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.