1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Higher order derivatives

  1. Aug 14, 2006 #1
    Let E, F be Banach spaces, and let [itex]L(E;F)[/itex] denote the space of linear, bounded maps between E and F. My goal is to understand better higher order derivatives.
    Let's take [itex]E=\mathbb{R}^2, F=\mathbb{R}[/itex]. Consider a function [itex]f:U\subset\mathbb{R}^2\rightarrow\mathbb{R}[/itex], where U is an open subset of [itex]\mathbb{R}^2[/itex]. Then [itex]D^2 f:U\rightarrow L(\mathbb{R}^2;L(\mathbb{R}^2;\mathbb{R}))[/itex].
    Now, I read that for [itex]u\in U, v,w\in\mathbb{R}^2[/itex] by definition [itex]D^2 f(u)\cdot (v,w):=D((Df)(.)\cdot w)\cdot v[/itex]. My question now is: Why was this defined precisely this way?
    Does it have something to do with "using the product rule", which would amount to [itex]D((Df)(.)\cdot w)=D^2 f(.)\cdot w+Df(.)\cdot D(w)=D^2 f(.)\cdot w[/itex]?
    Thanks for any help. Best regards...Cliowa
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted