Line integral

A line integral of a vector function f along a line or curve segment C is the following (all three formulations are equivalent):

$\int _{\mathbf {C} }{\vec {f}}({\vec {s}})\cdot d{\vec {s}}$

$\int _{\mathbf {C} }f_{1}s_{1}dx+f_{2}s_{2}dy+f_{3}s_{3}dz$

$\int _{\mathbf {C} }\langle {\vec {f}}({\vec {s}}),d{\vec {s}}\rangle$

Example: The work done on a particle to move it from one point to another is the line integral of the force on the particle along the curve of its motion:

$\int _{\mathbf {C} }{\vec {F}}({\vec {r}})\cdot d{\vec {r}}$

Note that if the contour is a closed curve (one that wraps around itself without intersecting) and if the vector field is a conservative, then the line integral must be zero. This is the case in physics whenever a particle is moved and then returns to original position: its line integral for its force field is the work performed and it is zero.

A line integral of a non-vector function is the summation of the values taken by the function (its integral) over the domain defined by the line.