Algebra and Geometry | Part IA, 2006

Express the unit vector er\mathbf{e}_{r} of spherical polar coordinates in terms of the orthonormal Cartesian basis vectors i,j,k\mathbf{i}, \mathbf{j}, \mathbf{k}.

Express the equation for the paraboloid z=x2+y2z=x^{2}+y^{2} in (i) cylindrical polar coordinates (ρ,ϕ,z)(\rho, \phi, z) and (ii) spherical polar coordinates (r,θ,ϕ)(r, \theta, \phi).

In spherical polar coordinates, a surface is defined by r2cos2θ=ar^{2} \cos 2 \theta=a, where aa is a real non-zero constant. Find the corresponding equation for this surface in Cartesian coordinates and sketch the surfaces in the two cases a>0a>0 and a<0a<0.

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