**Unformatted text preview: **θ = π 2 or 3 π 2 . Verify also that the four “petals” are traversed in the order i , ii , iii , iv as indicated in Figure 2.17 . (b) Now consider the vector-valued function −→ q ( σ ) de±ned in pieces by b x = sin 2 σ cos σ = 2sin σ cos 2 σ y = sin 2 σ sin σ = 2sin 2 σ cos σ − π 2 ≤ t ≤ π 2 b x = − sin 2 σ cos σ = − 2sin σ cos 2 σ y = − sin 2 σ sin σ = − 2sin 2 σ cos σ π 2 ≤ t ≤ 3 π 2 . (2.25) Verify that this function is regular (the main point is di²erentiability and continuity of the derivative at the crossings of the origin). (c) Verify that the image of −→ q ( σ ) is also the four-leaf rose. In what order are the loops traced by −→ q ( σ ) as σ goes from 0 to 2 π ?...

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