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Pastean
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Have you studied any calculus? Any fluid statics?Pastean said:I do not understand why this notation means ∂p/∂z
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.
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,
∂z^{2}/∂x (2nd order/degree partial derivative with respect to x in function f(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.Pastean said:Oh I see, I was used to the following notation, hence the confusion: ƒ'_{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: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. As another member mentions, no prime symbol (') is used with partial derivatives.Pastean said:, or in the notation I have learned, f'_{x} (notice no parentheses on x), given the function right above it
Again, no. ∂z/∂y means the partial derivative of z (not f(z)) with respect to x.Pastean said:∂y/∂x means the partial derivative with respect to y in the function f(z), or f'_{y}
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.Pastean said:For the 2nd order partial derivatives,
∂z^{2}/∂x (2nd order/degree partial derivative with respect to x in function f(z) )
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.
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.
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.
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.
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.