In my reading on critical zone science (the interface between earth atmosphere, surface, and subsurface processes), I come across a mathematical relationship in a graduate text that caused me a bit of consternation last week. After several attempts, I unsuccessfully tried to derive the following Taylor series expansion to the second term:
f(x + \delta x) = f(x) + \dfrac{df}{d x} \delta x + ...While my undergrad classes certainly taught me the Taylor and Maclaurin series, I did not see anything quite like this. Just to be sure I consulted all of my undergrad texts. So I did what all (or most) people normally do, consult Google. It took me a couple hours of digging just to learn how to search for this equation. After another couple of hours of digging I finally come up with the proper formula for one dimension:
f(x + \delta x) = f(x) + (\delta x \cdot \nabla)f(x) + \dfrac{1}{2!} (\delta x \cdot \nabla)^2 f(x) + \dfrac{1}{3!} (\delta x \cdot \nabla)^3 f(x) + ...I was not able to “reverse engineer” the typical approach to Taylor series presented in my undergrad classes to derive this form. However, I do see something noteworthy. The function
f(x + \delta x)
denotes a movement from the datum to some infinitesimally small distance beyond the datum. This implies the presence of a gradient. The Taylor series expansion brings out this gradient very explicitly.
The two-dimensional form is much more complicated:
f(x + \delta x, y + \delta y) = f(x, y) + (\delta x \cdot \nabla)f(x) + (\delta y \cdot \nabla)f(y) + \dfrac{1}{2!} (\delta x \cdot \nabla)^2 f(x) + \dfrac{1}{2!} (\delta y \cdot \nabla)^2 f(y) + (\delta x \delta y \cdot \nabla)^2 f(x,y) + ...