Fréchet derivative of $W in C^2(Omega times Q)$
up vote
0
down vote
favorite
I'm a reading a book about PDE. There is a function $W: Omega times Q to mathbb{R}$, where $Omega subseteq mathbb{R}^3$ is an open set and $Q$ is the set of $3times 3$ matrices with positive determinant. But I don't understand what means that $W in C^2(Omega times Q)$. I don' know how can I define derivative of Fréchet for $W$, because from definition of that derivative I need that $Q$ to be a Banach space. Also there is the expression $frac{partial W}{partial F_{ij}}(x, F(x))$, where $F:=(F_{ij}) in Q$ is the Jacobian matrix of a function $f: Omega to mathbb{R}^3$, and I don't know why I can do that.
Thanks for any help.
pde partial-derivative frechet-derivative
asked Nov 25 at 14:02
Minysh
13010
add a comment |
up vote
0
down vote
favorite
I'm a reading a book about PDE. There is a function $W: Omega times Q to mathbb{R}$, where $Omega subseteq mathbb{R}^3$ is an open set and $Q$ is the set of $3times 3$ matrices with positive determinant. But I don't understand what means that $W in C^2(Omega times Q)$. I don' know how can I define derivative of Fréchet for $W$, because from definition of that derivative I need that $Q$ to be a Banach space. Also there is the expression $frac{partial W}{partial F_{ij}}(x, F(x))$, where $F:=(F_{ij}) in Q$ is the Jacobian matrix of a function $f: Omega to mathbb{R}^3$, and I don't know why I can do that.
Thanks for any help.
pde partial-derivative frechet-derivative
asked Nov 25 at 14:02
Minysh
13010
You just need $Q$ to be an open set of a normed space to define derivative.
– Will M.
Nov 26 at 4:47
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm a reading a book about PDE. There is a function $W: Omega times Q to mathbb{R}$, where $Omega subseteq mathbb{R}^3$ is an open set and $Q$ is the set of $3times 3$ matrices with positive determinant. But I don't understand what means that $W in C^2(Omega times Q)$. I don' know how can I define derivative of Fréchet for $W$, because from definition of that derivative I need that $Q$ to be a Banach space. Also there is the expression $frac{partial W}{partial F_{ij}}(x, F(x))$, where $F:=(F_{ij}) in Q$ is the Jacobian matrix of a function $f: Omega to mathbb{R}^3$, and I don't know why I can do that.
Thanks for any help.
pde partial-derivative frechet-derivative
asked Nov 25 at 14:02
Minysh
13010
I'm a reading a book about PDE. There is a function $W: Omega times Q to mathbb{R}$, where $Omega subseteq mathbb{R}^3$ is an open set and $Q$ is the set of $3times 3$ matrices with positive determinant. But I don't understand what means that $W in C^2(Omega times Q)$. I don' know how can I define derivative of Fréchet for $W$, because from definition of that derivative I need that $Q$ to be a Banach space. Also there is the expression $frac{partial W}{partial F_{ij}}(x, F(x))$, where $F:=(F_{ij}) in Q$ is the Jacobian matrix of a function $f: Omega to mathbb{R}^3$, and I don't know why I can do that.
Thanks for any help.
pde partial-derivative frechet-derivative
pde partial-derivative frechet-derivative
asked Nov 25 at 14:02
Minysh
13010
asked Nov 25 at 14:02
Minysh
13010
asked Nov 25 at 14:02
Minysh
13010
asked Nov 25 at 14:02
Minysh
13010
asked Nov 25 at 14:02
Minysh
13010
13010
You just need $Q$ to be an open set of a normed space to define derivative.
– Will M.
Nov 26 at 4:47
add a comment |
You just need $Q$ to be an open set of a normed space to define derivative.
– Will M.
Nov 26 at 4:47
You just need $Q$ to be an open set of a normed space to define derivative.
– Will M.
Nov 26 at 4:47
You just need $Q$ to be an open set of a normed space to define derivative.
– Will M.
Nov 26 at 4:47
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3012874%2ffr%25c3%25a9chet-derivative-of-w-in-c2-omega-times-q%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
You just need $Q$ to be an open set of a normed space to define derivative.
– Will M.
Nov 26 at 4:47