 On successful completion of the module, students should be able to:  Obtain a coordinateinduced 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
 Determine whether a particular map is a tensor by either checking multilinearity or by showing that the components transform according to the tensor transformation law
 Construct manifestly chartfree 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
 Compute, explicitly, the covariant derivative of an arbitrary tensor
 Define parallel transport, derive the geodesic equation and solve problems invloving parallel transport of tensors
 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
 Define the metric, the LeviCivita connection and the metric curvature tensor and compute the components of each of these tensors given a particular lineelement
 Define tensor densities, construct chartinvariant volume and surface elements for curved Lorentzian manifolds and hence construct welldefined covariant volume and surface integrals for such manifolds
