Fréchet derivative of $W in C^2(Omega times Q)$
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
13510
add a comment |
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
13510
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 |
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
13510
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
13510
asked Nov 25 at 14:02
Minysh
13510
asked Nov 25 at 14:02
Minysh
13510
asked Nov 25 at 14:02
Minysh
13510
asked Nov 25 at 14:02
Minysh
13510
13510
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 |
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You just need $Q$ to be an open set of a normed space to define derivative.
– Will M.
Nov 26 at 4:47