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I need to apply Euler's formula to the following, 1st order PDEs which are boundary conditions for a numerical physics problem in magneto-statics:

$$\partial_r{A_\phi}^n = \partial_r{A_\phi}^m$$ $$[\partial_z{A_\phi}^n = \frac{\mu_m}{\mu_n}\partial_z{A_\phi}^m]\hat{\phi}$$

To provide some context, these are meant to ensure continuity across a boundary on the $\pm \hat{z}$ surfaces of a cylinder material in a vacuum. This is in cylindrical coordinates where the $n$ and $m$ superscripts indicate the material and vacuum space around it (On either side of the boundary), the vector function is representing. I can simplify them down to:

$$\partial_r({A_\phi}^n - {A_\phi}^m)=0$$ $$\partial_z({A_\phi}^n - \frac{\mu_m}{\mu_n}{A_\phi}^m)\hat{\phi}=0$$

I know what the Euler formula is and have seen a few examples of how it's applied, but for these specific equations, I am not sure how it is applied. This is because for two reasons:

  1. These equations have vector functions A that are distinctly different on either side of the boundary, so I can't just solve for $\frac{\partial f}{\partial x}$ directly I think? Also, this implies that I cannot just forward/backward solve with the previous solution point, because it's saying I need to take into account both the forward and backward solution points (Indicated by n and m) and so it makes me think I can't use Euler's formula at all because of that but apparently I can and so my logic is flawed somewhere.

  2. It looks like these equations are essentially in the form needed for Euler's formula: $\frac{\partial f}{\partial x}= f(x,y)$, but then that would mean that my $f(x,y)=0$ (After dividing out the LHS terms).

Some clarity is appreciated.

Sophia
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    It is hard to parse your Question because the mention of boundary condition is made without describing the region whose boundary is involved. Partial differential equations would suggest more than one dimension, and your notation suggests two dimensions given by $z$ and $r$, but also mention is made of the "superscripts [that] indicate which region (On either side of the boundary)," so Readers will naturally wonder what the geometry is. – hardmath Jun 17 '23 at 12:14
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    Hi @hardmath I realized that the equations I posited were incorrect to begin with and fixed them (Hence why it was a false question to ask in the first place). I fixed them and the equations are simpler, but the problems/questions I have still remain. I also added some more context of what the BCs actually represent in the problem. – Sophia Jun 18 '23 at 03:14

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