We discuss in what sense the scalar line integrals do not depend on the orientation of the curve.
Let us start with integrals where is a Riemann integrable function. Let and let be such that let for and We can now form the Riemann sum
By considering the limit where such that we obtain the Riemann integral
Notice that we now have only defined the integral for the case when This definition does not apply to integrals of the form where Hence we are still free to choose what we mean by for It is customary, however, to define
This corresponds to reversing the partition in the sense that we now have that is, we now have instead of
Let now If for all then for all and hence it follows that Let now that is, we reverse the orientation in the sense that and Then we still have What happens now to Using the substitution we obtain
since . Thus, in this sense, integrating the function from to yields the same result as when integrating from to
Scalar line integrals
A similar result holds for scalar line integrals. Assume we have a curve with end points and and a continuous function defined on the curve Let where be a continuously differentiable parameterisation of such that and Then, if we define by then and
Hence, as goes from to the parameterisation traverses the curve from to whereas the parameterisation traverses the curve in the opposite direction from to Then, by using the substitution we obtain
since In this sense the scalar line integral is independent of the orientation of the curve For instance, this implies that if and for all points on the curve, then
regardless in which direction the parameterisation traverses the curve Hence, when computing scalar line integrals, we do not need to specify the direction in which the parameterisation traces out the curve
However, we have
In this case the orientation of the curve is also reversed, but the minus sign comes from the convention (1) and not from changing the orientation of the curve, in the sense that if we left out the minus sign in (1), we would also get
Note that this implies that the scalar line integral needs to be set up such that we have where
Line integrals of vector fields
We now consider line integrals of vector fields. In this case the situation is different. We use the same notation as in the part on scalar line integrals.
Let be a vector field defined on the curve We have Hence, if we now go through the same steps as above we obtain
Hence, changing the orientation of the curve for line integrals over vector fields also changes the sign in the result.
Both of these results are also clear from the interpretation of scalar line integrals as the area of a fence and the interpretation of line integrals of vector fields as the work done by a force.