ABSTRACT
We have come across curves in our investigations in many different settings, e.g., as the graph of a function of one variable, as the boundary of an open set and as the level set of a function of two variables. We used the term loosely and as an aid to our intuition but in each case we could, if pressed, fall back on a mathematical definition. We provide, in the next few chapters, a systematic study of these objects by considering mappings from R to R2. A new feature that we could previously afford to ignore is introduced here. This is the idea of direction along a curve. This concept is fundamental in integration theory in higher dimensions and while it can appear initially as an unnecessary complication it is a good idea to master this concept by studying elementary curves in simple settings. Keeping track of directions and checking that parametrizations give the correct sense of direction from the very beginning leads to a better appreciation of the overall picture.
