General properties

1. The first property involves multiplication by a scalar. Let be a scalar, be a vector field and let C be an oriented curve. Since we have the simple result that

   

2. The next property is a version of the rule that . If and are two vector fields defined on the same curve C, then

   

3. This next one involves a little notation. If the curve C is an oriented curve, the nthe curve -C is a curve that is the same as C, but oriented in the opposite direction. It stands to reason that if you go the opposite direction on C you should get the opposite answer for your line integral, and, indeed,

   

4. The last property I will list is useful if the curve C is not a smooth curve, but is a piecewise smooth curve. Such a curve may have kinks and corners in it, but between each kink or corner, the curve is smooth. Thus, the curve C can be thought of as the sum of finitely many smooth curves, . The picture below should help clarify this point.

   (illustration of C = sum(many c's))

   The line integral around the total path is, logically, the sum of the line integrals around each of the smaller curves. Thus, we are led to the statement