Lets assume block motion for a fluid. From the Navier-Stokes equation we get $\vec \nabla p = \rho (\vec g - \vec a)$. Let us say s is the direction where pressure has the steepest increase. How do you calculate the change of pressure in the direction s ?
According to my book, the answer is, $$ dp=(\vec \nabla p \cdot\ d\vec s). $$
My question is: Instead of this, shouldn't we simply project the gradient of pressure on an unit vector of path s ?
To calculate work, you do $dW=\vec F \dot\ d\vec r $ , so you’re getting F multiplied by the magnitude of dr on the path, but that’s the definition of work. You apply a force and when you multiply it by the distance you get WORK. In my case, shouldn’t we just do the dot product between the gradient of pressure with the unit vector of path?