Seeking an intuition linking definitions of contravariant and covariant

[nomadreid]

TL;DR Summary

There are three definitions for the difference between contra/co/variant vectors; I would like an intuition of their equivalence.

This is an elementary question in Analysis. I thought it might belong in "General Math", but finally settled on this rubrique. If a moderator wishes to move it to its proper rubrique, I would be grateful.

I have come across three explanations for the distinction between covariant and contravariant vectors.

[1] If the vector is the sum of (quantities times the base vectors) (or, otherwise put, the sum of the projections which are perpendicular to the respective axes), then these quantities are the components of the contravariant vector. If, on the other hand, the vector equals the sum of the (quantities times the dual base vectors) (or, otherwise put, the sum of the projections which are anti-parallel to the axes) , then those quantities are the covariant vectors

[2] If there is a coordinate transformation, if the vectors are transformed in the opposite direction to the direction of the coordinate system, then the vectors are contravariant, but if they are transformed in the same direction, they are covariant (hence the names).

[3] If there is a coordinate transformation so that

x'ⁱ=fⁱ(x¹….xⁿ)
xⁱ=Fⁱ(x'¹….x'ⁿ)

so that , for i=1,2....n,
dx'ⁱ= Σ_j=1ⁿ(∂x'ⁱ/∂x^j)dx^j
dxⁱ= Σ_j=1ⁿ(∂xⁱ/∂x'^j)dx'^j

Then quantities that which transform as A'ⁱ= (∂x'ⁱ/∂x^j)A^j,for i,j=1,2....n,
are components of a contravariant vector,
and if they transform as A'_i= (∂x^j/∂x'ⁱ)A_j, they are components of a covariant vector.

The third definition seems to be the one that is most used in applications. I do not have an intuition for this definition ( let alone mastering the technical part -- for example, from the symmetry of the definitions, it seems to my untrained eye that a contravariant vector with respect to f might be a covariant vector with respect to F, but that is probably wrong....) it might help if I could relate it to the more intuitive first and second definitions. The correspondence is assuredly elementary, but I do not really see it. Could someone give me an intuitive explanation (or send me an appropriate link: Wikipedia's explanation somehow doesn't sink in) as to why the three definitions are equivalent?

Many thanks for the help, and for the patience of those who contribute.

 

[jedishrfu]

There are a few videos on Youtube that might give you the necessary intuition:



From my memory, contravariant vectors arose from base vectors that were defined as tangents to the curvilinear coordinates, whereas as covariant vectors arose from base vectors defined from normals to the curvilinear coordinates, as shown in this diagram:

The problem in visualizing them is that our common XYZ coordinate system masks the uniqueness of the covariant base vectors, which align with the XYZ contravariant base vectors.

We need to consider spherical coordinates as an example to imagine how they differ from one another.