I've heard many times that they are somehow similar and in some cases mean the same thing. Consider this function: $$f(x,y)=x^2y$$ I have to calculate the Fréchet derivative $f'(x_0,y_0)$ and some "remainder" $r(\binom{h}{k})$. I've been told that, in this case, Fréchet derivative and total derivative mean the same thing.

However, I can't seem to remember that this remainder is also present in the total derivative.

Therefore, what exactly is the difference between Fréchet and total derivative? Are they the same thing in my example?

  • $\begingroup$ The Fréchet derivative in finite-dimensional spaces is the usual derivative. In particular, it is represented in coordinates by the Jacobian matrix. en.wikipedia.org/wiki/Fr%C3%A9chet_derivative#Finite_dimensions $\endgroup$ – reuns Jul 20 '15 at 1:04
  • $\begingroup$ Remainder is not a part of the derivative, it shows that the difference between the function change and the differential keeps small with a correct order (higher than one). Think of the Taylor expansion up to the first order plus the remainder. $\endgroup$ – A.Γ. Jul 20 '15 at 2:09

Frechet derivative is a generalization of total derivative to the normed linear spaces. They have analogous definition. For example, the Frechet/total differential $df(x,h)$ of a functional/function $f:D\rightarrow Y$ at $x$ with increment $h$ is a bounded linear operator such that $$ \lim_{\lVert h\rVert\rightarrow 0}\frac{\lVert f(x+h)-f(x)-df(x,h)\rVert}{\lVert h\rVert}=0 $$

If $f$ is defined on an open set in $\mathbb{R}^n$ with $\lVert \cdot\rVert$ being the Euclidean norm, then $df(x,h)$ is called the total differential.

If $f$ is defined on an open set in a norm linear space $(X,\lVert \cdot\rVert)$ with some arbitrary norm $\lVert \cdot\rVert$, then $df(x,h)$ is called the Frechet differential.

A differential is a function of the increment $h$. In both cases, we may have the representation $$ df(x,h)=A(x)h. $$ In $\mathbb{R}^n$, $A(x)$ maps $x$ to the space of $m\times n$ matrices called the Jacobians, $m$ is the dimension of $Y$. In arbitrary normed linear space $(X,\lVert\cdot\rVert_X)$ and $(Y,\lVert \cdot\rVert_Y)$. $A(x)$ maps $x$ to the space of bounded linear operators between $X$ and $Y$. Therefore, the mapping $A(x)$ is called total/Frechet derivative accordingly.

Since $(\mathbb{R}^n,\lVert \cdot\rVert_2)$ is a norm linear space, total derivative is always Frechet derivative, which corresponds to the case you mentioned.

For motivation, it turns out a lot of the Classical calculus results can be extended to Banach spaces (e.g. space of infinite sequences, space of integrable functions, etc.). It is really powerful for solving Calculus of Variantion problems such as finding the time path (continuous function) of a particle that minimizes energies or finding the optimal investment sequence over a long horizon for an economy.

  • 1
    $\begingroup$ I think you want to say "The Frechet/total differential of a functional/function f:D→Y at x is a bounded linear operator $df(x)$ such that ... $\endgroup$ – zhw. Jul 20 '15 at 1:28
  • $\begingroup$ I have made the change. Thank you for the suggestion! $\endgroup$ – Xiao Yang Jul 20 '15 at 3:01

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