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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The ridges (or the ridge set) of a smooth function of two variables is a set of curves whose points are, loosely speaking, local maxima in at least one dimension. For a function of N variables, its ridges are a set of curves whose points are local maxima in N 1 dimensions. (A more precise definition is given below). In this respect, the notion of ridge points can be seen as an extension of the concept of a local maximum. Correspondingly, the notion of valleys for a function can be defined by replacing the condition of a local maximum with the condition of a local minimum. The union of ridge sets and valley sets, together with a related set of points called the connector set form a connected set of curves that partition intersect or meet at the critical points of the function. This union of sets together is called the function''s relative critical set.