## Question

###### (5 Consider the two surfaces Mi and Mz defined by the parametrizations:q1 (t,0)(sinh & cos t, sinh 0 sin t,t) (0, 0,t) + sinh & (cos t, sin t, 0) _ (cosh t cos &, cosh t sin &,+) _92 (t,0)Show that 91 R x R - M is a one-to-one parametrization of a helicoid (see section 4.1 exercise 9). Show that 92 is a parametrization that is not one-to-one Show that Mz is rotationally symmetric (see section 4.1 exercise 4) and can also be described by the equation 22 +y? cosh? z _ Show further

(5 Consider the two surfaces Mi and Mz defined by the parametrizations: q1 (t,0) (sinh & cos t, sinh 0 sin t,t) (0, 0,t) + sinh & (cos t, sin t, 0) _ (cosh t cos &, cosh t sin &,+) _ 92 (t,0) Show that 91 R x R - M is a one-to-one parametrization of a helicoid (see section 4.1 exercise 9). Show that 92 is a parametrization that is not one-to-one Show that Mz is rotationally symmetric (see section 4.1 exercise 4) and can also be described by the equation 22 +y? cosh? z _ Show further that this equation defines surface: It is called the catenoid Define a map F Mi 5 Mz by Foq1 (t,0) = 92 (t,0). Show that this map is smooth; not one-toone. but locally diffeomorphism.

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