The line integral over a scalar field f can be thought of as the area under the curve C along a surface z = f(x,y), described by the field.

Stoke's Theorem: The value of the line integral along a simple, closed, piecewise-smooth, positively oriented curve C, is related to the area of the surface C encloses. F must have continuous partial derivatives on a region in ℝ³. Stoke's theorem can be used to find the magnetic field strength a given distance from a straight wire (Ampere's law). C would represent the circumference of an imaginary circle at a constant distance around the wire, and the right side of the equation would be…

Maxwell’s Equations in Differential and Integral form.

The trajectory of a particle (in red) along a curve inside a vector field. Starting from a, the particle traces the path C along the vector field F. The dot product (green line) of its displacement vector (red arrow) and the field vector (blue arrow) defines an area under a curve, which is equivalent to the path's line integral.

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Introduction to the line integral

Line integral of scalar field In mathematics, a line integral is an integral where the function to be integrated, be it a scalar field as here or a vector field, is evaluated along a curve. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve).

The trajectory of a particle along a curve inside a vector field. At the bottom are the vectors of the field seen by the particle as it travels along the curve. The sum of the dot products of these vectors with the tangent vector of the curve at each point of the trajectory results in the line integral. Click thru to animation.

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