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:
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)
(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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