Infinitesimal work

Question

It is usually to find that $\delta W=f\cdot dr$ is the definition of infinitesimal work. My question is:

(1) if $f$ is a smooth vector field (force), and

(2) $dr$ is the differential of a function $r:I\to \mathbb{R}^3$

How is defined the product $f\cdot dr$?

Many thanks!

Answer 1

In $\mathbb{R}^3$ using Cartesian coordinates,the displacement $\vec{dr}$ is a vector: $$ \vec{dr}=(dx,dy,dz) $$ and the force is a vector $$ \vec f=(f_x,f_y,f_z) $$

so $\vec f \cdot \vec{dr}=f_xdx+f_ydy+f_zdz$ is a differential 1-form.