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: In a 2D Euclidean space with polar coordinates ((r,\theta)), the metric is ( ds^2 = dr^2 + r^2 d\theta^2 ). (a) Write the metric tensor ( g_ij ) and its inverse ( g^ij ). (b) Compute the Christoffel symbols ( \Gamma^r_\theta\theta ) and ( \Gamma^\theta_r\theta ). (c) Find the covariant derivative ( \nabla_\theta V^\theta ) for a vector field ( \mathbfV = r^2 \partial_r + \sin\theta , \partial_\theta ). Show that ( T^ij = A^i B^j ) is a tensor of rank (2,0)

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Show that ( T^ij = A^i B^j ) is a tensor of rank (2,0).