Intro
Attractors and repellers characterize the tendency of a system to gravitate toward or away from some state respectively. These concepts help us predict the behavior of advanced systems.
Think for example of big cities tend to gathering more and more people through the urban revolution. Hence, they act as attractors, improving our odds of guessing how people will be distributed in a future society.
An examples that paints the picture yet clearer is a pendulum, where the bottom position is an attractor and the top one is a repeller.
Just looking at the equations of a complex system, it is often hard to tell whether some state is an attractor or repeller. It is then useful to turn to differentials, looking at tiny changes in the equations' variables and observing the effect.
Concept
Let be a function of two variables. If you take a small step along the two axes
then, the value of will change by . There are two parts that contribute to , one coming from and the other from .
Remember that the derivative measures how much the function changes per unit traveled.
So if you walk the length along the -axis, the function will change by
Similarly, if you walk the length along the -axis, the function will change by
Thus, the total change is
Math
The differential of a function is
An interpretation is that if we take a small step , then the function change depends on the partial derivatives.
Differentials
Definition of differentials
In multivariable calculus, we refer to differentials as an approximation of a change in function value for a function , as we take a small step in the direction. This gives the difference in function value:
The differential, on the other hand, we refer to as , and it's defined as follows:
The differential is an approximation to the change in function value
consists of all the partial derivatives, multiplied by the corresponding component in the vector. Intuitively, the approximation of the change consists of = (rate of change in direction) (length of small step ), in each direction .
Adding all these small, approximated changes, we get , the approximated change in the function value. If all the are small, we get a pretty good approximation of the change in function value. A small note of warning: we require that is differentiable for this to hold.
In two variables
Let's bring it down to our favourite space , where things are pretty. Consider the function surface:
Let's call it . Zooming in to the small square with in the corner, we'll construct the differential through pictures.
Below, we have indicated the change in function value as we make the displacement in the -plane. You can also see the approximation of the change, .
But how did we get this approximation , then? Well, the differential of our two-variable function is:
So, we take the partial derivatives in the and directions at , and multiply them with the step components and respectively. Adding these two terms up, we get :
Geometrically, we can thus interpret the differential as an approximation of by its tangent plane.
Differentiability
For a single-variable function , differentiability refers to the mere existence of the derivative .
When it comes to functions of more variables though, the existence of all first partial derivatives in a point is not enough to make them differentiable there. Instead, what is required to exist there is a linear approximation.
Intuitively, this makes sense since we must be able to find a function's slope not only in the - and -directions, but also in any other direction in the whole -plane.
A function of two variables is differentiable wherever there is a non-vertical tangent plane
The linear approximation of a function of two variables at a point is formed by a tangent plane:
Now moving any short enough distances in the -direction and in the -direction, away from , the difference between the function's real value at and the one obtained by its linear approximation is negligible.
More rigorously, we have:
A function is differentiable at if
Analogous to this definition is to say that is differentiable at if and only if its surface has a non-vertical tangent plane there. This implying that the function first needs to be continuous before it can be differentiable.
The definition above can be extended to differentiability of functions in more variables too, where linear approximations consist of hyperplanes which are hard to visualize.
