Given a constant, the matrix A [3] column vector cos sin 0 on the left, like this - sin 0 cos is a counterclockwise rotation by about the origin ie if you multiply A into the cos sin 0 20] [2] = [29 cos then the result is the rotation of the point (x, y). The inverse of A is obtained by replacing by its negative ie by rotating backwards by 0. Let r(t) = (x(t), y(t)) be a C¹ curve and F (x, y) = (P(x, y),Q(x, y)) be a C¹ vector field (C¹ means that the derivatives exist and are continuous). The rotated curve is f(t) = Ar(t) and the rotated vector field is f(x, y) = AF (A-¹(x,y)). To understand that last formula, think of drawing the vector field as arrows at places (x,y), and rotating that whole picture. To calculate the rotated vector field at (x, y), you have to calculate the rotation of the original vector field at the place A-¹(x,y). - sin 0] I cos - y sin 0 sin 0 + y cos 0 (A) Prove, in general, Green's theorem for the isosceles triangle enclosed by the three lines y = mx, y = -mx, and x =a, where m > 0 and a > 0. Start with the double integral and judiciously choose the order of integration. (B) For the special case of F= (0, 1) and r(t) = (t, t²), oriented with increasing 0 ≤ t ≤ 1, sketch the original vector field and curve, and the π/4 rotated vector field and rotated curve, and verify that the resulting two line integrals are the same. (C) Prove, in general, for any curve, vector field, and rotation angle, that the two line integrals are the same. ƏQ (D) Do the analogous general proof (that the value is the same before and after rotation) for the right side 32 theorem. You will need the chain rule and the general change of variables formula for multivariable integration. (E) Using (A), (C) and (D), prove Green's theorem for any regular polygon centered at the origin. ар მყ (dx dy) of Green's