This article defines a property that can be evaluated for a trajectory on the space of functions on a manifold
Let be a manifold and be a function , where:
- denotes the time parameter, and varies in
- denotes the spatial parameter, and varies in
In other words, is a trajectory (or path) in the space of all functions from to .
Then, is said to be max-decreasing if the function:
is a monotone decreasing function. (The function defined above is termed the timewise-max function for ).
The corresponding notion is of a min-increasing trajectory -- viz a trajectory where the minimum (or infimum) keeps increasing.