Solutions To Differential Geometry (Parametric Angle Variables for Surfaces ) Problems

 The Problems:

1) For the given cone of revolution (Example 2) write the corresponding Jacobian (matrix).

Solution:

J =

( cos u ² , -u¹  sin u ² )

( sin u ² ,  u¹  cos u ² )

(a .......................0  )

2) Find a parametric representation for each of the following:

a) Ellipsoid:  x ( u¹,  u ² )  =   (a cos u² cos u¹ , b cos u² sin u¹, c sin u ²)

b)    Hyperbolic paraboloid:  

x ( u¹ ,  u ²)  =  (au¹ cosh u²,  bu¹ sinh u²,  (u¹ ) ²)

Solutions:

a)   x ₁² / a ²   +     x ₂²   / b²  +    x ₃²  / c²  -  1  = 0

b) x ₁² / a ²   -   x ₂²   / b²  -   x ₃ =  0