A line integral, also known as a path integral or contour integral, is an integral that calculates the accumulation of a scalar or vector quantity along a curve. Line integrals are essential in multivariable calculus and are used to generalize the concept of a definite integral from a single-variable function to a function of multiple variables.
Line integrals have two main types: scalar line integrals and vector line integrals. Scalar line integrals calculate the accumulation of a scalar function along a curve, while vector line integrals involve the dot product of a vector field and a differential displacement vector along a curve.
Line integrals have numerous applications in physics, engineering, and mathematics. They are used to calculate work done by a force field, mass and center of mass of a wire, fluid flow along a curve, and more. Line integrals are also closely related to the fundamental theorems of calculus in higher dimensions, such as Green’s theorem, Stokes’ theorem, and the divergence theorem.