Thursday, July 14, 2022

Solutions to Differential Geometry Problems (Part 5)

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

Solution:

J =

cos 2 , -u1  sin 2 )

( sin 2 ,  u1  cos 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 ,  2 ) = ( h 1 (u1 )h 2 (u1 )h 3 (u1 ) + 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,  2 )  =  
(a cos ucos u1 , b cos u2 sin u1c 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

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