Explore BrainMass
Share

Explore BrainMass

    Gaussian Curvature of the Unit Sphere

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Compute the curvature of the unit sphere in R^3.
    I started the problem with r = [sin u cos v
    sin u sin v
    cos v. ]
    From there, how do you get the principal curvatures k_1 and k_2?

    © BrainMass Inc. brainmass.com October 10, 2019, 3:39 am ad1c9bdddf
    https://brainmass.com/math/curves/gaussian-curvature-unit-sphere-429253

    Solution Preview

    The Gaussian curvature of a surface is given by

    K = (LN - M^2)/(EG - F^2),

    where

    E = r_u . r_u
    F = r_u . r_v
    G = r_v . r_v

    and

    L = n . r_uu
    M = n . r_uv
    N = n . r_uv

    where n = r_u X r_v / |r_u X r_v| is the unit normal vector to the surface. We have

    ( cos u cos v )
    r_u = ( cos u sin v )
    ( -sin u )

    ( -sin u sin v )
    r_v = ( sin u cos v )
    ( 0 )

    whence

    E = r_u . r_u = cos^2 u (cos^2 v + sin^2 v) + sin^2 u = cos^2 u + sin^2 u = 1,
    F = r_u . ...

    Solution Summary

    We use the first and second fundamental forms to show that the Gaussian curvature of the unit sphere is equal to 1; this solution includes a detailed step-by-step process for solving this problem as well as references to consult for further questions.

    $2.19