Tensor Analysis : Lecture 02

In this lecture I examine the concept behind covectors (which were also called covariant vectors earlier). I begin by examining why the dot product of two vectors yields a scalar, and then point out that this is true only because the underlying coordinate transformation for standard 3D vectors - rotations - are orthogonal. Then I go on to show that we can rescue the "dot product" , even for non-orthogonal transformations, by introducing a new kind of n-component object that transforms differently from a vector - the covector. I also explain why high school physics does not mention the existence of covectors at all - and why, the familiar gradient of a scalar, which is actually a covector, had been introduced to us as a vector! After briefly discussing 4-covectors, i.e. covectors under Lorentz transformations, I go on to discuss covectors under general coordinate transformations.