Line Integrals in Space

Line integrals in space provide a mathematical method for accumulating quantities along a three-dimensional path, such as a thin coiled spring represented by the trajectory curve. The path is defined by a position vector r, which represents the x, y, and z coordinates in terms of a single parameter t, usually time or an angle. As this parameter changes, the vector traces a smooth and continuous curve through space, defining the trajectory.

Line Integrals: Scalar-Valued Functions

For scalar-valued functions, representing the line integral with respect to arc length in terms of t involves the usual incorporation of the magnitude of r’(t). This can be used, for example, to integrate a scalar density to compute the curve's total mass.

Line Integrals: In Vector Fields

Alternatively, when interested in integrating the component of a vector field that lies in the direction of the curve’s motion, we project the field onto the unit tangent vector, T(t).

\begin{equation*}\mathbf{T}(t) = \jfrac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}\end{equation*}

Integrating this tangential component with respect to arc length causes the arc length terms to cancel:

\begin{equation*}\int_C \mathbf{F} \cdot \mathbf{T} \, ds=\int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot\Biggl(\jfrac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}\Biggr)|\mathbf{r}'(t)| \, dt=\int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) \, dt\end{equation*}

This cancellation allows the formula to be simplified into its standard, compact notation:

\begin{equation*}\int_C \mathbf{F} \cdot d\mathbf{r}\end{equation*}

For instance, when a particle travels along a helical path within a vector field, the work done is computed by summing the tangential component of the field at every point. Together, these integrals are used to calculate the accumulation of a specific quantity along various spatial paths.