Vector Line Integrals

A vector field is defined as a set of vectors associated with each point in space. For instance, the velocity ${\bf v}({\bf r})$ in a moving liquid (e.g., a whirlpool) constitutes a vector field. By analogy, a scalar field is a set of scalars associated with each point in space. An example of a scalar field is the temperature distribution $T({\bf r})$ in a furnace.

Consider a general vector field ${\bf A}({\bf r})$. Let $d{\bf r} \equiv (dx,\,dy,\,dz)$ be the vector element of line length. Vector line integrals often arise as

$$\int_P^Q {\bf A}\cdot d{\bf r} = \int_P^Q (A_x\,dx+A_y\,dy + A_z\,dz).$$ (1.81)

For instance, if ${\bf A}$ is a force-field then the line integral is the work done in going from $P$ to $Q$.

As an example, consider the work done by a repulsive inverse-square central field, ${\bf F} = - {\bf r}/ \vert r^{\,3}\vert$. The element of work done is $dW={\bf F}\cdot d{\bf r}$. Take $P=(\infty, 0, 0)$ and $Q=(a,0,0)$. Route 1 is along the $x$-axis, so

$$W = \int_{\infty}^a \left(-\frac{1}{x^{\,2}}\right)\,dx = \left[\frac{1}{x}\right]_{\infty}^a =\frac{1}{a}.$$ (1.82)

The second route is, firstly, around a large circle ($r=$ constant) to the point ($a$, $\infty$, 0), and then parallel to the $y$-axis. See Figure A.16. In the first part, no work is done, because ${\bf F}$ is perpendicular to $d{\bf r}$. In the second part,

$$W = \int_{\infty}^0 \frac{-y\,dy}{(a^{\,2} + y^{\,2})^{3/2}} = \left[\frac{1}{(y^{\,2}+a^{\,2})^{1/2}} \right]^0_\infty = \frac{1}{a}.$$ (1.83)

In this case, the integral is independent of the path. However, not all vector line integrals are path independent.

Figure A.16: An example vector line integral.