It is noontime, and Johnny is practicing with his new jetpack by flying back and

Question

It is noontime, and Johnny is practicing with his new jetpack by flying back and forth between a spot on the ground at (1, 0, 0) and a platform at the point (1, 0, 3). On his first trip from the ground to the platform he goes a little heavy on the right thruster, resulting in a helical path, making three full counterclockwise revolutions before (miraculously) landing on the platform. As he flies, his shadow on the ground satisfies the equation x + y^2/4 = 1. On his return trip from the platform to the ground he manages to fly along a helical path that makes two full counterclockwise revolutions, with his shadow this time described by x^2 + y^2 = 1. Oh, and on Johnny's planet gravity is described by the vector field F = yi - xj + xyk. (a) Parameterize the curve representing Johnny's upward path. Don't forget, to include the limits on time. (b) Compute the work (flow) done by the vector field F as Johnny flies up to the platform. (c) Parameterize the curve representing Johnny's downward path. Don't forget to include the limits on time. (d) Compute the arc length of Johnny's return trip to the ground. There is no need to simplify your result.

Explanation / Answer

upward path :
x^2 + y^2/4 = 1
(1,0,0) to (1,0,3)

Clearly, we can parameterize as
x = cost , y = 2sint
with 0 <= t <= 6pi because we have 3 counterclockwise revolutions

And we can parameterize z using a different variable, s
as z = s
and z gors from 0 to 3
we can write
z = s with 0 <= s <= 3

So, the parameterization is:

<cost , 2sint , s>
with 0 <= t <= 6pi
and 0 <= s <= 3

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Downward path :
x^2 + y^2 = 1
So, this is x = cost , y = sint
with 0 <= t <= 4pi(2 counterclockwise revolutions)

And z must go from 3 to 0

This can be : z = 3 - 3s
with 0 <= s <= 1

So, the parameterization can be :

r = <cost , sint , 3 - 3s>
with 0 <= t <= 4pi
and 0 <= s <= 1

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