Total differential of general function mapping

[birulami]

I am looking for an explanation and derivation of a total differential of a 2nd order function, i.e. a function that maps one function to another.

To be more specific, let's say I have a function $l:ℝ^n\to ℝ$ that I use to define a 2nd order function $L:(ℝ^k\to ℝ^n) \to (ℝ^k\to ℝ)$ as $L(f) := l\circ f$ for every $f:ℝ^k\to ℝ^n$.

Given that I have sufficiently useful norms defined on a function space $ℝ^i\to ℝ^j$, I assume that the total differential, $D(L,f)$ is well defined and is, for each "point" $f:ℝ^k\to ℝ^n$ of the function space a linear function (map) with the same algebraic type as $L$, i.e. $D(L,f):(ℝ^k\to ℝ^n) \to (ℝ^k\to ℝ)$.

My hunch is that $D(L,f)$ can be expressed in terms of the total differentials of $l$ and $f$, if they are uniformly continuous, but I just cannot derive the solution.

Can someone confirm, that the total differential of $L$ is a well defined concept?
What is the solution in terms of $l$ and $f$?

 

[jvicens]

Thank you for your question. The total differential of a 2nd order function is indeed a well-defined concept, and can be derived using the chain rule for multivariable functions. Let's break down the notation in your question to better understand the solution.

First, we have the function l:ℝ^n\to ℝ, which maps n variables to a single variable. This is a first order function, meaning it takes in a single function as its input. Then, we have the 2nd order function L:(ℝ^k\to ℝ^n) \to (ℝ^k\to ℝ), which takes in a function f:ℝ^k\to ℝ^n as its input and outputs a function of the same type.

To find the total differential of L, we need to use the chain rule. The total differential of L at a point f is defined as the linear function D(L,f):(ℝ^k\to ℝ^n) \to (ℝ^k\to ℝ) that best approximates the change in L as f changes at that point. In other words, it measures the sensitivity of L to small changes in f at that point.

Now, using the chain rule, we can express the total differential of L as follows:

D(L,f) = D(l\circ f) = D(l)\circ D(f)

Where D(l) is the total differential of the first order function l, and D(f) is the total differential of the function f. This expression shows that the total differential of L is indeed a well-defined concept, as it can be expressed in terms of the total differentials of l and f.

I hope this helps clarify your understanding of the total differential of a 2nd order function. Let me know if you have any further questions or need any additional explanation. Best of luck with your research!