タグ: vector form

VECTOR NOTATION FOR LINE INTEGRALS

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)

where \bold{A} = A_1\bold{i} + A_2\bold{j} + A_3\bold{k} and d\bold{r} = dx\bold{i} + dy\bold{j} + dz\bold{k}. The line integral (14) is a special case of this with z = 0.

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

\displaystyle \int_{C} \bold{F}\cdot d\bold{r}\cdots(18)

represents physically the total work done in moving the object along the curve C.