Concept of contra-variance and co-variance tensor !!

I was reading some one’s notes on tensors and metrics. [Some intuition for the use of metric tensors to raise and lower indices] I am coming across these definitions after a decade or so of first introduction so can’t say I knew these or not. But here is what I realized howto remember these easily.

Co-variance: variance/change in direct-accordance with the change in x-coordinate. For a vector transformation its always the vector index which is summed up with the x-coordinate index. Hence when x-coordinate change is direct with the vector-transformation the x has to be in the numerator. This is co-with-x. Or direct proportionality.

Also since x and vector index is summed over the resultant vector index is the z or y coordinate’s index. That means that remaining index shows up on the resulting vector. That index occurs on z/y etc which is in the denominator.

That means a co-variant vector and tensor (tensor is more than one vector multiplied) has its index on the subscript of the vector or tensor. And this index is of y/z etc.

Contra-variance: variance/change in inverse-accordance with the change in x-coordinate. For a vector transformation its always the vector index which is summed up with the x-coordinate index. Hence when x-coordinate change is inverse with the vector-transformation the x has to be in the denominator. This is contrary-with-x. Or inverse proportionality.

Also since x and vector index is summed over the resultant vector index is the z or y coordinate’s index. That means that remaining index shows up on the resulting vector. That index occurs on z/y etc which is in the numerator.

That means a contra-variant vector (and tensor; tensor is more than one vector multiplied) has its index on the superscript of the vector or tensor. And this index is of y/z etc.

Thats all you need to know about Contra-variance and Co-variance for their elemental concept.

here is a simple notational example.

[Is bar called a transpose? and A are vectors and when occur in union eg * or A*B would be called tensors of 2nd order, more vectors there and order increases: ABC is a tensor of 3rd order]

[this example is contravariant as x is inverse to A and , also the x and A index are summed up so k index which is on top or superscript goes to ]

You can then easily see whats the co-variant case.

I am an experimental particle physicist, traveler, teacher, researcher, scientist and communicator of ideas. I am a quarkist and a bit quirky ! Hypothesis non fingo, eppur si muove, dubito cogito ergo sum are things that turn me on ! Researcher in experimental high energy physics (aka elementary particle physics; like “quarks, leptons & mesons and baryons”) … Teacher of Physics (and occasionally chemistry and maths) Blogger (check my website; mdashf.org) ! Love to read read and read but only stuff that interest me. Love to puff away my time in frivolities, just dreaming and may be thinking. Right now desperately trying to streamline myself.

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