Simpliying this partial differential equation

[Safinaz]

Homework Statement

How to simplify this partial differential equation?

Relevant Equations

Hello, I need help in this simple example:

Consider for instance a partial differential equation:

$(x + y ) \delta_{ij} + \partial_i \partial_j x = \partial_i \partial_j y + z \delta_{ij} ,$

where $\delta_{ij} = (-1,1,1,1)$ is a diagonal metric, and $x$ , $y$, and $z$ are functions in $i,j$ .

Does this equation mean that:

$x+y =z$, and
$x = y$?

I mean $\delta_{ij}$ terms in the LHS of the eqaution equal those at the RHS ?

with knowing that $\partial_i \partial_j x$ term dose not vanish for $\delta_{ij}=1$


Any help is appreciated!

 

[renormalize]

Any help is appreciated!

I think your notation is confusing. Can you clarify:

• What range does the index $i$ run over: $\left( 1,2,3\right)$ or $\left( 0,1,2,3\right)$ or something else?
• What explicitly are the variables $w^i$ that you are differentiating with respect to, i.e., $\partial_{i}=\partial/\partial w^{i}$?
• How are these variables related to $x,y,z$?

 

[Safinaz]

I think your notation is confusing. Can you clarify:

• What range does the index $i$ run over: $\left( 1,2,3\right)$ or $\left( 0,1,2,3\right)$ or something else?
• What explicitly are the variables $w^i$ that you are differentiating with respect to, i.e., $\partial_{i}=\partial/\partial w^{i}$?

How are these variables related to $x,y,z$?

Hi, thanks for reply.

• What range does the index $i$ run over: $\left( 1,2,3\right)$ or $\left( 0,1,2,3\right)$ or something else? Ans.: $\left( 0,1,2,3\right)$
• What explicitly are the variables $w^i$ that you are differentiating with respect to, i.e., $\partial_{i}=\partial/\partial w^{i}$? Ans.: It is $\partial_{i}=\partial/\partial w^{i}$?
• How are these variables related to $x,y,z$? Ans.: consider for instance $\partial_{i} x =\partial x /\partial w^{i}$. So that x, y and z can be written explicitly as x(w), y(w), and z(w).

 

[renormalize]

OK, thanks for clearing things up!
So we have:$$\left(x+y\right)\delta_{ij}+\frac{\partial^{2}x}{\partial w^{i}\partial w^{j}}=\frac{\partial^{2}y}{\partial w^{i}\partial w^{j}}+z\delta_{ij}\;\Rightarrow\;\left(x+y-z\right)\delta_{ij}=\frac{\partial^{2}\left(y-x\right)}{\partial w^{i}\partial w^{j}}$$or:$$U\delta_{ij}=\frac{\partial^{2}V}{\partial w^{i}\partial w^{j}}\quad\text{where}\quad U\equiv x+y-z,\;V\equiv y-x\tag{1a,b,c}$$Contracting (1a) with $\delta^{ij}$ gives $4U=\partial^{2}V/\partial w^{i}\partial w_{i}\equiv\square V$, leading to:$$\frac{1}{4}\square V\delta_{ij}=\frac{\partial^{2}V}{\partial w^{i}\partial w^{j}},\quad U=\frac{1}{4}\square V\tag{2a,b}$$The off-diagonal ($i\neq j$) terms of (2a) are: $$0=\partial_{0}\partial_{1}V=\partial_{0}\partial_{2}V=\partial_{0}\partial_{3}V=\partial_{1}\partial_{2}V=\partial_{1}\partial_{3}V=\partial_{2}\partial_{3}V$$with the unique solution $V=k$, where $k$ is an arbitrary constant. Thus, $U=\frac{1}{4}\square V=0$, and from (1b,c) we get finally:$$y=x+k,\;z=2x+k\;\left( x\text{ an arbitrary function},\;k\text{ an arbitrary constant}\right)$$

 

[Safinaz]

Thanks so much for the answer : )