Skew Adjoint Frechet derivative?

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In summary, a skew adjoint Fréchet derivative is a type of derivative that is defined on a Banach space and maps a function to its derivative at a specific point in the space. It differs from a regular derivative in that it takes into account the geometry and topology of the space and applies to a wider range of functions. It has applications in mathematical physics, optimization and control theory, numerical analysis, and engineering. The calculation of a skew adjoint Fréchet derivative involves taking the limit of a difference quotient and may require the use of functional analysis techniques. However, it may not always be defined for a function, as it is only defined for differentiable and certain conditions must be satisfied.
  • #1
Anthony
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Hi all,

I've been trying to construct a set of nonlinear PDEs:

[tex]P_\nu=0, \qquad \nu = 1, \ldots , l[/tex]

that has skew-adjoint Frechet derivative, but with no luck. Is there any reason such a system of equations shouldn't exist? Here each [tex]P_\nu[/tex] is an analytic function of the coordinates on [tex]\sim\mathrm{pr}^s (x,u)[/tex], the s-th jet of [tex](x,u)[/tex], where [tex]x=(x^1, \ldots , x^n)[/tex] and [tex] u = (u^1, \ldots , u^l)[/tex].

Any help would be much appreciated!

Ant
 
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  • #2
Hold the phone - I think I've sorted it.
 

1. What is a skew adjoint Fréchet derivative?

A skew adjoint Fréchet derivative is a type of derivative that is defined on a Banach space, which is a complete normed vector space. It is a linear operator that maps a function to its derivative at a specific point in the space. The term "skew adjoint" refers to the property that the operator is equal to the negative of its own adjoint, which is similar to a transpose in linear algebra.

2. How is a skew adjoint Fréchet derivative different from a regular derivative?

A skew adjoint Fréchet derivative differs from a regular derivative in that it is defined on a Banach space rather than a finite-dimensional vector space. It also takes into account the geometry and topology of the space, rather than just the algebraic properties. Additionally, a skew adjoint Fréchet derivative is a more general concept that applies to a wider range of functions.

3. What are some applications of skew adjoint Fréchet derivatives?

Skew adjoint Fréchet derivatives are commonly used in mathematical physics, in particular in the study of quantum mechanics and quantum field theory. They are also used in optimization and control theory, as well as in numerical analysis for solving differential equations. Additionally, they have applications in engineering, specifically in the analysis and design of control systems.

4. How is a skew adjoint Fréchet derivative calculated?

The calculation of a skew adjoint Fréchet derivative involves taking the limit of a difference quotient, similar to how a regular derivative is calculated. However, since it is defined on a Banach space, the limit may be more complex and require the use of functional analysis techniques. The specific method for calculating a skew adjoint Fréchet derivative may vary depending on the function and space involved.

5. Can a skew adjoint Fréchet derivative always be defined?

No, a skew adjoint Fréchet derivative may not always be defined. It is only defined for functions that are differentiable on a Banach space. If a function is not differentiable, then a skew adjoint Fréchet derivative cannot be calculated. Additionally, the function must also satisfy certain conditions, such as continuity and boundedness, in order for a skew adjoint Fréchet derivative to exist.

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