A Multiresolution Adaptive Wavelet Method for Nonlinear Partial Differential Equations
If you are interested in the full lecture on the Multiresolution Adaptive Wavelet Method, I have given The Journal of Computational Physics lecture as the part of the Cassyni project.
Apples and oranges. Each element in a set is a pile containing some number of apples and some number of oranges. Adding two piles means putting them together, resulting in a pile in the set. Multiplying a pile and a real number r means finding in the set a pile r times the amount. We model each pile as a vector, and the set as a two-dimensional vector space over the field of real numbers.
A vector represents different objects as a single entity. A pile containing some number of apples and some number of oranges is a vector. The addition of two vectors does not require us to add apples and oranges. Rather, in adding two piles, we add apples to apples, and oranges to oranges. The addition of vectors generalizes the addition of numbers: adding two vectors corresponds to adding two lists of numbers in parallel.
Mass and volume. We can also list different physical quantities together as a single object. Consider a set, each element of which is a piece of some mass and some volume. Adding two pieces means putting them together, resulting in a piece in the set. Multiplying a piece and a real number r means finding in the set a piece r times the amount. This set is a two-dimensional vector space over the field of real numbers. We do not have any familiar name for this vector space, and will call it massvolume.
Spacetime. When we list apples and oranges together, or volume and mass together, the results do not surprise us. But when Einstein and Minkowski listed directed segments in space and directed intervals of time together, the result was shocking. Minkowski said, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” What makes spacetime, but not massvolume, so shocking and so enduring?
I'm curious how you think about it.