quality:corruption, structural:expansion, information:highlighting, structural:reformatting
Definition

For a vector field F: U ⊆ R → R, the line integral along a piecewise smooth curve C ⊂ U, in the direction of r, is defined as ∫ C F (r) ⋅ d r = ∫ a b F (r (t)) ⋅ r ′ (t) d t {\displaystyle \int _{C}\mathbf {F} (\mathbf {r})\cdot d\mathbf {r} =\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt} where · is the dot product, and r: → C is a regular parametrization (i.e: | | r ′ (t) | | ≠ 0 ∀ t ∈ {\displaystyle ||\mathbf {r} '(t)||\neq 0\;\;\forall t\in }) of the curve C such that r (a) and r (b) give the endpoints of C.

A line integral of a scalar field is thus a line integral of a vector field, where the vectors are always tangential to the line of the integration.

Line integrals of vector fields are independent of the parametrization r in absolute value, but they do depend on its orientation. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.

From the viewpoint of differential geometry, the line integral of a vector field along a curve is the integral of the corresponding 1-form under the musical isomorphism (which takes the vector field to the corresponding covector field), over the curve considered as an immersed 1-manifold.