Consider the surface S=\left\{(x, y, z) \in \mathbb{R}^{3} \mid 0<x<2,4 y^{2}+4(z+1)^{2}=x^{2}\right\} and consider the following parametrisation of S: \sigma:(0,1) \times \mathbb{R} \rightarrow S, \quad \sigma(u, v)=(2 u, u \cos v, u \sin v-1) (a) Sketch the image of o. Moreover, on your sketch, indicate (i) one path obtained by holding v constant and varying u, and (ii) one path obtained by holding u constant and varying v. \text { Find the tangent plane to } \mathrm{S} \text { at the point }\left(1, \frac{1}{2},-1\right) \text {. } (c) Compute the surface integral \iint_{\mathrm{S}} \mathbf{H} \cdot \mathrm{d} \mathbf{A}, where S has the outward-facing orientation, and where H is the vector field \mathbf{H}(x, y, z)=\left(x^{2},-z-1, y\right)_{(x, y, z)}, \quad(x, y, z) \in \mathbb{R}^{3}

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