1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: How to classify a quadratic surface?

  1. Mar 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Classify the quadratic surface:

    2x^2 + 4y^2 - 5z^2 + 3xy - 2xz + 4yz = 2

    2. Relevant equations

    A quadratic form is a second degree polynomial equation in three variables which has the form

    F(x,y,z) = ax^2 + by^2 + cz^2 + 2dxy + 2exz + 2fyz

    where coefficients a through i are real numbers.

    The curve F(x,y,z) = j can be written in the form (x^T)*A*x = j where

    x = [x, y, z]

    A = [a, d, e]
    [d, b, f]
    [e, f, c]

    3. The attempt at a solution

    A = [2, 3/2, -1]
    [3/2, 4, 2]
    [-1, 2, -5]

    Then, find the eigenvectors of A-tI...check how many are positive, negative, or 0 and classify using the information? Is that the right way to proceed?
  2. jcsd
  3. Mar 22, 2009 #2
    I think you have the right idea, but the wrong terminology: You are trying to find the eigenvalues of A by finding zeros of the polynomial det(A-tI).
  4. Mar 22, 2009 #3
    Oh ok. And then, the fact that F(x, y, z) is = 2 doesn't make a difference, right? Because, it is the j that F(x, y, z) can be set equal to such that (x^T)*A*x = j?
  5. Mar 22, 2009 #4
    The sign of the RHS does make a difference, in this case it tells you how many "sheets" the quadratic surface has (have you found the eigenvalues yet?).
  6. Mar 22, 2009 #5
    I get two positive and one negative eigenvalues: -5.647, 1.661, 4.986. This would be a hyperboloid of one sheet.
  7. Mar 22, 2009 #6
    That's correct. :smile:
  8. Mar 22, 2009 #7
    Thanks a bunch!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook