Curvilinear boundary coordinates

We express for in the form where is the outward normal coordinate, and is a (periodic) tangential location on the boundary , increasing in an anti-clockwise fashion (see Fig. I.1). I will assume is a curve differentiable to sufficient order. The corresponding locally-orthonormal unit vectors are (normal) and (tangential). In our curvilinear coordinates they are vector fields defined by and . For an introduction to such noncartesian systems see [7]. For the local components of the gradient of a scalar field I will use the abbreviations in the normal direction, and in the tangential direction. A general vector field will have the components with and .

Our task is to express (I.2) in terms of , ,
, and higher derivatives , etc.
We will choose a point
, so that these derivatives
are evaluated at the boundary.
The boundary conditions will later be applied.
We need rules for handling derivatives.
The metric is such that derivatives of the unit vectors obey

(I.3) | |||

(I.4) |

on the boundary (). The local inverse radius of curvature of the boundary is

(I.5) |

The four derivative terms on the RHS are the components of the

which we will use later. We will also need the divergence of a general vector field , which is

the trace of the covariant derivative tensor. Choosing finally gives the Laplacian

equivalent to the form in locally-cylindrical coordinates.