In the peridynamic state based model, the Frechet derivative is used, I would like to know why is this derivative is used, instead of the Gateaux or the ordinary derivative?
The Fréchet derivative f'(x) of a function f at x_0 is defined, if the Gâteaux derivative is defined and provides a "good linear approximation" in the sense that \|R(x)\| / \||x-x_0|\to 0, where the residuum is R(x)=f(x)-(f(x_0)+ f'(x) (x-x_0)). The Fréchet derivative might be considered as the "ordinary derivative" in any Banach space, and it is geometrically interpretable via a subspace tangent to the graph. For ODEs in a Banach space solutions are Fréchet differentiable with respect to parameters or inital values if everything else is Fréchet differentiable. With weaker notions of differentiability this is much less obvious and sometimes false. When you describe mechanical systems by ODEs, it's therefore easiest to deal with Fréchet derivatives.
By the way, what did _you_ mean by "the ordinary derivative"?
The ordinary is the classical definition given by lim dx-> 0 (f (x+dx) - f (x))/dx.
The ordinary is the classical definition given by lim dx-> 0 (f (x+dx) - f (x))/dx.
Well, this definition makes sense only for functions of one variable (you can't divide by a vector); your state-spaces, however, are multidimensional or possibly infinite-dimensional. If you try to generalize this 1-d definition of derivative by introducing convergence in norm and replacing the quotient by a condition on the residuum as sketched above you end up precisely with the notion of Fréchet derivative.
Now in particular, my question would be if the derivative of a real valued function f (x)=x^3 is f'(x) = 3x^2. Is the Frechet Derivative of f (x) also f'(x) = 3x^2?
Frechet derivative is generalization of ordinary derivative (for a function F:X \to Y, where X and Y are normed linear spaces). Particularly for X=Y=R with usual norm, the concept of Frechet derivative coincide with that of ordinary derivative.
Frechet derivation is actually generalization of ordinary derivation for functions on normeds spaces.
Yes definitely bc it is derivation of operator thanksbfor your valuable explanation dear prof pavlovic.
Frechet derivative is a generalization of the ordinary derivative and the first Frechet derivative is Linear operator. When you study differential calculus in Banach spaces you need to study Frechet and Gateaux derivatives. The different between Frechet and Gateaux derivatives is that the Frechet derivative is generalization of partial derivative for multivariable functions and Gateaux derivative is generalization of directional derivative. I think that the Frechet derivative is used in the peridynamic state based model to linearized the model.
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