Difference between "scalar line integral" and "line integral"
What is the difference between the phrases "scalar line integral" and "line integral"? If the phrases are equivalent, what purpose does the adjective "scalar" serve in the phrase; why is it there?
$\endgroup$ 21 Answer
$\begingroup$In the calculus textbook that I learned multivariable calculus from (Marsden),
they had two terms, the line integral along a curve $C$ of a scalar function $f(x,y,z)$ given by
$$\int_Cf(x,y,z)\,ds.$$
Given a parametrization of the curve $C=(x(t),y(t),z(t))$ this could be computed explicitly as a one-variable integral
$$\int_{t_0}^{t_1}f(x(t),y(t),z(t))\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2+\left(\frac{dz}{dt}\right)^2}\,dt.$$
And then there's the path integral of the vector field $\mathbf{F}(x,y,z).$ Which is denoted
$$\int\mathbf{F}(x,y,z)\cdot\,d\mathbf{s}.$$
It is a work computation, which only applies to vector fields, not to scalar functions. It is this latter integral which appears in the fundamental theorem of calculus, and Stokes' theorem.
I don't know source you used, or what your author had in mind, but my guess would be that what Marsden is calling a path integral, is what your author is calling a scalar line integral (it's the integral along a path of a scalar function). And what Marsden calls a line integral is what your author is calling a "line integral" (of a vector field).
Of course, if you give the name of your textbook, we can read the text to be sure of the answer.
$\endgroup$
