It is often convenient to express a line integral in vector form as an aid in physical or geometric understanding as well as for brevity of notation. For example, we can express the line integral (15) in the form
![\displaystyle \int_{C}[A_1dx + A_2dy + A_3dz] \\ = \int_{C} (A_1\bold{i} + A_2\bold{j} + A_3\bold{k}) \cdot (dx\bold{i} + dy\bold{j} + dz\bold{k})\\ = \int_{C} \bold{A}\cdot d\bold{r} \cdots (17)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cint_%7BC%7D%5BA_1dx+%2B+A_2dy+%2B+A_3dz%5D+%5C%5C++%3D+%5Cint_%7BC%7D+%28A_1%5Cbold%7Bi%7D+%2B+A_2%5Cbold%7Bj%7D+%2B+A_3%5Cbold%7Bk%7D%29+%5Ccdot+%28dx%5Cbold%7Bi%7D+%2B+dy%5Cbold%7Bj%7D+%2B+dz%5Cbold%7Bk%7D%29%5C%5C++%3D+%5Cint_%7BC%7D+%5Cbold%7BA%7D%5Ccdot+d%5Cbold%7Br%7D+%5Ccdots+%2817%29+&bg=T&fg=000000&s=0 "\displaystyle \int_{C}[A_1dx + A_2dy + A_3dz] \\ = \int_{C} (A_1\bold{i} + A_2\bold{j} + A_3\bold{k}) \cdot (dx\bold{i} + dy\bold{j} + dz\bold{k})\\ = \int_{C} \bold{A}\cdot d\bold{r} \cdots (17)")
where
and
. The line integral (14) is a special case of this with
.
If at each point (x, y, z) we associate a force F acting on an object (i.e. if a force field is defined), then
represents physically the total work done in moving the object along the curve C.
