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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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