A boundary condition which specifies the value of the normal derivative of the function is a Neumann boundary condition, or second-type boundary condition.
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There are generalizations of these sequences to H " k " ( ? ) involving higher powers of the normal derivative in the trace map:
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By the maximum principle for in and so by the Hopf lemma the normal derivatives of are negative on the outer curve and positive on the inner curve.
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Both the normal derivative of the indicator and the Laplacian of the indicator are supported by " surfaces " rather than " points ".
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The Neumann boundary conditions for Laplace's equation specify not the function ? itself on the boundary of " D ", but its normal derivative.
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More generally, the normal derivative of a simple layer supported on a surface is a double layer supported on that surface, and represents a laminar magnetic monopole.
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The derivative of the Heaviside step function can be seen as the'inward normal derivative'at the'boundary'of the domain given by the positive half-line.
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In higher dimensions, the derivative naturally generalises to the inward normal derivative, while the Heaviside step function naturally generalises to the indicator function of some domain " D ".
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The third equality follows by the divergence theorem and shows, again, a sum ( or, in this case, an integral ) of outward normal derivatives over all boundary locations.
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It becomes clear that the boundary must be sufficiently smooth such that the normal derivative can exist, since for example at corner points of the boundary the normal vector is not well defined.
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