# Gaussian Curvature of the Unit Sphere

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?

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.