Gradient of scalar function discontinuous on boundary

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SUMMARY

The discussion centers on the properties of a scalar function g(r) that is constant within a volume 'v' but exhibits discontinuity at the boundaries. The magnitude of this discontinuity is represented by a constant 'M'. The integral expression presented, involving the gradient of g(r) and the outward normal vector \hat{n}, is analyzed, concluding that the first integral evaluates to zero unless the discontinuity approaches infinity. The relationship between the second and third integrals is confirmed as valid.

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mfurqan
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suppose g(r) is a scalar function which is constant inside the volume 'v' but discontinuous at the boundaries of 'v'. The magnitude of discontinuity is given by constant 'M' then can we write the following expression
[itex]\int\nabla[/itex]g(r)dv=M[itex]\int\hat{n}\delta[/itex](r-rs)dv=M[itex]\hat{n}\int[/itex]d[itex]\delta[/itex]v

where [itex]\delta[/itex]v is the boundary of volume 'v'
rs[itex]\in\delta[/itex]v
[itex]\hat{n}[/itex] is the outward normal
 
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I think your first integral is zero, even though there is a discontinuity at the boundary. The only possible way that your first integral could be non-zero is if the discontinuity jumped to infinity.

As for your second integral equaling your third integral, that's correct.
 

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