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Simplifying the Algebra for Gaussian Curvature

  1. Jun 13, 2013 #1
    1. The problem statement, all variables and given/known data

    Hello everyone. :) I'm having trouble simplifying the last little bit of this question that deals with Gaussian curvature. I've taken all the required derivatives, and double checked with my professor to make sure that they're correct. I'm only have trouble with reducing the algebra to get the right answer. If anyone could point out where I'm going wrong with my steps, I'd be really grateful.

    κG = {[(-z2 - x2)/(z3)][(-z2 - (3-y)2)/z3] - [(x(3-y))/z3]2}/[1 + x2/z2 + (3-y)2/z2]2

    2. Relevant equations

    κG = [(fxx)(fyy) - fxy2]/[1+fx2+fy2]2

    This was the equation given for the Gaussian Curvature. The original function and subsequent derivatives are below. The question mentioned that when the derivatives are taken, they should be written in terms of x, y and z. So any time the derivative contained the term √(6y - x2 - y2 - 8), it was replaced with z.

    f(x,y) = z = √(6y - x2 - y2 - 8)

    fx = -x/√(6y - x2 - y2 - 8) = -x/z
    fy = (3-y)/√(6y - x2 - y2 - 8) = (3-y)/z

    fxx = -z2 - x2/z3
    fyy = -z2 - (3-y)2/z3
    fxy = x(3-y)/z3



    3. The attempt at a solution

    κG = {[(-z2 - x2)/(z3)][(-z2 - (3-y)2)/z3] - [(x(3-y))/z3]2}/[1 + x2/z2 + (3-y)2/z2]2

    κG = {[(z2 + x2)[z2 + (3-y)2] - x2(3-y)2]/z6}/(1 + (x2/z2) + ((3-y)2/z2))2

    κG = [(z4 + z2(3-y)2 + z2x2 + x2(3-y)2 - x2(3-y)2)/x6]/[(z2+x2+(3-y)2)/z2]2

    κG = [(z4 + z2(3-y)2 + z2(x2))/z6)][z4/(z2+x2+(3-y)2)2]

    κG = [z2(z2 + (3-y)2 + x2)/z6][z4/(z2 + x2 + (3-y)2)2]

    κG = 1/[z2 + x2 + (3-y)2]

    This is what I have so far. Every time I do the algebra, I get the answer 1/[z2 + x2 + (3-y)2]. The answer is listed as 1, and that makes sense, since the curvature for a sphere is constant irrespective where you are on the curve.

    I'm just wondering where I went wrong with my algebra. It seems correct to me, and I've double checked my derivatives as well.
     
  2. jcsd
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