Origin of Gradient Expansion Series

[maverick_starstrider]

"Gradient Expansion"

Hi,

I'm having trouble finding the origin of a series expansion of the form:

$$f(x)= A_i \partial_i f(x) + B_{ij} \partial_{i} \partial_j f(x) + C_i [\partial_i f(x)]^2 + \ldots$$

or the similar expansion

$$f(x)= A \nabla f(x) + B \nabla^2 f(x) + C \vert \nabla f(x) \vert^2$$

The C term is especially confusing to me. Could someone point me to a book (or simply explain) how such a series is obtained and its exact meaning (as well as what the higher order terms are)? Any help is greatly appreciated.

 

[HallsofIvy]

That looks like a Taylor's polynomial (or perhaps more accurately a MacLaurin polynomial) approximation to a function of several variables. The equation you write is not exact- f is not equal to that, only approximately equal.

 

[maverick_starstrider]

I forgot the ellipsis on the second equation but that's not any taylor series I've seen. Where does the C term (which is actually the most important term) come from? Could you possibly send me a link or refer me to a specific book if you know where this comes from.