- #1

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use the gauss map to find the gaussian curvature of a sphere of radius r at any point. also, use the gauss map to find the gassian curvature of a cylinder of radius r at any point.

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- Thread starter halvizo1031
- Start date

- #1

- 78

- 0

use the gauss map to find the gaussian curvature of a sphere of radius r at any point. also, use the gauss map to find the gassian curvature of a cylinder of radius r at any point.

- #2

- 139

- 12

The Gaussian curvature is defined as the product of the eigenvalues of the Gauss map. Thus, all you need to do is find the matrix of the Gauss map and calculate its determinant.

The way to go about this is to find the first and second fundamental forms. Let the surface be parametrized by

(E F)

(F G) = g,

where E =

(l m)

(m n) = A,

where l =

l =

m =

n =

(try verifying this yourself). The first definition of the second fundamental form is equivalent to saying that A = S

A = S

g

Thus, since similar matrices have equal determinants, we have K = det(W) = det(A)det(g

For the sphere, use the usual parametrization x = r sin(p) cos(t), y = r sin(p) sin(t), z = r cos(p) (where p = phi and t = theta). Find a usable expression for

As a final comment, it's interesting to note (perhaps you've already covered this) that the Gaussian curvature can in fact be calculated only using the components of g, the first fundamental form (usually called the Riemannian metric in this context). Specifically, there's a famous formula for K involving only the Christoffel symbols associated to the metric. This is usually

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