Formally stated, a curve C having continuous position vector  is piecewise smooth if  is continuous and different from  at all but possibly a finite number of values of t.

One such example is shown below.

Notice that there are points on the curve, namely the dots, at which there is no tangent vector.

To evaluate such a line integral, the rule we use is quite simple. If C is piecewise smooth, consisting of smooth curves , the line integral of  over C is defined to be the sum of the line integrals of  over each of the smooth curves making up C or written symbolically as

There are a few things to keep track of:
1. The orientation along C must be maintained over the curves  implying that the initial point of  is the terminal point of the smooth curve preceding it namely . Another way of looking at it is that the smooth curves must all be connected.

2. The vector field  is the same for all the individual smooth curves but the position vector  is different. It is the position vector of that specific curve in the subscript of the integral sign, i.e.,  in the integral must be the position vector describing .

3. The usual rules of calculating line integrals apply for each individual smooth curve.