
A recurring theme in mathematical physics is that while we consider that a physical system exists independent of how we choose to represent it mathematically, that choosing one particular mathematical representation vs another might make things look a lot simpler. Since these representations are different ways of looking at the same thing, you have to be able to go from one to the other by mathematical transformations (in principle  when you can't, there's often new physics hiding buried in the math and the math is trying to tell you that). Let's try this in a straightforward example: circular motion.
(a) Consider a vector describing an object's position in a two space:
. We want to consider the object moving around the origin in a circle, so it's distance from the origin is a constant, even though x(t) and y(t) are functions of time. Let's create a new polar coordinate basis, with the first unit vector along the direction towards the point and the second perpendicular to it as shown in the figure at the right. Express the unit vectors (note that these change as the position of the object changes) in terms of the fixed unit vectors. 
(c) Now consider that the particle is moving in the circle determined by the polar equations
where ω is a constant. Construct the vector velocity by differentiating with respect to time. Do it in two ways: first by differentiating the expression for the vector in terms of of the xy basis and second by differentiating the expression for the vector in terms of the rθ basis. Show that the answers you get are the same.
Last revision 4. November, 2004.