A Taylor Series expansion (neglecting higher order terms) and Newton’s second law are essential for the derivation of a pressure field and fluid at rest.

\[ \sum _{n=0} ^{\infty} \frac {f^{n} (a)} {n!} (x-a)^{n} = f(a) + \frac {f'(a)} {1!} (x-a) + … \]

where \( a \) is a point in the \(x\) direction, \(f(a)\) is the function of interest evaluated at \(a\), and \(f'(a)\) is the derivative of \(f(x)\) with respect to \(x\) and evaluate at point \(a\). \[ \sum \vec{F} = m \vec{a} \]Where \(\vec{F}\) is a force vector, \(\vec{a}\) is an acceleration vector, and \(m\) is mass.

Using a fluid element, the surface and body forces can be used to derive the equation of motion for a fluid with no shear stresses.

\[-\nabla P -\gamma \hat{k} = \rho \hat{a}\]

where \(\nabla P\) is the gradient of pressue, \(\gamma\) is specific weight of the fluid, \(\hat{k}\) is the unit vector in the z direction, \(\rho\) is the density of the fluid, and \(\hat{a}\) is an acceleration vector. 

When a fluid is at rest, the acceleration vector is zero. Thus, the only non-zero component is in the z direction. 

\[\frac{dP}{dz}=-\gamma\]