We now define some particular kinds of integrals as special cases of our theory of integrals over manifolds. And the first such special case is that of a line integral.
More explicitly, the pullback is evaluated as
That is, for a , we take the value of the -form at the point and the tangent vector and pair them off. This gives us a real-valued function which we can integrate over the interval.
So, why do we care about this particularly? In the presence of a metric, we have an equivalence between -forms and vector fields . And specifically we know that the pairing is equal to the inner product — this is how the equivalence is defined, after all. And thus the line integral looks like
Often the inner product is written with a dot — usually called the “dot product” of vectors — in which case this takes the form
We also often write as a “vector differential-valued function”, in which case we can write
Of course, we often parameterize a curve by a more general interval than , in which case we write
This expression may look familiar from multivariable calculus, where we first defined line integrals. We can now see how this definition is a special case of a much more general construction.