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1. It's against some people's belief that a tensor is just a pile of numbers with super - and subscripts. Most importantly, a tensor needs to follow certain rules to transform when the base space changes coordinates. A good example is to look at the relation of the "surface" of the human's head and the "hair". Let's assume that the "hair" is infinitely long so that it may be considered as a one dimensional vector space, while the "surface" has a "hair" attached (tangent) to it everywhere (human may look like that thousand years later). To repesent the "surface", you have many choices. You can use Descartes coordinates or whatever. But when you change the coordinates of the "surface", the vectors in the "hair" should transform accordingly, which is where you need to pay your attention to. This "hair" vector is the simplest contrvariant tensor, because it is going to transform contravariantly. What does "transform contravariantly" mean? Just look at how d/dx transforms when you perform a coordinate transformation. The simplest covariant tensor space is the space of the linear functionals over the "hair" space. All vectors in this space transform covariantly. Then what does "transform covariantly" mean? Just look at how dx transform when you perform a coordinate transformation.
Generally, a tensor has supercripts (contravariant indices) and subscripts (covariant indices), and they transform following the rules said above independently of each other. A very important operation of tensors is the "contraction", which amounts to make one superscript and one subscript equal and then sum over (using Einstein's convention) with that index. This operation will produce a new tensor with the total scripts minus two. The operation is a bit lenthy but the idea roughly comes from the cancelation of one action to transform d/dx and another action to transform dx. Now, here is the trouble if you make the two superscripts equal and then sum over. This operation fails to produce a new tensor, because it will not transform following the tensor's rules. Roughly speaking, you are doing two transform to d/dx and nothing can be cancelled out.
2. This can be easily seen. Since Aij is antisymmetric, then Aij=-Aji. You get zero when you sum over.
Wow, thank you and that's so nice of you, and I believe that everybody knows something the others don't. Just curious, what kind of subjects do you think you can help me with WITHOUT knowing the questions I would post in advance?