Understanding notation for hydrostatic equation

[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.

 

[paisiello2]

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.

 

[Pastean]

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

 

[SteamKing]

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.

 

[SteamKing]

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) )

 

[SteamKing]

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) )

This article may help you understand partial differentiation better:

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.

 

[HallsofIvy]

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 f_z for a partial derivative but not the two together.

 

[Mark44]

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)

, 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.

∂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.

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.