calc

    Cards (32)

    • What is the title of the presentation?
      EG-232: Multivariable Calculus for Medical Engineers
    • What is the main focus of the PDEs section?
      Constructing and solving PDEs for physical processes
    • What are the two types of PDEs mentioned?
      Wave equation and heat equation
    • What does the wave equation represent in 3D?
      Sound waves and displacement of air molecules
    • What does the 1D wave equation represent?
      Waves on a liquid surface or displacement of a wire
    • What law is applied to form the wave equation?
      Newton's 2nd law
    • What does Hooke's law relate to in the wave equation?
      The force on mass from adjacent springs
    • What happens to a discrete system of masses/springs for small h?
      It becomes a continuous system
    • What is the formula for density in the wave equation?
      Density, r = m/Ah
    • What does the spring constant k equal in the wave equation?
      k = EA/h
    • What does the expression inside the large brackets equal in the wave equation?
      The 2nd derivative of u with respect to position
    • What type of waves does the derived 1D wave equation represent?
      Longitudinal (sound) waves in a solid
    • What is the form of displacement u(x,t) that indicates wave properties?
      A pulse moving from left to right
    • What does the constant c represent in the wave equation?
      The wave speed
    • What assumption is made about the wave in the wave equation?
      There are no losses; it carries on forever
    • What are the boundary conditions for a flexible string in the wave equation?
      Fixed endpoints at x=0 and x=L
    • What are the initial conditions for the wave equation of a string?
      Initial deflection f(x) and initial velocity g(x)
    • What does separation of variables allow in solving PDEs?
      Reducing a PDE to easier ODEs
    • What form do the solutions take when using separation of variables?
      Factorized solutions X(x) and T(t)
    • What happens when separating variables in the wave equation?
      It must equal a constant
    • What type of functions are solutions to the wave equation?
      Combinations of sines and cosines
    • What does k represent in the context of the wave equation?
      It can be positive, negative, or zero
    • What is the general solution for second-order homogeneous ODEs?
      u = A1e^(m1x) + A2e^(m2x)
    • What is the auxiliary equation used for in second-order linear ODEs?
      To find roots for the general solution
    • What does the Helmholtz equation relate to in the wave equation?
      It is a constant not a function of x or t
    • What are eigenvalues and eigenfunctions in the wave equation?
      Discrete solutions with corresponding eigenvalues
    • How does the wave equation relate to Fourier Series?
      It reminds us of Fourier Series solutions
    • What are the steps involved in solving the wave equation using separation of variables?
      1. Assume factorized solutions X(x) and T(t)
      2. Rewrite the wave equation
      3. Separate variables to equal a constant
      4. Solve the resulting ordinary differential equations
      5. Combine solutions to form the general solution
    • What are the boundary and initial conditions for a vibrating string?
      • Boundary conditions: Fixed endpoints at x=0 and x=L
      • Initial conditions:
      • Initial deflection f(x) at t=0
      • Initial velocity g(x) at t=0
    • What is the significance of the wave speed constant c in the wave equation?
      • Represents the speed of wave propagation
      • Relates to waves traveling in both directions
      • Assumes no energy loss during propagation
    • What are the implications of k being positive, negative, or zero in the wave equation?
      • Positive k: Oscillatory solutions
      • Negative k: Non-oscillatory solutions
      • Zero k: Static solutions
    • What is the relationship between the wave equation and the concept of eigenvalues and eigenfunctions?
      • Eigenvalues correspond to discrete solutions
      • Each eigenvalue has a unique eigenfunction
      • Solutions are determined by boundary conditions
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