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9.1 Paths and Surfaces in Two Dimensions

In two dimensional space, such as the x - y plane, a linear equation determines a line. That is, the set of (locus of) points obeying it is a straight line. For example, x - y = 0 describes a forty five degree line passing through the origin. A curve or path is often described by similarly by one equation, which related x and y, but is not necessarily linear. A path is a directed curve, so that at any point it is aimed in some direction. Thus x2 + y2 = 4 characterizes a circle of radius 2 around the origin. We typically make this a path by going around it in the counterclockwise direction in the xy plane.
For the most part we consider simple smooth paths: a smooth path is one that can be described by a well behaved equation; a simple one is one that does not intersect itself.
When we use the word path we will usually mean a simple smooth path which starts at some point a and ends at some point b. When a and b are the same we call the path closed. Our paths will generally be connected but not always.
Thus the equation (x2 + y2 - 1)(x2 + y2 - 4) = 0 defines two concentric circles of radius 1 and 2. In orienting such a pair of curves into a path it is necessary to be consistent in how we go around them. We do this by introducing a zipper between them. When this zipper is closed it is invisible and the path behaves as two separate paths. When it is open one has to orient the two circles in a manner consistent with both forming one big cycle; thus if one goes around the outer one counterclockwise, when one reaches the zipper one goes down it and then must go around the inner one clockwise and then back up the zipper to form one consistently oriented path.
We often encounter such curves or paths through equations. We also meet them described in terms of a parameter. Thus our radius 2 circle path around the origin can be described by a parameter t as follows:

r = x i+y j; x = 2cos t, y = 2sin t, 0 t 2.