line integral (redirected from Tangential line integral)

line integral

In mathematics, the integral of a function of several variables defined on a line or curve that has been expressed in terms of arc length (see length of a curve). An ordinary definite integral is defined over a line segment, whereas a line integral may use a more general path, such as a parabola or a circle. Line integrals are used extensively in the theory of functions of a complex variable.

---

line integral [′līn ¦int·ə·grəl]

(mathematics)

For a curve in a vector space defined byx=x(t), and a vector functionVdefined on this curve, the line integral ofValong the curve is the integral overtof the scalar product ofV[x(t)] anddx/dt; this is written ∫V·dx.

For a curve which is defined byx=x(t),y=y(t), and a scalar function ƒ depending onxandy, the line integral of ƒ along the curve is the integral overtof ƒ[x(t),y(t)] · √(dx/dt)²+ (dy/dt)²); this is written ∫ ƒds, whereds= √(dx)²+ (dy)²) is an infinitesimal element of length along the curve.

For a curve in the complex plane defined byz=z(t), and a function ƒ depending onz, the line integral of ƒ along the curve is the integral overtof ƒ[z(t)] (dz/dt); this is written ∫ ƒdz.