# Hilbert Book Model Project/Slide S5

hbmp Hilbert Book Model Project S5

We separate all point-like discontinuities from the domain ${\displaystyle {\color {white}\Omega }}$ by encapsulating them in an extra boundary

${\displaystyle {\color {white}\,\int \limits _{\Omega -H}d\omega =\int \limits _{\partial \Omega \cup \partial H}\omega =\int \limits _{\partial \Omega }\omega -\sum _{k=1}^{N}\int \limits _{\partial H_{n}^{x}}\omega }}$

In fact, it is sufficient that ${\displaystyle {\color {white}\partial H_{n}^{x}}}$ surrounds the current location of the elementary module

We will select a boundary, which has the shape of a small cube of which the sides run through
a region of the parameter spaces where the manifolds are continuous.

If we take everywhere on the boundary the unit normal to point outward, then this reverses the direction of the normal on ${\displaystyle {\color {white}\partial H_{n}^{x}}}$ which negates the integral

Thus, in this formula, the contributions of boundaries ${\displaystyle {\color {white}\{\partial H_{n}^{x}\}}}$
are subtracted from the contributions of boundary ${\displaystyle {\color {white}\partial \Omega }}$

This means that ${\displaystyle {\color {white}\partial \Omega }}$ also surrounds the regions ${\displaystyle {\color {white}\{\partial H_{n}^{x}\}}}$

This fact renders the integration sensitive to the sequencing of the participating domains

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