Can anyone tell me what the heck the following means? Don't have to get technical, just what is it referring to? It has to do with water. At first I was thinking it had something to do with water pressure but really have no idea.

Using your clue of water.
I went to good ol' Wikipedia on the term Reynolds number. Within that article there's a list of equations.
Your equation appears to be an estimate of the incompressible flow. The Navier-Stokes equation.

considering we are in a Math forum
it is obviously something to do with Vectors

may imply either of velocity or loci
or neither :>

Pin~Jinx / anarchist

I could be wrong, but I think that one of the Clay Institute problems is to give insight into this equation . . . how it works, or something like that.

Peteeo is correct in identifying it as the Navier-Stokes Equation (Wiki. article) for the motion of fluids. The general equation describes the flow of gases and other compressible fluids, though the form given in Kendor's question is for incompressible fluids. It is derived from the principle of the conservation of momentum.

u is flow velocity, t is time, rho is fluid density, p is pressure, F is external forces such as gravity, and gamma is viscosity. The upside-down triangle is the del operator of vector calculus. Overall the equation is a non-linear differential equation typical for describing motion in 3 dimensions. The equation can be used to solve even turbulent flow, though not without difficulty!

Tsaeb is correct in making the connection to the Clay Institute problems. From the article cited above:

quote:

The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist, or that if they do exist they do not contain any infinities, singularities or discontinuities (smoothness). These are called the Navier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics, and offered a $1,000,000 prize for a solution or a counter-example.

wow Professor, you surely lived up to your handle!

hmn, so my stab at velocity and vectors wasn't so far out, afterall.

Ofcourse though, would have to admit, wasn't as specif as the others..

The little arrows over the variables u and F indicate that they are vectors. The Wikipedia article explains things better than I could.

It's easy to look smart after reviewing material online. In an earlier life -- as a college physics major -- I lived and breathed vector calculus, but never studied the Navier-Stokes eq.

Professor: What's the del operator of vector calculus--something telling us to differentiate using a particular differential calculus method?

Well, if you're familiar with basic calculus you know that a function y = f(x) has a derivative y' = dy/dx representing the rate of change of y with respect to x. This can be generalized to 3 dimensions, where the derivative becomes something called the gradient, denoted by del (upside-down triangle) or grad, defined for a function u(x,y,z) as:

grad(u) = (i∂/∂x + J∂/∂Y + k∂/∂Z) u
where i, j, and k are unit vectors along the x-, y-, and z- axes respectively and ∂ is the partial derivative symbol.

For example, if u represents an electrostatic potential field, then grad(u) is a vector pointing in the direction that the field is changing most strongly, whose length is the magnitude of the change.

Del can also be combined with a vector function using dot product and cross product, producing the so-called divergence and curl of the function, respectively. Del can be operated upon itself (del-dot-del = del-squared = Laplacian operator) to express additional properties of the field.

Wikipedia has an entry under "del". That article, and others it links to, can give you an introduction -- as can almost any introductory college physics text.

Professor: I think that the subject may be in my old (and fat) calculus textbook. However, I would like to know when you mention divergence and curl, what are they, and do they have anything to do with space-time curvature--before I refresh my memory on the dot product and cross product stuff?

In physics these concepts are most commonly applied to electromagnetism, with divergence and curl appearing in Maxwell's equations -- the elegant set of four equations unifying electricity and magnetism. The electric and magnetic fields have different properties. In particular, the divergence of magnetic field is always zero (which implies that there are no magnetic monopoles); the curl of electric field is always zero (which implies that it lacks rotation).

Not sure about space-time curvature. Although I studied special relativity, general relativity was beyond the scope of the undergraduate physics curriculum at Cornell. It uses tensor equations and a set of operators that are different from div, grad, and curl of vector calculus.

Also, this was was a really long time ago. I now resemble your calculus textbook.

Professor: Some day I will tackle Maxwell's equations before I look over any of Einstein's work. I'll start with the vector stuff, which is simple, relatively speaking that is.