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

 P: 1 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 $f(\boldsymbol{x}, t)$ where $\boldsymbol{x}$ is n-dimensional vector space with some boundary region defined by $A(\boldsymbol{x})=0$ where A is another scalar function. Then I want to specify a mixed (Robin-type) boundary condition in the following way: $a f(\boldsymbol{x}) + b \frac{∂f}{∂\boldsymbol{n}} = c(\boldsymbol{x})$, 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 $b$ in my example: from Wikipedia [ http://en.wikipedia.org/wiki/Neumann_boundary_condition ] it follows that $\frac{∂f}{∂\boldsymbol{n}} = ∇f \cdot \boldsymbol{n}$, 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_...ensional_space ]. So their product cannot be a scalar function that I need. What is wrong about it?