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The mathematics of the hydrostatic equation is that of the Momentum Equation (Newton's 2nd Law) applied to a small "element" of static fluid. Static does not mean "not moving." It means "not disturbed." It means the sum of forces acting on it equals zero. The sum of forces are zero with no motion and with uniform motion.
When the size of an "element" is imagined to become smaller and smaller... until its size vanishes, this limit requires the calculus definition of a derivative. This is the term, dp(z)/dz, in the equation below which is commonly called the Hydrostatic Equation. Here z (called the independent variable) is a measure of distance vertical-to-Earth with values increasing upward. Also, since most applications are for fluids on Earth and with but slight differences of elevation, the acceleration of gravity is expressed (not as a function of z but as the Earth-surface constant, g0,Earth.
The above form, though correct, is not used. The equation leads with "0" which means simply that the rate of change of momentum of every fluid particle is zero, d(mV)/dt = 0. Right of equality the equation expresses that the sum of forces (gravity force plus gradient of pressure-forces) equals zero. Almost always, the equation is algebraically manipulated. In usual developments of the hydrostatic equation, that it is a consequence of Newton's Second Law is obscured. Also it is presented in "integrated form.
To start a study by use of a differential equation is superior; in the diff-eq averaging has not been applied. It is always best to write a differential equation as explicitly as possible. An equation equivalent to the above which shows the dependence of variables is our preference:
Equation (3) above is an ordinary, first-order differential equation. re are a few distinctions in solution of this equation. The solutions are important to learn but much less important than the manner of solution. Also (but less so) are the resultant solutions which predict realities of fluid behaviors. Full understanding of the mathematics of solution are a path, a learning directly applicable to the mathematics of other physical systems. Therefore please take these solutions and how they are accomplished seriously.
Steps of Solution:
We have experience in solving first-order differential equations. It is simply a matter of separation of variables then integration of the equation. All equations are solve by the same set of steps. We begin with equation (1) written above.
Separate Variables of the Differential Equation. When equation (1) is multiplied by the differential quantity, dz, variables are separated.
Apply the integration operator: No thinking for this step. Integration is a "linear" operation. Apply integral signs to terms left-and-right of the equality.
Our next task is to specify the limits for the integrals. The limits of an integral are related to its differential. Thus for dp(z) and for dz the limits will be a pair of pressures and a pair of elevations, respectively. For a specific situation, limits could be specified "explicitly" with numbers with dimensions. Here to proceed in general, we use symbols to obtain an implicit solution.
Left-of-Equality: The lower limit is a pressure at a location, "1." The upper limit of that integral is the pressure at location "2."
Right-of-Equality: The lower and upper limits of this integral are the elevations, z1 and z2 respectively.
We are developing an implicit solution. The next step is to integrate both sides of our equation.
Integration ~ Left-of-Equality: This integral is accomplished "by definition." Upon this integration, our equation becomes as shown below left.
Integration ~ Right-of-Equality: The Mean Value Theorem of Calculus is applied to this integral. The effect is to assume the average value of the density over the limits is known. That constant is brought out leaving a simple leaving. Integration of both sides yields the below.
Further statements about solutions are made in the examples.