# Understanding notation for hydrostatic equation

• Pastean
In other words,∂z2/∂x2 = ∂/∂x ( ∂z/∂x )In summary, the notation ∂p/∂z represents the partial derivative of pressure in a fluid with respect to depth in the fluid. It is a way to express the change in pressure with change in depth below the surface of the fluid, and is related to Pascal's Law. It is a form of Leibniz notation and is used in calculus to represent partial derivatives. It can be expressed in multiple ways, including the "curly d" notation (∂) and regular differential notation (d).
Pastean
I understand everything (in a really simple way) that lies behind the hydrostatic equation, but I have no idea what this notation (light blue) means. I would really appreciate as much information as possible on this. I want to fully understand it.

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Why don't you start and layout your understanding first? You said you understand everything so I am confused about specifically what it is you have no idea about in the equation.

I do not understand what this notation means ∂p/∂z

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Pastean said:
I do not understand why this notation means ∂p/∂z
Have you studied any calculus? Any fluid statics?

That notation means the partial derivative of the pressure in a fluid with respect to depth in the fluid, or the change in pressure with change in depth below the surface of the fluid.

In other words, it's a fancy way to state Pascal's Law:

http://en.wikipedia.org/wiki/Pascal's_law

Pastean
Oh I see, I was used to the following notation, hence the confusion: ƒ'z
Now when I look back at it, I feel bad for posting a thread just for this, but I just couldn't figure it out.

Pastean said:
Oh I see, I was used to the following notation, hence the confusion: ƒ'z
Now when I look back at it, I feel bad for posting a thread just for this, but I just couldn't figure it out.

Well there is a subtle difference between f'(z) = df / dz and say f(x,y) = z and ∂z / ∂x or ∂z / ∂y. Can you spot it?

That's why I asked if you had studied calculus. Atmospheric pressure depends on other variables besides altitude:

http://en.wikipedia.org/wiki/Barometric_formula

Pastean
Let me see if I got it right, this notation still bugs me, but I've been searching the web for the Leibniz notation (as far as my powers of using google go, this is what it is called).

∂z/∂x means the partial derivative with respect to x in the function f(z), or in the notation I have learned, f'x (notice no parentheses on x), given the function right above it
∂y/∂x means the partial derivative with respect to y in the function f(z), or f'y

For the 2nd order partial derivatives,
∂z2/∂x (2nd order/degree partial derivative with respect to x in function f(z) )

Pastean said:
Let me see if I got it right, this notation still bugs me, but I've been searching the web for the Leibniz notation (as far as my powers of using google go, this is what it is called).

∂z/∂x means the partial derivative with respect to x in the function f(z), or in the notation I have learned, f'x (notice no parentheses on x), given the function right above it
∂y/∂x means the partial derivative with respect to y in the function f(z), or f'y

For the 2nd order partial derivatives,
∂z2/∂x (2nd order/degree partial derivative with respect to x in function f(z) )

http://en.wikipedia.org/wiki/Partial_derivative

Note the multiple ways in which the same partial derivative can be expressed. The "curly d" notation (∂) is just one way to do this, being similar to the regular differential notation (d) employed to signify derivatives of functions of a single variable.

Pastean said:
Oh I see, I was used to the following notation, hence the confusion: ƒ'z
That is NOT a standard notation. f ' is often used for a regular derivative and fz for a partial derivative but not the two together.

Pastean said:
Let me see if I got it right, this notation still bugs me, but I've been searching the web for the Leibniz notation (as far as my powers of using google go, this is what it is called).

∂z/∂x means the partial derivative with respect to x in the function f(z)
No. ∂z/∂x means the partial derivative of z (not f(z)) with respect to x. Since we're talking about partial derivatives, we can infer that z is probably, but not necessarily some function of two, or possibly more, variables. IOW, z = f(x, y)
Pastean said:
, or in the notation I have learned, f'x (notice no parentheses on x), given the function right above it
No. As another member mentions, no prime symbol (') is used with partial derivatives.
Pastean said:
∂y/∂x means the partial derivative with respect to y in the function f(z), or f'y
Again, no. ∂z/∂y means the partial derivative of z (not f(z)) with respect to x.
Pastean said:
For the 2nd order partial derivatives,
∂z2/∂x (2nd order/degree partial derivative with respect to x in function f(z) )
The second partial of z with respect to x. To get this, take the partial with respect to x of the partial of z (not f(z)) with respect to x.

## 1. What is the hydrostatic equation?

The hydrostatic equation is a fundamental equation in fluid mechanics that describes the relationship between pressure, depth, and density in a static fluid. It states that the pressure at any point in a fluid at rest is equal to the weight of the fluid above that point, divided by the area over which the weight is distributed.

## 2. Why is notation important for understanding the hydrostatic equation?

Notation is important because it allows us to express complex mathematical relationships in a concise and standardized way. In the context of the hydrostatic equation, notation allows us to represent the variables of pressure, depth, and density in a clear and consistent manner, making it easier to manipulate and understand the equation.

## 3. What are the key symbols used in the notation for the hydrostatic equation?

The key symbols used in the notation for the hydrostatic equation are P for pressure, ρ for density, g for acceleration due to gravity, and h for depth. These symbols are typically used in combination with subscripts and superscripts to represent specific points or depths within the fluid.

## 4. How does the hydrostatic equation relate to the principles of buoyancy?

The hydrostatic equation is closely related to the principles of buoyancy, as it explains why objects float or sink in a fluid. The equation shows that the pressure at the bottom of an object submerged in a fluid is greater than the pressure at the top, resulting in a net upward force that causes the object to float.

## 5. Are there any real-world applications of the hydrostatic equation?

Yes, the hydrostatic equation has numerous real-world applications, particularly in engineering and environmental sciences. It is used to design and analyze structures that are in contact with fluids, such as dams and pipelines. It is also used to understand and predict changes in ocean and atmospheric pressure, as well as in the development of weather forecasting models.

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