Gaussian Curvature of (x^2+y^2+1)^-2

  • Thread starter foxjwill
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  • #1
foxjwill
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1. Homework Statement
Is the gaussian curvature at a point on the surface
[tex]
\frac{1}{(x^2+y^2+1)^2}?[/tex]

2. Homework Equations
shape operator: [tex]
S(\textbf{x})=-D_\textbf{x}\hat{\textbf{n}}=\frac{\partial (n_x, n_y)}{\partial (x,y)}[/tex]

Gaussian Curvature = [tex]
|S(\textbf{x})|[/tex]

[tex]
\hat{\textbf{n}}=\frac{\nabla g}{\|\nabla g\|}[/tex]

3. The Attempt at a Solution

I basically plugged stuff into the above equations. I'm not sure if they're all correct.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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971
1. Homework Statement
Is the gaussian curvature at a point on the surface
[tex]
\frac{1}{(x^2+y^2+1)^2}?[/tex]
I have no idea what you mean by this that is an equation, not a surface. It's graph, in the xy-plane is a curve, not a surface. What surface do you mean?
[tex]z= \frac{1}{(x^2+y^2+1)^2}?[/tex]?

2. Homework Equations
shape operator: [tex]
S(\textbf{x})=-D_\textbf{x}\hat{\textbf{n}}=\frac{\partial (n_x, n_y)}{\partial (x,y)}[/tex]

Gaussian Curvature = [tex]
|S(\textbf{x})|[/tex]

[tex]
\hat{\textbf{n}}=\frac{\nabla g}{\|\nabla g\|}[/tex]

3. The Attempt at a Solution

I basically plugged stuff into the above equations. I'm not sure if they're all correct.
 

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