By setting the function " g " equal to one, it follows that the inward normal derivative of the indicator integrates to the surface area of " D ".
22.
The Schwarzian derivative describes how a complex function is approximated by a fractional-linear map, in much the same way that a normal derivative describes how a function is approximated by a linear map.
23.
Just as the single layer potentials are continuous at the boundary with a jump in the normal derivative, so the double layer potentials have a jump across the boundary while their normal derivatives are continuous.
24.
Just as the single layer potentials are continuous at the boundary with a jump in the normal derivative, so the double layer potentials have a jump across the boundary while their normal derivatives are continuous.
25.
By setting the function " f " equal to one, it follows that the inward normal derivative of the indicator integrates to the numerical value of the surface area " S ".
26.
When considering a finite " d "-dimensional domain " D ", the sum over outward normal derivatives is expected to become an " integral ", which can be confirmed as follows:
27.
This form expresses the well-known property of harmonic functions, that " if the value or normal derivative is known on a bounding surface, then the value of the function inside the volume is known everywhere ".
28.
Proceeding, it can be derived that the inward normal derivative of the indicator gives rise to a'surface delta function', which can be indicated by ? " S " ( "'x "'):
29.
If " S " is a rectifiable hypersurface completely enclosing " K ", then the harmonic capacity can be equivalently rewritten as the integral over " S " of the outward normal derivative of " u ":
30.
Although the induced metric of g _ 1 and g _ 2 will agree on \ Sigma _ 2, the normal derivative of g _ 1 at \ Sigma _ 2 will not in general be equal to that of g _ 2 at \ Sigma _ 2.
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