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Question

(a) Show that the following set is a curve: \mathbf{M}=\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}-3 x y+2 y^{2}=1\right\} Briefly explain your reasoning. O Compute the arc length of the curve X=\left\{(x, y, z) \in \mathbb{R}^{3} \mid 0<z<1, x=2 e^{z}, y=e^{2 z}\right\} :) Let F be a vector field on R$. Dr Mistake wishes to compute the curl of the divergence of F (i.e. V x (V - F)). Why is Dr Mistake not allowed to do this? (d) Let C denote the unit circle about the origin, C=\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{2}=1\right\} R? given bywith the anticlockwise orientation, and let W be the vector field on \mathbf{W}(x, y)=(a x+b y, c x+d y)_{(x, y)} where a, b, c, dER are fixed constants. Apply Green's theorem to compute \int_{C} \mathbf{W} \cdot \mathrm{ds} State your answer in terms of the constants a, b, c, d.