Can a Single Scalar Variable Adequately Label All Particles in a 3D Space?

[J Hill]

Okay, so I've recently been reading through C. Pozrikdis' Introduction to Theoretical and Computational Fluid Dynamics, and came across an interesting exercise: "Discuss whether it is possible to label all point particles within a finite three-dimensional parcel using a single scalar variable, or even two scalar variables."

Now, mathematically speaking, it should be possible-- the set that contains the volume of a fluid should have an uncountably infinite number of points, but its cardinality should be the same as, say, the unit interval. Therefore, it should be possible to create a mapping from the scalar quantity to the region of space, and should be sufficient (though perhaps impractical) for a Lagrangian coordinates. Does this seem like a valid argument, though?

An argument against this that was posed is that that mapping is not necessarily continuous, but does this condition have to be met?

 

[Valerie Park]

Is continuity necessary for this exercise? I'm not sure how to explain the concept of continuity in this context, so any help would be appreciated.Yes, continuity is important in order for a single scalar variable (or two scalar variables) to label all point particles within a finite three-dimensional parcel. Without continuity, it is impossible to create a meaningful mapping from the scalar quantity to the region of space. To explain further, continuity means that points close together in the space should be labeled with values close together in the scalar variable. This is necessary because the values of the scalar variable will determine how the particles interact with each other, and so the values must be consistent for all points in the region. Without continuity, the values could jump around unpredictably, making it impossible to predict the behavior of the particles.