9) Gauss’s Theorem Egregium

Question 1

The Gaussian curvature of a surface depends only on

Answer 1

Question 2

Smooth orthonormal basis of tangent plane?
Use to form basis of R^3

Answer 2

Question 3

Form partial derivatives of smooth orthonormal basis of R^3

Answer 3

Where e’ and e’’ are smooth functions of surface parameters (u,v) where N = e’ X e’’ (cross product)

Question 4

Using smooth orthonormal basis of R^3 show that

Answer 4

Follows immediately from (below) since e’ , e” and N are perpendicular unit vectors

Question 5

Using smooth orthonormal basis of R^3 show that

Answer 5

(1) is the linear combinations of partial derivatives of orthonormal bases elements

Question 6

Using smooth orthonormal basis of R^3 show that

Answer 6

Question 7

Use (below) to prove Gauss’s theorem egregum

Answer 7

Question 8

Gaussian curvature if F = 0

Answer 8

Question 9

Gaussian curvature if F=0 and E=1

Answer 9

Question 10

Prove

Answer 10

Question 11

Gaussian curvature for surface of revolution using only matrix of first fundamental form

Answer 11

Question 12

Prove that why Map of any region of the earth’s surface must distort distances

Answer 12