https://mathlets.org/mathlets/ * Home * About * Training * Mathlets * Courses * Activities * Demos Mathlets * Affine Coordinate Changes [affine-coordinate-changes] Affine Coordinate Changes The graph of the function f(mx+b) is related to the graph of f(x) in interesting ways. * Amplitude and Phase: First Order [amplitude-and-phase-first-order] Amplitude and Phase: First Order The tide in a harbor lags behind that of the open ocean, and is controlled by a first order linear equation. Bode and Nyquist plots illustrate the steady state and method of solution. * Amplitude and Phase: Second Order I [amplitude-and-phase-second-order-i] Amplitude and Phase: Second Order I A spring drives sinusoidally a spring/dashpot/mass system. The predictable amplitude and phase lag of the sinusoidal system response can be understood using Bode and Nyquist plots. * Amplitude and Phase: Second Order II [amplitude-and-phase-second-order-ii] Amplitude and Phase: Second Order II A dashpot drives sinusoidally a spring/dashpot/mass system. The predictable amplitude and phase lag of the sinusoidal system response can be understood using Bode and Nyquist plots. * Amplitude and Phase: Second Order III [amplitude-and-phase-second-order-iii] Amplitude and Phase: Second Order III Both the spring and the dashpot drive sinusoidally a spring/ dashpot/mass system. The predictable amplitude and phase lag of the sinusoidal system response can be understood using Bode and Nyquist plots. * Amplitude and Phase: Second Order IV [Screen-Shot-2015-02-13-at-1] Amplitude and Phase: Second Order IV A sinusoidally varying force acts directly on the mass in a spring/dashpot/mass system. The predictable amplitude and phase lag of the sinusoidal system response can be understood using Bode and Nyquist plots * Amplitude Response: Pole Diagram [amplitude-response-pole-diagram] Amplitude Response: Pole Diagram The exponential response of an LTI system is determined by its transfer function W(s), and roughly by the pole diagram of W(s). The amplitude response or gain is the restriction to the imaginary axis of |W(s)|. * Ballistic Trajectory [Screen-Shot-2015-01-24-at-11] Ballistic Trajectory A thrown stone responds to the forces of gravity and drag. * Beats [Screen-Shot-2015-02-13-at-1] Beats Beats occur when two sinusoids superimpose. The beat frequency is captured by an envelope. * Beats with Sound [Screen-Shot-2017-10-07-at-10] Beats with Sound Beats with sound, zoom, and units. * Beta Distribution [Screen-Shot-2015-01-07-at-8] Beta Distribution The beta-distribution depends on two parameters. * Bode and Nyquist Plots [Screen-Shot-2015-02-13-at-1] Bode and Nyquist Plots The system or transfer function determines the frequency response of a system, which can be visualized using Bode Plots and Nyquist Plots. The pole/zero diagram determines the gross structure of the transfer function. * Complex Arithmetic [Screen-Shot-2015-03-30-at-9] Complex Arithmetic Complex numbers and operations on them can be visualized on the complex plane. * Complex Exponential [complex-exponential] Complex Exponential The complex exponential function sends straight lines through the origin to spirals. * Complex Roots [complex-roots] Complex Roots Any nonzero complex number has n distinct nth roots. * Confidence Intervals [Screen-Shot-2015-01-07-at-8] Confidence Intervals Confidence intervals are range estimates computed from data. * Conjugate Priors [Screen-Shot-2015-01-07-at-8] Conjugate Priors Bayesian updating is easy with conjugate priors. * Convolution: Accumulation [convolution-accumulation] Convolution: Accumulation The convolution integral is the superposition of unit impulse responses. * Convolution: Flip and Drag [convolution-flip-and-drag] Convolution: Flip and Drag Convolution at t is computed by integrating the signal weighted by the time reversal of the unit impulse response dragged to start at time t. * Coupled Oscillators [Screen-Shot-2015-01-04-at-10] Coupled Oscillators Two masses and three springs make an interesting dance. For equations, see the Theory page. * Creating the Derivative [creating-the-derivative] Creating the Derivative The graphs of f'(x) and of f"(x) reflect interesting features of the graph of f(x). * Damped Vibrations [damped-vibrations] Damped Vibrations The decay from initial condition to equilibrium of an unforced second order system can be understood using the roots of the characteristic polynomial and the phase diagram. * Damped Wave Equation [Screen-Shot-2016-01-11-at-10] Damped Wave Equation The vibration of a plucked string dies off because of damping, but can still be understood via Fourier series. * Damping Ratio [damping-ratio] Damping Ratio The damping ratio and natural frequency of a second order LTI system are determined by the roots of the characteristic polynomial. Initial conditions determine the phase plane trajectory. * Discrete Fourier Transform [discrete-fourier-transform] Discrete Fourier Transform * Eigenvalue Stability [eigenvalue-stability] Eigenvalue Stability An eigenvalue analysis of stability of difference schemes leads to a variety of relationships between the gain and the discrete eigenvalue. * Euler's Method [eulers-method] Euler's Method Given an initial condition and step size, an Euler polygon approximates the solution to a first order differential equation. * Forced Damped Vibration [forced-damped-vibration] Forced Damped Vibration The solution to a sinusoidally driven LTI system depends on the initial conditions, and is the sum of a steady state solution and a transient. * Forced Damped Vibration: Phasors [forced-damped-vibration-phasors] Forced Damped Vibration: Phasors A solution of the ODE representing a driven spring/mass/dashpot system represents a balance of forces. If the driving force is sinusoidal, these various forces also vary sinusoidally, and the balance may be represented using phasors (i.e. complex numbers). * Fourier Coefficients [fourier-coefficients] Fourier Coefficients The initial terms of a Fourier series give the root mean square best fit. Symmetry properties of the target function determine which Fourier modes are needed. * Fourier Coefficients: Complex with Sound [fourier-coefficients-complex-with-sound] Fourier Coefficients: Complex with Sound The coefficients in a Fourier series, when it is viewed as a sum of complex exponetials, are best thought of in terms of their magnitude and argument. You can only hear the magnitudes, even though the arguments greatly influence the waveform. * Graph Features [graph-features] Graph Features Portions of graphs of cubic polynomials rise, fall, or are convex or concave. * Graphing Rational Functions [Screen-Shot-2015-01-07-at-9] Graphing Rational Functions The poles and zeros of a rational function let you make a rough sketch of its graph. * Harmonic Frequency Response: Variable Input Frequency [harmonic-frequency-response-variable-inpu] Harmonic Frequency Response: Variable Input Frequency The periodic frequency response of a harmonic oscillator to a periodic signal depends upon the frequency of the signal. * Harmonic Frequency Response: Variable Natural Frequency [harmonic-frequency-response-variable-natu] Harmonic Frequency Response: Variable Natural Frequency The periodic response of a tunable harmonic oscillator to a periodic signal depends upon its natural frequency. * Heat Equation [heat-equation] Heat Equation The evolution of the temperature distribution on an insulated bar can be understood in terms of the Fourier decomposition of the initial condition. * Hypocycloids [Screen-Shot-2018-01-30-at-11] Hypocycloids Wheels within (or outside of) wheels! * Impulse Response: Natural Frequency [impulse-response-natural-frequency] Impulse Response: Natural Frequency The natural angular frequency and damping ratio determine the system responses to delta, step, and ramp input signals. * Impulse Response: Spring System [impulse-response-spring-system] Impulse Response: Spring System The system parameters determine the system response of a spring system to delta, step, and ramp input signals. * Isoclines [isoclines] Isoclines Graphs of solutions of a first order equation can be understood in terms of the slope field and isoclines. * Linear Phase Portraits: Cursor Entry [linear-phase-portraits-cursor-entry] Linear Phase Portraits: Cursor Entry The phase portrait of a homogeneous linear autonomous system depends mainly upon the trace and determinant of the matrix, but there are two further degrees of freedom. * Linear Phase Portraits: Matrix Entry [linear-phase-portraits-matrix-entry] Linear Phase Portraits: Matrix Entry The type of phase portrait of a homogeneous linear autonomous system -- a companion system for example -- depends on the matrix coefficients via the eigenvalues or equivalently via the trace and determinant. * Linear Programming [Screen-Shot-2015-05-18-at-7] Linear Programming How do you maximize a linear objective function subject to linear constraints? * Linear Regression [Screen-Shot-2015-04-06-at-3] Linear Regression Data can be fit by a class of functions using least squares. * Linearized Trigonometry [linearized-trigonometry] Linearized Trigonometry For small angles the sine is approximately the angle, but there are higher terms in the Maclauren expansion. * Matrix Vector [matrix-vector] Matrix Vector The product of a matrix and a vector depends in interesting ways on the entries of each. Eigenvectors represent a coincidence of direction. * Periodic Box [Screen-Shot-2017-10-07-at-10] Periodic Box An impulse train is approximated by "periodic box functions," and the system response to the box function converges to the response to the impulse train as the boxes get narrower. * Phase Lines [phase-lines] Phase Lines The nonlinear autonomous equation x' = g(x) can be understood in terms of the graph of g(x) or the phase line. As a parameter in g (x) varies, the critical points on the phase line describe a curve on the bifurcation plane. * Poles and Vibrations [poles-and-vibrations] Poles and Vibrations A wide range of waveforms occur as the sum of two damped oscillations. The long-term behavior is reflected by the pole diagram of the Laplace transform. * Probability Distributions [Screen-Shot-2015-01-07-at-8] Probability Distributions Different probability distributions are useful in different situations. * Riemann Sums [riemann-sums] Riemann Sums An integral can be approximated as a sum in many ways. * Secant Approximation [secant-approximation] Secant Approximation Secants converge to tangent lines. * Series RLC Circuit [series-rlc-circuit] Series RLC Circuit System responses of a sinusoidally driven RLC circuit can be understood by means of phasors. * Sinusoids [Screen-Shot-2018-07-20-at-6] Sinusoids Any sinusoidal function is a distorted cosine. * Solution Targets [solution-targets] Solution Targets Sometimes solutions converge as time increases, and sometimes they diverge, making the Uniquenss Theorem surprising. * T Distribution [Screen-Shot-2015-01-07-at-9] T Distribution The t distribution depends on one parameter. * Tangent Approximation [Screen-Shot-2015-05-27-at-9] Tangent Approximation Smooth graphs are close to linear when you blow them up enough. * Taylor Polynomials [taylor-polynomials] Taylor Polynomials Most functions are well approximated near any point by a sequence of polynomials. * Trigonometric Identity [trigonometric-identity] Trigonometric Identity Any linear combination of cosine and sine (with the same frequency) is again sinusoidal. The amplitude and phase lag of the sum are related to the coefficients of cosine and sine by means of polar coordinates. * Vector Fields [vector-fields] Vector Fields Nonlinear autonomous systems can have complicated solutions, which can be represented with some loss of information by their trajectories. Usually they behave nearly linearly near equlibria. * Wave Equation [Screen-Shot-2015-01-24-at-12] Wave Equation A plucked string can be analyzed following either Fourier or d'Alembert. * Wheel [wheel] Wheel A light on a wheel rim traces a curve which may be understood by vector addition. Copyright (c) 2009--2015 H. Miller | Powered by WordPress Except where otherwise noted, Attribution-NonCommercial-ShareAlike content on this site is licensed 3.0 Unported under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license Accessibility