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On successful completion of the module, students should be able to:
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Obtain a coordinate-induced basis for the tangent space and cotangent space at points of a differentiable manifold, construct a coordinate induced basis for arbitrary tensors and obtain the components of tensors in this basis
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Determine whether a particular map is a tensor by either checking multi-linearity or by showing that the components transform according to the tensor transformation law
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Construct manifestly chart-free definitions of the Lie derivative of a function and a vector, to compute these derivatives in a particular chart and hence compute the Lie derivative of an arbitrary tensor
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Compute, explicitly, the covariant derivative of an arbitrary tensor
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Define parallel transport, derive the geodesic equation and solve problems invloving parallel transport of tensors
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Obtain an expression for the Riemann curvature tensor in an arbitrary basis for a manifold with vanishing torsion, provide a geometric interpretation of what this tensor measures, derive various symmetries and results involving the curvature tensor
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Define the metric, the Levi-Civita connection and the metric curvature tensor and compute the components of each of these tensors given a particular line-element
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Define tensor densities, construct chart-invariant volume and surface elements for curved Lorentzian manifolds and hence construct well-defined covariant volume and surface integrals for such manifolds
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