Green's first identity at the boundary

[Jeffrey Yang]

As required by the Green's identity, the integrated function has to be smooth and continuous in the integration region Ω.

How about if the function is just discontinuous at the boundary? Actually, this function is an electric field. So its tangential component is naturally continuous, but the normal component is discontinuous due to the abrupt change of refractive index in these two regions. However, a boundary condition is hold that is

## n_1 E_{n1} = n2 E_{n2}##

In this case, can I still use the Green's first identity to the normal component, by treating the integration region as an open region?

If I can, what kind of surface divergence I should use at the boundary?

Thanks for your help

 

[jvicens]

!
Thank you for your question. In this case, you can still use the Green's first identity to the normal component of the electric field. The integration region should be treated as an open region, and you can use the surface divergence at the boundary. The boundary condition that you have mentioned, ## n_1 E_{n1} = n2 E_{n2}##, is important and should be taken into consideration when using the Green's first identity.

The Green's first identity is a powerful tool in solving problems related to electric fields, and it can still be applied even if the function is discontinuous at the boundary. However, it is important to note that the discontinuity should not be too severe, as it may affect the accuracy of the solution. In such cases, it may be necessary to use other methods or techniques to solve the problem.

I hope this helps to answer your question. If you have any further questions, please do not hesitate to ask. Thank you for your interest in the Green's first identity and its application to electric fields.