Question 25 Using Green's Theorem, compute the counterclockwise circulation of F around the closed curve C. (3.0).(-7), and (0.7) 0, F = sin 5yi + cos 9xj; C is the rectangle with vertices at (0, 0), O O 25 O TT 4|5

(4) Consider the set C = {(x, y) = R² | y² = xº(1 − x²)}. E (a) Is C a curve? If yes, why yes? If no, why not? (b) Find a parametrisation of C and sketch C. Be sure to specify the domain of your parametrisation. Hint: a nice set of identities to consider are cos² (¹) sin t = 2 sin () COS ; (§). (c) Find the tangent line to C at - 1+cost, sin² () 2 - 1-cost 2 " and

Let a, b be nonzero numbers. Prove that •2π dt a² cos² (t) + b² sin² (t = 2π ab' by integrating on two homotopic (in C\{0}) curves: 7₁ (t) = a cos(t) + i b sin(t), t = [0, 27], and 72 the unit circle.

5. The function r(t) moving in 3D space. = (2 sin (¹), 3, 2 cos (t)) parametrizes (draws) the path of a particle (a) Evaluate r(t) at t = 0, 1, 2, 3, 4.