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

[aditya ver.2.0]

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.

 

[jedishrfu]

You're doing Vector Analysis with components and indices... The index approach can be extended to N dimensions and is useful in a lot of contexts.

http://en.wikipedia.org/wiki/Tensor_calculus

 

[Matterwave]

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.