What is Tensor Calculus and How is it Related to Differential Geometry?

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Tensor calculus is fundamentally linked to differential geometry, focusing on tensors as multi-linear operators on tangent or cotangent spaces of manifolds. It involves the manipulation of vectors and one-forms, with the rank of a tensor indicating how many of each it can take as arguments. This mathematical framework extends the index approach to N dimensions, making it applicable in various fields. Understanding tensor calculus requires familiarity with vector analysis and its components. Overall, it serves as a crucial tool in modern theoretical physics and geometry.
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I have started to learn a bit about Tensor calculus and it all going above my head. May anyone give a brief outline about the topic (preferably theoretical) and the supplementary concepts attached to it.
 
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Tensor calculus is, in modern times, basically subsumed into differential geometry. It is the study of tensors which are multi-linear operators existing on the tangent (or cotangent) spaces to a manifold. They take any number of vectors or one forms as arguments (the rank of a tensor being how many vectors and one forms it takes) and gives a scalar (coordinate independent) number.
 

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