active passive rotation reject

active passive rotation reject I don't think my rotation example really adds anything. It cannot show the distinction, and the co-varying idea is very dim, and the picture is confusing regarding point of rotation. So I park it here for possible repair some day. ********************************************* The notion of "covariant" in the case of Cartesian x-space and x'-space and rotations can be illustrated by this picture, which shows the passive view of a vector rotation: the vector stays put and the coordinate system moves: [ In this picture, the rotation is about the point at the tail of the V arrow. The unit vectors should be drawn with their tails at that point as well, but then the picture gets too cluttered, so they have been displaced graphically to the coordinate system origin. ] // passive rotation The following equations apply: V = Vn = Vx + Vy = V'n ' = V'x ' + V'y ' V'n = Rnm Vm Vx' = cosθVx - sinθVy Vy' = sinθVx + cosθVy ' = Rnm ' = cosθ - sinθ ' = sinθ + cosθ The basis vectors are rotated clockwise, and the axes, which by projection determine vector components, are also rotated clockwise (they co-vary). As a result, the vector V is the same in either expansion. Of course for rotations, we know that Rab = Rab since g = g' = 1, so "contravariant" then has the same picture and there is no distinction. The active view of vector rotation is this, where one thinks of V'n = Rnm Vm as V' = RV, // active rotation In this picture of rotation, one has V = Vn = Vx + Vy V' = V'n = V'x + V'y All the previous equations are still true, including V'n = Rnm Vm , but there is no need to display the ' vectors. This picture is not useful, however, for the purpose of illustrating the covariant idea that two objects are moving in the same direction.