|
The terms "covariant" and "contravariant" are sometimes useful for describing a change in coordinate system. Suppose that the new basis vectors are e1 = (3,4,5), e2 = (1,4,4) and e3 = (2,7,1). Then it follows that the new basis vectors are much larger than the old basis vectors, namely (1,0,0), (0,1,0), and (0,0,1). These new basis vectors make acute angles with each other, but all lie in the first octant of the first coordinate system, and so aren't hard to visualize. Clearly any point in 3 space could be represented equally well by using either the new coordinates or the old coordinates. How will a given distance be affected by changing from the old coordinate system to the new one? Consider for example the distance between (0,0,0) and (2,7,1) in the old coordinate system (recalling from above that the latter is e3 in the new system). Well, in the old system, the distance is Sqrt(2^2 + 7^2 + 1) or about 7.28. But in the new system, the distance is 1. That distances in the new system are smaller than distances in the old is because (at least in the simple example under consideration) the new basis vectors are larger than the old. In this example, it's easy to see that distance varies oppositely to changes in basis vectors, which by definition makes distance a "contravariant" quantity. Now if the new basis vectors had been smaller, then the nominal distance would have been larger, and this is the meaning of "Contra-variant". But if a quantity grows or shrinks along WITH the new basis vectors, then that quantity is "covariant". A temperature gradient is the typical example of a covariant vector, e.g., suppose that the temperature varies two degrees per unit distance in some direction (using the old coordinates). Then it will vary a lot more (in this example) in the new coordinates per unit, since the unit of distance is larger. Thus the quantity is "co-variant". eof |