Skip to main content
deleted 7 characters in body
Source Link

Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.

I need to study the PDE problem in $ x \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x}{\partial t} \cdot a = 0 \\ \displaystyle \frac{\partial x}{\partial u} \cdot b = 0 \\ \displaystyle \frac{\partial x}{\partial t} \cdot b = \frac{\partial x}{\partial u} \cdot a \end{cases} $$ (summations over $i$ understood) and the auxiliary condition $$x(\cdot,0)= \frac{\partial x}{\partial u}(\cdot,0)=0\,.$$

Question: how would you go about solving this problem, i.e. representing its solution set?

Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.

Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.

I need to study the PDE problem in $ x \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x}{\partial t} \cdot a = 0 \\ \displaystyle \frac{\partial x}{\partial u} \cdot b = 0 \\ \displaystyle \frac{\partial x}{\partial t} \cdot b = \frac{\partial x}{\partial u} \cdot a \end{cases} $$ (summations over $i$ understood) and the auxiliary condition $$x(\cdot,0)= \frac{\partial x}{\partial u}(\cdot,0)=0\,.$$

Question: how would you go about solving this problem, i.e. representing its solution set?

Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.

Let $a,b$ two smooth functions from the open square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.

I need to study the PDE problem in $ x \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x}{\partial t} \cdot a = 0 \\ \displaystyle \frac{\partial x}{\partial u} \cdot b = 0 \\ \displaystyle \frac{\partial x}{\partial t} \cdot b = \frac{\partial x}{\partial u} \cdot a \end{cases} $$ and the auxiliary condition $$x(\cdot,0)= \frac{\partial x}{\partial u}(\cdot,0)=0\,.$$

Question: how would you go about solving this problem, i.e. representing its solution set?

Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.

added 2 characters in body
Source Link

Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and let $a^{i},b^{i}$ denote the$b(t,u)$ be linearly independent for all $i$-th component functions$(t,u) \in I^{2}$.

I need to study the PDE problem in $ x = (x^{i})_{i=1}^{4} \in C^{\infty}(I^{2},\mathbb{R}^{4})$$ x \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x^{i}}{\partial t}a^{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial u}b^{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial t}b^{i} = \frac{\partial x^{i}}{\partial u}a^{i} \end{cases} $$$$ \begin{cases} \displaystyle \frac{\partial x}{\partial t} \cdot a = 0 \\ \displaystyle \frac{\partial x}{\partial u} \cdot b = 0 \\ \displaystyle \frac{\partial x}{\partial t} \cdot b = \frac{\partial x}{\partial u} \cdot a \end{cases} $$ (summations over $i$ understood) and the auxiliary condition $$x^{i}(\cdot,0)= \frac{\partial x^{i}}{\partial u}(\cdot,0)=0\,.$$$$x(\cdot,0)= \frac{\partial x}{\partial u}(\cdot,0)=0\,.$$

Question: how would you go about solving this problem, i.e. representing its solution set?

Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.

Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$, and let $a^{i},b^{i}$ denote the $i$-th component functions.

I need to study the PDE problem in $ x = (x^{i})_{i=1}^{4} \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x^{i}}{\partial t}a^{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial u}b^{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial t}b^{i} = \frac{\partial x^{i}}{\partial u}a^{i} \end{cases} $$ (summations over $i$ understood) and the auxiliary condition $$x^{i}(\cdot,0)= \frac{\partial x^{i}}{\partial u}(\cdot,0)=0\,.$$

Question: how would you go about solving this problem, i.e. representing its solution set?

Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.

Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$. In particular, assume $a(t,u)$ and $b(t,u)$ be linearly independent for all $(t,u) \in I^{2}$.

I need to study the PDE problem in $ x \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x}{\partial t} \cdot a = 0 \\ \displaystyle \frac{\partial x}{\partial u} \cdot b = 0 \\ \displaystyle \frac{\partial x}{\partial t} \cdot b = \frac{\partial x}{\partial u} \cdot a \end{cases} $$ (summations over $i$ understood) and the auxiliary condition $$x(\cdot,0)= \frac{\partial x}{\partial u}(\cdot,0)=0\,.$$

Question: how would you go about solving this problem, i.e. representing its solution set?

Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.

edited body
Source Link

Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$, and let $a^{i},b^{i}$ denote the $i$-th component functions.

I need to study the PDE problem in $ x = (x_{i})_{i=1}^{4} \in C^{\infty}(I^{2},\mathbb{R}^{4})$$ x = (x^{i})_{i=1}^{4} \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x^{i}}{\partial t}a_{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial u}b_{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial t}b_{i} = \frac{\partial x^{i}}{\partial u}a_{i} \end{cases} $$$$ \begin{cases} \displaystyle \frac{\partial x^{i}}{\partial t}a^{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial u}b^{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial t}b^{i} = \frac{\partial x^{i}}{\partial u}a^{i} \end{cases} $$ (summations over $i$ understood) and the auxiliary condition $$x^{i}(\cdot,0)= \frac{\partial x^{i}}{\partial u}(\cdot,0)=0\,.$$

Question: how would you go about solving this problem, i.e. representing its solution set?

Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.

Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$, and let $a^{i},b^{i}$ denote the $i$-th component functions.

I need to study the PDE problem in $ x = (x_{i})_{i=1}^{4} \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x^{i}}{\partial t}a_{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial u}b_{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial t}b_{i} = \frac{\partial x^{i}}{\partial u}a_{i} \end{cases} $$ (summations over $i$ understood) and the auxiliary condition $$x^{i}(\cdot,0)= \frac{\partial x^{i}}{\partial u}(\cdot,0)=0\,.$$

Question: how would you go about solving this problem, i.e. representing its solution set?

Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.

Let $a,b$ two smooth functions from the open square square $I^{2}$ in $\mathbb{R}^{2}$ to $\mathbb{R}^{4}$, and let $a^{i},b^{i}$ denote the $i$-th component functions.

I need to study the PDE problem in $ x = (x^{i})_{i=1}^{4} \in C^{\infty}(I^{2},\mathbb{R}^{4})$ given by the underdetermined system $$ \begin{cases} \displaystyle \frac{\partial x^{i}}{\partial t}a^{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial u}b^{i} = 0 \\ \displaystyle \frac{\partial x^{i}}{\partial t}b^{i} = \frac{\partial x^{i}}{\partial u}a^{i} \end{cases} $$ (summations over $i$ understood) and the auxiliary condition $$x^{i}(\cdot,0)= \frac{\partial x^{i}}{\partial u}(\cdot,0)=0\,.$$

Question: how would you go about solving this problem, i.e. representing its solution set?

Before somebody legitimately asks for motivation, let me finally say that this problem is linked to the isometric flexibility of a surface in $\mathbb{R}^{4}$.

Source Link
Loading