to PDF. Requires no Office software and uploads nothing to the cloud.