Theory

Introduction

The aligned vector line integral is the integral of a vector field over a curve according to its alignment with the curve. It measures how much the vector field flows along the curve. It does not depend on the chosen parametrization (i.e. on the speed), but its sign depends on the direction traveled along the curve. It is given for a parametric curve $\mathbf{r}(t)$ that traverses its image $\mathcal{C}$ one time, together with a vector field $\mathbf{F}$, by the formula:

$${\int }_{\mathcal{C}}\mathbf{F}\cdot d\mathbf{s}={\int }_{\mathcal{C}}\mathbf{F}(\mathbf{r}(t))\cdot {\mathbf{r}}^{\prime }(t)dt.$$

Here we use the infinitesimal vector $d\mathbf{s}={\mathbf{r}}^{\prime }(t)dt$. This is the tangent vector scaled by $dt$; alternatively it is the vector of infinitesimal travel of the particle during $dt$.

The same line integral is sometimes written with a different notation:

$${\int }_{\mathcal{C}}\mathbf{F}\cdot d\mathbf{s}={\int }_{\mathcal{C}}{F}_{1}dx+F_{2}dy+F_{3}dz.$$

This notation is produced by writing $d\mathbf{s}=(dx,dy,dz)$ and expanding the dot product.

To evaluate a line integral in this notation, first write a parametrization $\mathbf{r}(t)=(x(t),y(t),z(t))$ that traverses $\mathcal{C}$ exactly once, then substitute $x^{\prime }(t)dt$ for $dx$, and $y^{\prime }(t)dt$ for $dy$, and $z^{\prime }(t)dt$ for $dz$, then factor out the $dt$ and evaluate the integral in $t$.