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.