I have a question and I'm not sure which equation to apply to it:
A fluid is at rest in a gravitational field of strength $\mathbf g = −g \underline k$, where $g$ is a positive constant, and both the unit vector $\underline k$ and the $z -axis$ point vertically upward. The fluid pressure, $p$ , and the fluid density, $\rho$ , are related by $∇p = \rho g$.
Show that in the fluid $$\frac {dp}{dz} = −\rho g$$.
Thanks
$$\nabla p=\left(\begin{matrix}\frac{\partial p}{\partial x}\\\frac{\partial p}{\partial y}\\\frac{\partial p}{\partial z}\end{matrix}\right)$$
Then $$\frac{\partial p}{\partial z}=\nabla p\cdot \vec k=\rho \vec g\cdot\vec k=-\rho g \vec k\cdot\vec k=-\rho g$$ There is no eliminating terms, this is quite easy to derive from the properties given.