I want to understand the type of stress tensor $\mathbf{P}$ in classical physics.
Usually in physics it is said that the force $\text d \boldsymbol F$ (vector) acting on an infinitesimal area $\text d \boldsymbol s$ (vector) equals
$\text d \boldsymbol F = \mathbf{P} \cdot \text d \boldsymbol s$
where $\cdot$ is a "scalar product".
How can it be rigourised? I guess directed area can be $\star s$ where $s$ is a 2-form, but can I avoid using $\star$ by employing the volume form for example? The force should be 1-form.
How is the power of surface forces is written? Usually it is given by
$\frac{dA}{dt} = \int_S \boldsymbol v \cdot \text d \boldsymbol F$
$\boldsymbol v$ being the speed of the surface of the deformed body.
What would be the corresponding local form, that is the power density of surface forces?
UPDATE 1
If it helps, I found a whole appendix "The Classical Cauchy Stress Tensor and Equations of Motion" in the book "The Geometry of Physics: An Introduction" by Theodore Frankel. Particularly it says
The Cauchy stress should be a vector-valued pseudo-$(n - 1)$-form.
However currently I don't know what does it mean. Further development in the book is rather obscure and I'm afraid of that "pseudo". If a thing called "pseudo-something" I would prefer it stated as "actual another thing".
UPDATE 2
Stress tensor can also be viewed as a (molecular) flux of momentum. Then the equation for balance of momentum would be the Newton's second law. Probably this approach would be more fruitful, analogues can be made with the flux of density.