Solutions to Differential Geometry Problems (Part 5)

 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) Specify a parametric representation for a cylinder generated by a straight line, L, which moves along a curve,

C: x(s)  =   ( h ₁ (s), h ₂ (s), h ₃ (s))

And which is always parallel to the x ₃ - axis.  Give the corresponding matrix (Jacobian).

Solution:

x ( u¹ ,  u ² ) = ( h ₁ (u¹ ), h ₂ (u¹ ), h ₃ (u¹ ) + u ² )

J =  

(h ₁   ....0)

(h' ₂ ......0)

(h' ₃......1)

3) 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)  = 

(a u¹ cosh u²,  b u¹ 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