I'm reading the book "Mathematical Models in Biology" by Leah Edelstein-Keshet and in page 70 the following explanation appears. Here, F(x,y) is a function with P = F(X0 + Y0) and the idea is to calculate the value of F using a Taylor series expansion. Then, the book says that, because the serie involves partial derivates (as x and y must be taken into account), it is convenient to define:
I'm a bit pluzzled with the notation and can't find a concrete explanation on the internet. I know the triple bar symbol means "identical equality between two functions", but I don't how "def" changes its meaning. Moreover, I don't get what the "delta hat", and the h and k (they weren't defined before) have to do here. It seems like the author just added a lot of new notation without much explaining.
Is someone familiar with this notation and willing to give a hint of how both sides of the equation connect?
The notation $\hat{\Delta} F$ is evidently being introduced as shorthand for a first-order approximation to the amount by which $F(x,y)$ differs from $F(x_0,y_0)$.
That is, if the first argument is perturbed by $h=x-x_0$ and the second argument by $k=y-y_0$, then the approximate change in $F(x,y)$ will be:
$$ F(x,y) - F(x_0,y_0) \approx h \frac{\partial F}{\partial x} + k \frac{\partial F}{\partial y} $$
where the partial derivatives are evaluated at $(x_0,y_0)$.
Such a notation is perhaps being used to discuss the accuracy of numerical methods that approximate a function $F(x,y)$ at uniformly spaced grid points, where $h,k$ can be taken to be those spacings in the $x$ and $y$ directions respectively.
Added: A preview of the book Mathematical Models in Biology by Leah Edelstein-Keshet (courtesy Google Books) shows that after introducing the notation $\hat\Delta$ as above, it is used to express the Taylor series for $F(x,y)$ expanded around $(x_0,y_0)$ as follows:
$$ F(x,y) = F(x_0,y_0) + \hat\Delta F(x_0,y_0) + \frac{1}{2!} \hat\Delta^2 F(x_0,y_0) + \ldots + \frac{1}{n!} \hat\Delta^n F(x_0,y_0) + \ldots $$
The author then remarks, "the function $F$ can be approximated by retaining the first two terms of a Taylor series expansion" and gives an illustration of this (a curved surface) to close the chapter. The next chapter, "Applications of Nonlinear Difference Equations to Population Biology," may exploit the notation to justify numerical methods for approximating solutions to these.