Simplifying the Algebra for Gaussian Curvature

[InertialRef]

Homework Statement

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.

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

Homework Equations

κ_G = [(f_xx)(f_yy) - f_xy²]/[1+f_x²+f_y²]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)

f_x = -x/√(6y - x² - y² - 8) = -x/z
f_y = (3-y)/√(6y - x² - y² - 8) = (3-y)/z

f_xx = -z² - x²/z³
f_yy = -z² - (3-y)²/z³
f_xy = x(3-y)/z³

The Attempt at a Solution

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

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

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

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

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

κ_G = 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.

 

[mighty2000]

Any help would be appreciated.

Dear student,

Thank you for reaching out for help with your question. The first thing I noticed is that you have a typo in your attempted solution. In the line where you have "κG = [(z4 + z2(3-y)2 + z2x2 + x2(3-y)2 - x2(3-y)2)/x6]/[(z2+x2+(3-y)2)/z2]2", the denominator should be z6 instead of x6. This may have affected your final answer.

Other than that, your steps seem to be correct. However, I would suggest simplifying the expression further by factoring out a z2 from the numerator, which would give you κG = z2/[z2 + x2 + (3-y)2]. Then, since z2 is present in both the numerator and denominator, it can be cancelled out, leaving you with the final answer of 1.

I hope this helps and good luck with your studies!