These integrals are known as \sword {line integrals over vector fields}. By contrast, the line integrals we dealt with in Section \ref {sec:line_int_intro} are sometimes referred to as \sword {line ...

A vector field on two or three dimensional space is a function $\vec{F}$ that assigns to each point $(x,y)$ or $(x,y,z)$ a two or three dimensional vector given by $\vec{F}(x,y)$ or $\vec{F}(x,y,z)$.