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?
[Math] Difference between “scalar line integral” and “line integral”
curvesintegrationmultivariable-calculus
Best Answer
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.