Vector Calculus

Four different vector fields on (x,y)-plane A vector field on a sphere ♣ Curves. A smooth curve is a continuous curve (no gaps or discontinuities) with no corners (no abrupt changes in slope at a point such as you would get from the intersection of two lines). A continuous and differentiable produces a smooth function. A path C is called closed if its initial and final points are the same point. For example a circle is a closed path. A path C is simple, if it doesn't cross itself. A circle is a simple curve while a curve in the shape of a figure 8 is not simple. A curve C has a positive orientation, if it is traced out in a counterclockwise direction. A piecewise curve is a union of finite number of smooth curves C 1 , C 2 , C 3 , · · · C n−1 , C n , where the initial point of C k+1 is the end point of C k .