Line integral

A line integral integrates a scalar or tangential vector-field component along a curve.

A line integral is path dependent when two admissible paths with the same starting and ending points give different values. This disproves path independence and, for a vector field, shows that it is not a conservative vector field on the relevant domain.

If $h:[\alpha ,\beta ]\to [a,b]$ is an increasing piecewise smooth change of parameter, with endpoints mapped to endpoints, the chain rule gives $(\gamma \circ h{)}^{\prime }=({\gamma }^{\prime }\circ h)h^{\prime }$. Change of variables in the defining line integral proves the displayed identity. A decreasing change of parameter reverses the endpoints and the sign. The result concerns the same traversal with the same multiplicity; retracing a loop additional times changes the integral in general.

A line integral can be defined as an improper integral of its parametrized integrand when that integrand is undefined or unbounded at isolated parameter values. At an interior singular parameter, split the integral and require both one-sided limits to exist separately; their cancellation in a symmetric Cauchy principal value is a different convention. For a scalar arc length integral, use $f(\mathbf{r}(t))|{\mathbf{r}}^{\prime }(t)|$ in place of the vector-field dot product.

If the restricted integrand has a removable singularity, integrating its continuous extension gives the same value. This does not require the original vector field to extend continuously to a full neighbourhood of the curve. Likewise, a differential identity may hold only on the regular segments: a zero integrand there has improper integral zero, although its printed classical expression may not be defined at the omitted points.

For an oriented parametric curve $\mathbf{r}(t)$ and scalar field $V$, define the componentwise line integralIts value is a vector and reversing orientation changes its sign. It differs from the scalar arc length integral ${\int }_{C}Vds=\int V(\mathbf{r}(t))|{\mathbf{r}}^{\prime }(t)|dt$, which is independent of orientation. A request to integrate a scalar field along a curve should specify which differential is intended.

For the counterclockwise circle $\mathbf{r}(\varphi )=(R\cos \varphi ,R\sin \varphi ,z_0)$, suppose the restricted field has Fourier series $V(\varphi )=A_0+{\sum }_{n\ge 1}(A_n\cos n\varphi +B_n\sin n\varphi )$. Orthogonality givesThe oriented vector integral selects the first angular harmonics, whereas the scalar arc-length integral selects the mean.