Frechet v Gateaux Derivative and the calculus of variations

[observer1]

Good Morning

Could someone please distinguish between the Frechet and Gateaux Derivatives and why one is better to use in the Calculus of Variations?

In your response -- if you are so inclined -- please try to avoid the theoretical foundations of this distinction (as I can investigate that by scoping out sources on the internet).

I am hoping here for a SIMPLE and PRACTICAL distinction that I can use in my head as I read about these issues. I am hoping for a dinstintion in words: "We use the Frechet deriviative which came about because we needed Y and it provided an easy out. And then we extend it to the Gateaux derivative in cases where Z or something and it is useful in cases where W... And we use these in directional derivatives which are different because of Q "- kind of thing

You see, for me, a derivative is a derivative and I know how to take a derivative (of a function or of a vector in the case of a directional derivative). My background is not mathematics: it is engineering: and my math education has been deficient, and I am trying to FIRST understand the distinction before I proceed to its theoretical foundation.

 

[fresh_42]

The terms were long forgotten to me, so maybe it could help you, how I would memorize them (or perhaps not).

• Fréchet derivatives as well as Gâteaux derivatives are concepts for infinite dimensional, normed spaces.
• Fréchet derivatives are a generalization of total differentials.
• Gâteaux derivatives are a generalization of directional differentials.
  
• Fréchet $\Longrightarrow$ Gâteaux (alphabetic)

Gâteaux derivatives are of better use in variation calculus for it investigates different behaviors in some directions rather than the overall behavior and variation calculus is about the variation of single parameters and not only the variation of all at once. Overall stability is only one aspect.

For a visualization you may think of the difference in finite dimensions, since infinite dimensions are hard to imagine. Then consider the graph of a function, e.g. a real valued function in two variables, so that it fits in our understanding of space. This gives you a kind of mountain. Now imagine you are placed at one point, then the total differential is like positioning a board upon this point, tangential in each direction, whereas the directional differentials are only the tangents along one fixed direction. They don't have to build the board when sampled all together.
$$ f(x,y) = \begin{cases} \frac{xy^3}{x^2+y^4} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $$
is an example of a function that has directional differentials in all directions but doesn't have a total differential (at $(0,0)$). For the image see: http://www.wolframalpha.com/input/?i=f(x,y)=(xy^3)/(x^2+y^4)

 

[observer1]

The terms were long forgotten to me, so maybe it could help you, how I would memorize them (or perhaps not).

• Fréchet derivatives as well as Gâteaux derivatives are concepts for infinite dimensional, normed spaces.
• Fréchet derivatives are a generalization of total differentials.
• Gâteaux derivatives are a generalization of directional differentials.
  
• Fréchet $\Longrightarrow$ Gâteaux (alphabetic)

Gâteaux derivatives are of better use in variation calculus for it investigates different behaviors in some directions rather than the overall behavior and variation calculus is about the variation of single parameters and not only the variation of all at once. Overall stability is only one aspect.

For a visualization you may think of the difference in finite dimensions, since infinite dimensions are hard to imagine. Then consider the graph of a function, e.g. a real valued function in two variables, so that it fits in our understanding of space. This gives you a kind of mountain. Now imagine you are placed at one point, then the total differential is like positioning a board upon this point, tangential in each direction, whereas the directional differentials are only the tangents along one fixed direction. They don't have to build the board when sampled all together.
$$ f(x,y) = \begin{cases} \frac{xy^3}{x^2+y^4} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $$
is an example of a function that has directional differentials in all directions but doesn't have a total differential (at $(0,0)$). For the image see: http://www.wolframalpha.com/input/?i=f(x,y)=(xy^3)/(x^2+y^4)

wow... exactly what I was looking for. I have been busy filling in gaps and this is now clear.

OK... may I follow up with one more question?

(I am an engineering filling in math gaps)

So now I understand the basic definition of the derivative of a function of one variable, many variables, etc. from fundamental calculus.
And now I understand that there are these two other ones.

Are there any more?

If there are any more can you categorize maybe one or two just so I can have a scaffold to contextualize this issue.

 

[fresh_42]

So now I understand the basic definition of the derivative of a function of one variable, many variables, etc. from fundamental calculus.
And now I understand that there are these two other ones.

They are not really other ones. It is simply a generalization to infinite dimensional spaces where one has to be a bit more careful with ordinary things like, e.g. summations. Or "matrices" for the linear operators which the derivatives are, because "coordinates" aren't as simple as in the finite dimensional case. But regarding the definitions, it's not much of a difference.

Are there any more?

This depends on what a derivative is to you.
In general it is nothing more than a linear approximation that obeys the product rule $d(fg) = (df)g+f(dg)$.
Now one can ask: "Linear, in which sense?" That leads to: linear along the coordinates $\rightarrow$ partial differentiation, linear in any direction $\rightarrow$ directional differentiation, overall linearity $\rightarrow$ total differentiation and Fréchet and Gâteaux as simply the infinite dimensional versions of them.
One can define derivatives as purely algebraic structure which leads to other fields in mathematics.

So the answer here is: no, but it depends. (At least I'm not aware of others in calculus.)

 

[observer1]

They are not really other ones. It is simply a generalization to infinite dimensional spaces where one has to be a bit more careful with ordinary things like, e.g. summations. Or "matrices" for the linear operators which the derivatives are, because "coordinates" aren't as simple as in the finite dimensional case. But regarding the definitions, it's not much of a difference.

This depends on what a derivative is to you.
In general it is nothing more than a linear approximation that obeys the product rule $d(fg) = (df)g+f(dg)$.
Now one can ask: "Linear, in which sense?" That leads to: linear along the coordinates $\rightarrow$ partial differentiation, linear in any direction $\rightarrow$ directional differentiation, overall linearity $\rightarrow$ total differentiation and Fréchet and Gâteaux as simply the infinite dimensional versions of them.
One can define derivatives as purely algebraic structure which leads to other fields in mathematics.

So the answer here is: no, but it depends. (At least I'm not aware of others in calculus.)

I am sorry. Now I am confused again. I see several things here...
1. traditional in one variable calculus
2, partial
3. directional (which sort of is like partial)
4. total (with Gateaux and Frechet as generalizatoins of total)

I am sorry. I am very dense and stubborn when it comes to naming categories.

Without getting anymore theoretical than you already have, can you recast these names into categories and subcategories?

 

[fresh_42]

I am sorry. Now I am confused again. I see several things here...
1. traditional in one variable calculus
2, partial
3. directional (which sort of is like partial)
4. total (with Gateaux and Frechet as generalizatoins of total)

I am sorry. I am very dense and stubborn when it comes to naming categories.

Without getting anymore theoretical than you already have, can you recast these names into categories and subcategories?

I'll try, but I'm not sure I understood you correctly.

1. traditional in one real variable calculus (partial = directional = total for there is only one direction)
2. in $\mathbb{R}^n$ finite dimensional real calculus, so called vector calculus:

• partial (along coordinate directions)
• directional (along any, but some fixed direction; generalization of partial)
• total (all directions at once)

3. in infinite dimensional normed spaces like, e.g. function spaces

• no partial version, because there are no natural coordinates and if, it's a special case of Gâteaux anyway
• Gâteaux (generalization of directional differentiation to $\infty$)
• Fréchet (generalization of total differentiation to $\infty$)

Things are a bit more complicated over complex numbers, because the two real dimensions of $\mathbb{C}=\mathbb{R}+i\cdot \mathbb{R}$ are connected by an arithmetic rule: $\, i\,\cdot\,i = -1$.

 

[observer1]

I'll try, but I'm not sure I understood you correctly.

1. traditional in one real variable calculus (partial = directional = total for there is only one direction)
2. in $\mathbb{R}^n$ finite dimensional real calculus, so called vector calculus:

• partial (along coordinate directions)
• directional (along any, but some fixed direction; generalization of partial)
• total (all directions at once)

3. in infinite dimensional normed spaces like, e.g. function spaces

• no partial version, because there are no natural coordinates and if, it's a special case of Gâteaux anyway
• Gâteaux (generalization of directional differentiation to $\infty$)
• Fréchet (generalization of total differentiation to $\infty$)

Things are a bit more complicated over complex numbers, because the two real dimensions of $\mathbb{C}=\mathbb{R}+i\cdot \mathbb{R}$ are connected by an arithmetic rule: $\, i\,\cdot\,i = -1$.

Thank you!
And I appreciate the last line, too: that helps, also.
Thanks

 

[Stephen Tashi]

1. traditional in one variable calculus
2, partial
3. directional (which sort of is like partial)
4. total (with Gateaux and Frechet as generalizatoins of total)

Should you add the concept of "gradient" to that list? - or do you mean one of those categories to include it ?

"Total derivative" may be a mathematical term that had various interpretations. For example, the definition given by http://mathworld.wolfram.com/TotalDerivative.html doesn't suggest (to me) the idea of "in all directions". Do we wish to make a distinction between the terms "total derivative" and "total differential" ?

One concept that is used to unify many different types of derivatives is the idea that a (particular type of) derivative of a function is the "closest linear transformation" to the function in some given set of linear transformations. It does require some mental gymnastics to make the idea fit the various types of derivatives.