line integral

An ordinary integral sums a function over a straight interval on the x-axis. A line integral sums a function along a curved path through space instead — imagine walking a winding mountain trail and adding up the altitude (or the steepness, or the headwind) you experience, weighted by the length of each step you take. The integral threads along the curve rather than along an axis.

There are two flavors. The line integral of a scalar field f along a curve C, written integral over C of f ds, adds up the values of f times tiny arc-length pieces ds; it computes things like the total mass of a wire of varying density, and does not care which direction you walk. The line integral of a vector field F along C, written integral over C of F dot dr, adds up only the part of F pointing along the path — the dot product F dot dr at each step — and this one does flip sign if you reverse direction. To compute either, you parametrize the curve as r(t), substitute, and reduce everything to an ordinary one-variable integral in t: for the vector form, integral of F(r(t)) dot r'(t) dt.

Line integrals are the calculus of accumulation along a route. The vector line integral is exactly the work done by a force along a path, the central quantity that opens the door to conservative fields, potentials, and the gradient theorem. Circulation — the line integral of a velocity or force field around a closed loop — is the quantity that Green's and Stokes' theorems connect to curl. Without line integrals there is no way to even state the great integral theorems.