Mathlets

  • 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
    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
    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
    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
    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

    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
    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

    Ballistic Trajectory
    A thrown stone responds to the forces of gravity and drag.

  • Beats

    Beats
    Beats occur when two sinusoids superimpose. The beat frequency is captured by an envelope.

  • Beats with Sound

    Beats with Sound
    Beats with sound, zoom, and units.

  • Beta Distribution

    Beta Distribution
    The beta-distribution depends on two parameters.

  • Bode and Nyquist Plots

    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

    Complex Arithmetic
    Complex numbers and operations on them can be visualized on the complex plane.

  • Complex Exponential

    Complex Exponential
    The complex exponential function sends straight lines through the origin to spirals.

  • Complex Roots

    Complex Roots
    Any nonzero complex number has n distinct nth roots.

  • Confidence Intervals

    Confidence Intervals
    Confidence intervals are range estimates computed from data.

  • Conjugate Priors

    Conjugate Priors
    Bayesian updating is easy with conjugate priors.

  • Convolution: Accumulation

    Convolution: Accumulation
    The convolution integral is the superposition of unit impulse responses.

  • 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

    Coupled Oscillators
    Two masses and three springs make an interesting dance. For equations, see the Theory page.

  • 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
    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

    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
    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

  • 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

    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
    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
    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
    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
    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
    Portions of graphs of cubic polynomials rise, fall, or are convex or concave.

  • Graphing Rational Functions

    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 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 Natural Frequency
    The periodic response of a tunable harmonic oscillator to a periodic signal depends upon its natural frequency.

  • 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

    Hypocycloids
    Wheels within (or outside of) wheels!

  • 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
    The system parameters determine the system response of a spring system to delta, step, and ramp input signals.

  • 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
    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
    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

    Linear Programming
    How do you maximize a linear objective function subject to linear constraints?

  • Linear Regression

    Linear Regression
    Data can be fit by a class of functions using least squares.

  • 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
    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

    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
    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
    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

    Probability Distributions
    Different probability distributions are useful in different situations.

  • Riemann Sums

    Riemann Sums
    An integral can be approximated as a sum in many ways.

  • Secant Approximation

    Secant Approximation
    Secants converge to tangent lines.

  • Series RLC Circuit

    Series RLC Circuit
    System responses of a sinusoidally driven RLC circuit can be understood by means of phasors.

  • Sinusoids

    Sinusoids
    Any sinusoidal function is a distorted cosine.

  • Solution Targets

    Solution Targets
    Sometimes solutions converge as time increases, and sometimes they diverge, making the Uniquenss Theorem surprising.

  • T Distribution

    T Distribution
    The t distribution depends on one parameter.

  • Tangent Approximation

    Tangent Approximation
    Smooth graphs are close to linear when you blow them up enough.

  • Taylor Polynomials

    Taylor Polynomials
    Most functions are well approximated near any point by a sequence of polynomials.

  • 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
    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

    Wave Equation
    A plucked string can be analyzed following either Fourier or d'Alembert.

  • Wheel

    Wheel
    A light on a wheel rim traces a curve which may be understood by vector addition.