Vector Line Integral

Given a vector field \( F : R^n \to R^m \), and a parametric curve \( r : R \to R^n \), we define the vector line integral:

\[ \int_C F \cdot dr = \int_a^b F(r(t)) \cdot r'(t) dt \]

Intuitively, it can be thought of being a measure of how much the curve agrees with the vector field: at each point on the curve we are taking the dot product of the tangent vector and the vector defined by the vector field at that point.

We show another common notation for a vector line integral:

\begin{align} \int_C F \cdot dr &= \int_C \langle P(x, y), \ Q(x, y) \rangle \cdot dr = \int_C \langle P(x, y), Q(x, y) \rangle \cdot \langle \frac{dx}{dt}, \frac{dy}{dt} \rangle \ dt \\ &= \int_C P(x, y)dx + Q(x, y)dy \end{align}

However, note the endpoints of the curve \( C \) are given in \( t \) values, so conversion of \( x \) and \( y \) may be necessary to evaluate the integral (alternatively we can apply the Gradient theorem if \( F \) is nice enough or Green's theorem if the curve is closed).

The FTC for line integrals, also known as the Gradient Theorem, states that the line integral through a vector field which is the Gradient of some scalar function is simply that scalar function evaluated at the lines endpoints.

Ie, if \( \vec{F} : R^n \to R^n \) and \( \vec{r} : R \to R^n \), and further \( \vec{F} = \nabla G \) for some scalar function \( G:R^n \to R \), then:

\begin{align} \int_a^b \vec{F}(r(t)) \cdot \vec{dr} &= \int_a^b \nabla G(r(t)) \cdot \vec{dr}\\ &= \int_a^b \nabla G(r(t)) \cdot r'(t) \ dt\\ &= G(b) - G(a) \end{align}

We call vector fields which have this property conservative.

The proof of theorem relies on the multivariate chain rule from which we deduce:

\[ \frac{d}{dt} G(r(t)) = \nabla G(r(t)) \cdot r'(t) \]

Let \( \theta (t) \) denote the angle between \( F(t) \) and \( r(t) \) at \( t \), now rewriting using dot products:

\[ \int_a^b F(r(t)) \cdot r'(t) dt = \int_a^b |F(r(t))| \ |r'(t)| \cos{(\theta(t))} \ dt \]

Now suppose \( r(t) \) is always tangent to \( F(r(t)) \), leaving \( \theta (t) = 1 \) ,for all \( t \):

\begin{align} \int_a^b F(r(t)) \cdot r'(t) dt &= \int_a^b |F(r(t))| \ |r'(t)| \ dt \\ &= \int_a^b g(r(t)) \ |r'(t)| \ dt \\ &= \int_C g(r) ds \end{align}

With \( g(r) = |f(r)| \). Hence we can conclude that the scalar line integral is just a special case of the vector line integral.