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1. In addition to the vector/dual vector distinction, there is also the distinction between contravariant/covariant forms of vectors. I believe covariant vectors and dual vectors transform the same way. Are they essentially the same thing?

2. Someone gave the electric field as an example of a dual vector because it is found as the gradient of the potential field and transforms as a covariant vector. One colloquial defining property of dual vectors is that they "eat vectors and produce a scalar." An example is -E

_{μ}dx

^{μ}= work done. But one can take any two regular vectors and form the sum F

^{μ}G

^{μ}which would also be a scalar. So how are dual vectors different in this regard?