Question

Sketch the surface S defined geometrically z=-\left(x^{2}+y^{2}\right), \quad x \geqslant 0, \quad y \geqslant 0, \quad x^{2}+y^{2} \leqslant 1 Sketch the projection of S in the xy plane, as seen from above. Find a parametric representation for each section of the boundary of S. Choose param-eterisations such that the direction followed along each section is anti-clockwise in your answer to part (b). ) For the vector field F = (z, 0, 0), use your answer to part (c) to evaluate the line integral \int \mathbf{F} \cdot \mathrm{dr} along each section of the boundary of S. Evaluate the surface flux integral \iint_{S}(\boldsymbol{\nabla} \times \mathbf{F}) \cdot \widehat{\mathbf{n}} \mathrm{d} S \iint_{S}(\nabla \times \mathbf{F}) \cdot \widehat{\mathbf{n}} \mathrm{d} S where is the unit normal with positive z-component.

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