Exercises

Show that each parameterization is orthogonal and then determine the normal curvature of the surface. Then determine the principal curvatures, the Mean curvature, and the Gaussian curvature of the surface. Which surfaces are minimal surfaces? Which are Gaussian flat?

1. r( u,v) = á v, sin( u) ,cos( u) ñ 2. r( u,v) = á cos( u), v ,sin( u) ñ 3. r( u,v) = á u, v,u ñ 4. r( u,v) = á u, v2, u ñ 5. r( u,v) = á u, v,v2 ñ 6. r( u,v) = á u2, v, u ñ 7. r( u,v) = á u, v,cosh( u) ñ 8. r( u,v) = á u, v, ln| sec( v) | ñ 9. r( u,v) = á v, u cos( v),  u sin( v) ñ   10. r( u,v) = á u2cos( v), u2sin( v),u ñ 11. r( u,v) = á u cos( v),  u sin( v),  u ñ 12. r( u,v) = áeu, e-u, v2 ñ 13. r( u,v) = á cosh(v) sin( u), cosh(v)cos( u) , v ñ 14. r( u,v) = á eucos(v), eusin(v), u ñ 15. r( u,v) = á sinh( v) cos( u), sinh( v) sin( u) ,u ñ 16. r( u,v) = á cosh( v)cos( u), cosh( v) sin( u), u ñ

Use the theorem Egregium to determine the Gaussian curvature of a surface with the given fundamental form.

17. ds2 = du2 + e4udv2 18. ds2 = v2du2 + v2dv2 19. ds2 = v2du2 + dv2 20. ds2 = 4u2du2 + 8v2dv2 21. ds2 = v4du2 + ( 4v2+1) dv2 22. ds2 = du2 + sec2( v) dv2 23. ds2 = du2 + ( 2+cos u) 2dv2 24. ds2 = du2 + ( 4+cos u) 2dv2

25. Use the result in example 5 to find the curvature of z = x²-y². What is the curvature at ( 0, 0, 0) ? Does the function have an extremum or a saddle point at ( 0, 0, 0) ?

26. Use the result in example 5 to find the curvature of

z =   4-x2-y2

27. In spherical coordinates, a sphere of radius R centered at the origin is parameterized by

ds2 = R2df2 + R2sin2( f) dq2

Apply the Theorem Egregium to this form to determine the curvature of the sphere intrinsically from this form.

28. In latitude-longitude coordinates, a sphere of radius R centered at the origin is parameterized by

ds2 = R2dj2 + R2cos2( j) dq2

Apply the Theorem Egregium to this form to determine the curvature of the sphere intrinsically from this form.

29. A helicoid can be parameterized by

r( u,v) = á sinh( v) cos(u), sinh( v) sin( u) , u ñ

Show that the helicoid is a minimal surface.

30. A catenoid is the surface parametrized by

r( u,v) = á cosh( v) cos(u), cosh( v) sin( u) , v ñ

Show that the catenoid is a minimal surface.

31. Suppose a surface has a fundamental form of

ds2 = du2 + e2kudv2

where k is a constant. What is the Gaussian curvature of the surface?

32. A Mobius strip can be constructed from a strip of paper by giving one end a half-twist and then ''gluing'' the ends together. A parameterization of a Mobius strip is given by

r( u,v) =   cos( u) + vsin æ è  u 2 ö ø , sin( u) + vsin æ è  u 2 ö ø , vcos æ è  u 2 ö ø

What is the Gaussian curvature of a Mobius strip? (Hint: the parameterization is not orthogonal).

33. The surface of revolution of y = f( x) about the x-axis can be parameterized by 

r( u,v) = á v,f( v) cos(u) ,f( v) sin( u) ñ

Find the fundamental form and then use the Theorem Egregium to show that the curvature of a surface of revolution is given by

K =  -f'' ( v) f( v) (1+[ f' ( v) ] 2)

34. The pseudosphere is a surface of revolution parameterized by

r( u,v) =  sin( u) cos(v) , sin( u) sin( v) , cos( u)+ln é ë tan æ è  u 2 ö ø ù û

Determine the fundamental form and then use the Theorem Egregium to show that K = -1.

35. Euler's Formula: Show that if  k₁ occurs along r_u (as it does along the cylinder, for instance), then

kn( q) = k1cos2( q) + k2sin2( q)

36. Write to Learn: Write an essay in which you prove mathematically that a minimal surface that is Gaussian flat must be a region in a plane. Then use sketches and concepts to explain in your own words why Gaussian flat minimal surfaces must be planar.

37. Write to Learn: Write a short essay in which you use the following steps to prove that the Gaussian curvature K satisfies

( nu×nv) · n = K || ru×rv||

1. Explain why r_u· n = 0, and then show that this implies that

   ruu· n = -ru· nu,   and    ruv· n = -ru· nv

   Also, show that r_v· n = 0 implies that r_vu· n = -r_v· n_u and r_vv·n = -r_v· n_v.
2. A cross product identity says that if A, B, C, and D are vectors, then

   ( A×B) · ( C×D) = ( A·C) ( B·D) -( B·C) ( A·D)

   Apply this identity to the quantity ( n_u×n_v) · ( r_u×r_v)
3. Use (a) and (b) to show that

   ( nu×nv) ·( ru×rv) = ( ruu·n) ( rvv·n) -( ruv·n) 2

   and that ( r_u×r_v) · ( r_u×r_v) = || r_u||²|| r_v|| ²-( r_u· r_v) ².
4. Conclude by explaining why

   ( nu×nv) ·( ru×rv) || ru×rv|| 2  = K

   and then derive the desired result.

38. Show that if we define a function K to satisfy the relationship

nu×nv = K( u,v) ( ru×rv)

then K is the Gaussian Curvature of the surface. You may assume that r( u,v) is orthogonal, and you may want to use the steps (b) and (c) in exercise 37.

Exercises 39-44 explore conformal metrics, which are fundamental forms in which g₁₁ = g₂₂ and g₁₂ = 0. A conformal metric has the form

ds2 = g( u,v) ( du2+dv2 )

 where g = g₁₁ = g₂₂. Conformality implies that angles are preserved, thus connecting the study of conformal metrics to a number of other fields of mathematics including group theory, complex variables, and analytic geometry.

39. Show that the Theorem Egregium for a conformal metric is

K =  -1 2g é ë  ¶2ln( g) ¶u2 +  ¶2ln( g) ¶v2 ù û

40. (Extends Exercise 41 in section 3-2). Stereographic projection of a sphere of radius R leads to a parameterization of the form

r( u,v) =  2Ru u2+v2+1 ,   2Rv u2+v2+1 ,   R( u2+v2-1) u2+v2+1

(5)

Show that || r( u,v) || = R for all (u,v) . Then show that the fundamental form is conformal, and use exercise 39 to compute the curvature of the surface.

41. Write to Learn: Suppose that r( u,v) is a conformal parameterization of a surface S. For fixed q, the curve r( t) = r( cos(q) t+p,sin( q) t+q) is the image of the line through ( p,q) in the uv-plane that crosses the u-axis at an angle q.

conformpic

In a short essay, calculate the angle between r_u and  r' (t) and use the result to explain what is meant by the statement ''Conformality implies that angles are preserved''.

42. Suppose that T( u,v) = á x(u,v) ,y( u,v) ñ is a conformal coordinate transformation (i.e., if r( u,v) = á x( u,v) ,y( u,v) ,0 ñ , then r_u· r_u = r_v· r_v  ). Define r and q such that x_u = r cos( q) and y_u = r sin( q) (i.e., transform r_u into polar coordinates). Show that the Jacobian matrix must be of the form

J( u,v) = r é ê ë cos( q) -sin( q) sin( q) cos( q) ù ú û

(i.e., the Jacobian matrix is a similarity transformation, which is a rotation followed by a scaling).

43. Any sufficiently differentiable parameterization of a surface can be transformed into a conformal parameterization. Let's explore this idea for surfaces of revolution with parameterizations of the form

r( u,v) = á v,f( v) cos(u) ,f( v) sin( u) ñ

In particular, show that if y( v) satisfies the differential equation

y¢ =  f( y) 1+[ f'  (y) ]2

then the parameterization

r( u,y( v) ) = á y( v), f( y( v) ) cos( u),  f( y(v) ) sin( u) ñ

is a parameterization of the original surface that is also conformal.

44. Use the result in exercise 43 to find a conformal parameterization of the right circular cone, which is a surface of revolution of the form

r( u,v) = á v, vcos( u) , vsin( u) ñ

Then use the result in exercise 39 to compute the Gaussian curvature of the right circular cone.

Note: Additional exercises similar to 43 and 44 can be found in the worksheet.