Courses

Precalculus

  • 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.

  • Beats

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

  • Graphing Rational Functions

    Graphing Rational Functions
    The poles and zeros of a rational function let you make a rough sketch of its graph.

  • Linear Programming

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

  • 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.

Calculus

    1. Differentiaton

    • 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.

    • Creating the Derivative

      Creating the Derivative
      The graphs of f'(x) and of f"(x) reflect interesting features of the graph of f(x).

    • Secant Approximation

      Secant Approximation
      Secants converge to tangent lines.

    • Tangent Approximation

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

    2. Higher derivatives

    • Graph Features

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

    • Linearized Trigonometry

      Linearized Trigonometry
      For small angles the sine is approximately the angle, but there are higher terms in the Maclauren expansion.

    • Taylor Polynomials

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

    3. Integration

    • Riemann Sums

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

    4. Vectors and matrices

    • 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.

    • Wheel

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

    Differential Equations

    1. First order equations

    • Ballistic Trajectory

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

    • Isoclines

      Isoclines
      Graphs of solutions of a first order equation can be understood in terms of the slope field and isoclines.

    • Solution Targets

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

    2. More first order equations

    • 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.

    • 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.

    • 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.

    3. Complex Numbers

    • 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.

    • 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.

    4. Second Order Linear Equations

    • 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

    • 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.

    • 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.

    • Series RLC Circuit

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

    5. Convolution, impulse response

    • 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.

    • 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.

    6. Fourier series

    • 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.

    • 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.

    7. Laplace Transform

    • 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)|.

    • 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.

    8. Linear systems

    • Coupled Oscillators

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

    • 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.

    • 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.

    9. Partial Differential Equations

    • 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.

    • 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.

    • Wave Equation

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

    Advanced applications of differential equations

    • 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)|.

    • 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.

    • 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

    • 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.

    • 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.

    • Nyquist Plot

      Nyquist Plot
      As frequency sweeps through a range of values, the corresponding complex gain sweeps out a curve in the complex plane which reveals stability characteristics of the underlying system.

    • Series RLC Circuit

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

    Probability and Statistics

    • Beta Distribution

      Beta Distribution
      The beta-distribution depends on two parameters.

    • Confidence Intervals

      Confidence Intervals
      Confidence intervals are range estimates computed from data.

    • Conjugate Priors

      Conjugate Priors
      Bayesian updating is easy with conjugate priors.

    • Linear Regression

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

    • Probability Distributions

      Probability Distributions
      Different probability distributions are useful in different situations.

    • T Distribution

      T Distribution
      The t distribution depends on one parameter.