A Quick Introduction to Vectors, Vector Calculus, and Coordinate Systems

For the present discussion, we define a “coordinate system” as a system for describing positions in space. Any physical law should be expressible in a form that is invariant with respect to our choice of coordinate systems; we certainly do not expect that the laws of physics change when we switch from spherical coordinates to cartesian coordinates! It follows that we should be able to express physical laws without making reference to any coordinate system. Nevertheless we do often use coordinate systems when discussing physical laws, and so we need to understand how the laws can be expressed in different coordinate systems, and in particular how various quantities “transform” as we change from one coordinate system to another.