We begin with physics because the idea of contravariance and covariance has its origins in the study of vectors. The concept of a vector has undergone a great amount of refinement and generalization in its history. Depending on the mathematical spaces one is considering, such generalization is necessary. But additionally, it is also useful to have a variety of ways of thinking about the vector concept. To quote Feynman:

"Theories of the known, which are described by different physical ideas, may be equivalent in all their predictions and hence scientifically indistinguishable. However, they are not psychologically identical when trying to move from that base into the unknown. For different views suggest different kinds of modifications which might be made and hence are not equivalent in the hypotheses one generates from them in one's attempt to understand what is not yet understood." -- Feynman [1]

(That is also a good quote to keep in the back of one's mind when we come to category theory.) Therefore, we will enumerate some of the common vector definitions. However, no matter how much the concept is generalized, in engineering and physics, the geometric notion of an arrow that obeys a superposition principle is fundamental. And in mathematics, the algebraic notion contained in a linear vector space, is fundamental. In both the arrow and vector space notions, there is a natural idea of contravariance and covariance. In computer science, the concept of a vector (i.e., an ADT) refers to a completely unrelated idea and in no way should be confused with the concepts of vector we are considering here.

Vector Concept 1 - Directed Line Segment: The vector concept one first studies in school is usually that of an arrow, or a directed line segment. Forces and displacements are examples of quantities that can be modeled by directed line segments. Experimentally, it is found that both forces and displacements can be combined graphically by the parallelogram method of addition. The discovery of this method seems to have been first made by the Dutch mathematician Simon Stevinus (1548-1620).

Figure 2  Addition of vectors by the parallelogram method

Abstracting, an elementary definition of a vector is that a vector is an object that can be represented by a directed line segment and that combines with other vectors via the parallelogram method of addition. The magnitude of the vector is given by the length of the arrow.

This definition is of great utility in physics and engineering, but it is not capable of being used in more abstact spaces than Rⁿ  without modification. However, those abstract spaces are typically manifolds, and a manifold "locally looks like" Rⁿ  (where n is the dimension of the manifold) So, even if we want to jump ahead to a more advanced definition, it is quite useful to understand how vectors are defined in Rⁿ.

Rectangular and Oblique Coordinates Systems in R²
      Without loss of generality, we restrict our discussion now to mostly R². Extending to Rⁿ is not difficult, but complicates the example. Shown below is a rectangular coordinate system and the resolution of a vector A into its rectangular coordinates.

Figure 3 Rectangular Components of Vector A

In rectangular coordinates, the contravariant and covariant components of a vector are usually the same, as will become clear later. 
      We now extend the class of coordinate systems to consider those systems whose axes are not perpendicular, or in other words, oblique coordinates. Let OX be a line through the points O and X, and OY be a line through the points O and Y, chosen so that the lines are not colinear and they meet at an oblique angle. Then, with O chosen as the origin, this constitutes an oblique coordinate system for R².(Figure 4)

Figure 4 An Oblique Coordinate System

The parallel projection onto the OX axis gives the x coordinate, or abscissa, A^x, of the vector A. The parallel projection onto OY gives the y coordinate, or ordinate, A^y, of the vector A.(Figure 5)

Figure 5 Oblique Coordinates of Vector A

The components A^x and A^y are called the contravariant components of A. It is important to note that x and y are not exponents, but their raised position relative to A is the standard notation for contravariant components that is used to distinguish them from the covariant components, which we will now introduce.

Reciprocal Lattice

Figure 6 The Reciprocal Lattice in R2

     To define the covariant components of A, we have to introduce the concept of the reciprocal lattice. Rotating the x axis by 90 degrees counter clockwise gives the y^'-axis of the reciprocal lattice, and rotating the y axis by clockwise by 90 degrees gives the x^'-axis of the reciprocal lattice. The original coordinate system is called the direct coordinate system, or direct lattice. The reciprocal lattice is shown, in green, in figure 6.

This procedure works in R²only. Generally in Rⁿ, the reciprocal lattice is defined by introducing non-zero "basis" vectors, not necessarily of unit length, along each axis x¹,...,xⁿ. (Because the number of axes can in general exceed the number of distinct letters in the alphabet, we now switch to labeling the axis not as x,y,z but as x¹,...,xⁿ) Let  e₁ be a non-zero vector along the x¹ axis, e₂  a non-zero vector along x ² axis,..., and e_n be a non-zero vector along xⁿ , then this set of vectors set {e₁,...,e_n} is a "basis" for the direct coordinate system. If we are given the basis, we can form the coordinate axes, and vice-versa. The reciprocal lattice "basis" , {e¹,...,eⁿ} then is defined by the requirement
 eⁱe_k cos[angle(i,k)] =1 if i=k, and 0 otherwise, where angle(i,k) is the angle between  e ⁱ and e_k.

Figure 7 Covariant Components of Vector A

     Having constructed the reciprocal lattice, the parallel projections of A onto the  x^'-axis and the y^'-axis give the covariant components of A, as shown in Figure 7.

For completeness, it should be noted it is also possible to give an interpretation of the covariant components as orthogonal projections on the direct coordinate sytem, rather than parallel projections on the reciprocal lattice. In fact, this interpretation was given by the founders Ricci and Lev-Civita. However, this interpretation requires  multiplication of the components by a factor, which destroys the symmetry of the formulation.

The reciprocal lattice seems to have its origins in crystallography, but it also has great application in solid state physics.

Next Week -- Part 2 -- Contravariance and covariance in abstract vector spaces. Vectors as elements of vector spaces, change of basis, multilinear algebra

---

References:

[1] Richard Feynman, The Development of the Space-Time View of Quantum Field Theory. Nobel Lecture, 1966.