I'm having trouble finding out how to set up Neumann (or, rather, "Robin") boundary conditions for a diffusion-type PDE. More specifically, I have a scalar function [itex]f(\boldsymbol{x}, t)[/itex] where [itex]\boldsymbol{x}[/itex] is n-dimensional vector space with some boundary region defined by [itex]A(\boldsymbol{x})=0[/itex] where A is another scalar function. Then I want to specify a mixed (Robin-type) boundary condition in the following way:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]a f(\boldsymbol{x}) + b \frac{∂f}{∂\boldsymbol{n}} = c(\boldsymbol{x})[/itex],

where a and b are some scalar numbers and c is a scalar function. What I don't understand is what is the nature of the directional derivative which is multiplied by [itex]b[/itex] in my example: from Wikipedia [ http://en.wikipedia.org/wiki/Neumann_boundary_condition ] it follows that

[itex]\frac{∂f}{∂\boldsymbol{n}} = ∇f \cdot \boldsymbol{n}[/itex],

where the first term, gradient of the scalar function, is a covariant vector, and the second, the normal vector to the boundary, also seems to be not a "true" vector but a covector (covariant vector) given by ∇A [ http://en.wikipedia.org/wiki/Normal_vector#Hypersurfaces_in_n-dimensional_space ]. So their product cannot be a scalar function that I need. What is wrong about it?

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# How to set up Neumann boundary condition for a PDE in a coordinate-invariant form?

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