Curvilinear boundary coordinates

We express ${\mathbf r}$ for $d=2$ in the form $(z,s)$ where $z$ is the outward normal coordinate, and $s$ is a (periodic) tangential location on the boundary $\Gamma$, increasing in an anti-clockwise fashion (see Fig. I.1). I will assume $\Gamma$ is a curve differentiable to sufficient order. The corresponding locally-orthonormal unit vectors are ${\mathbf n}$ (normal) and ${\mathbf t}$ (tangential). In our curvilinear coordinates they are vector fields defined by ${\mathbf n} \equiv \partial{\mathbf r}/\partial z$ and ${\mathbf t} \equiv \partial{\mathbf r}/\partial s$. 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 $\partial \psi/\partial z \equiv \partial_n \psi \equiv \psi^n$ in the normal direction, and $\partial \psi/\partial s \equiv \partial_t \psi \equiv \psi^t$ in the tangential direction. A general vector field ${\mathbf b}$ will have the components $(b_n,b_t)$ with $b_n \equiv {\mathbf n}\cdot{\mathbf b}$ and $b_t \equiv {\mathbf t}\cdot{\mathbf b}$.

Our task is to express (I.2) in terms of $\psi$, $\psi^n$, $\psi^t$, and higher derivatives $\psi^{nn}$, $\psi^{nt}$ etc. We will choose a point ${\mathbf r} \in \Gamma$, 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

$$\partial_n {\mathbf n} = {\mathbf 0},\hspace{0.2in}\hspace{0.2in}\; \partial_n {\mathbf t} \ = \ {\mathbf 0},$$ (I.3)

$$\partial_t {\mathbf n} = \alpha {\mathbf t}, \hspace{0.2in}\hspace{0.2in} \partial_t {\mathbf t} \ = \ -\alpha {\mathbf n},$$ (I.4)

on the boundary ($z=0$). The local inverse radius of curvature of the boundary is

$$\alpha(s) \; \equiv \; \frac{1}{R(s)} \; \equiv \; -{\mathbf n}(s) \cdot \frac{\partial {\mathbf t}}{\partial s} .$$ (I.5)

Thus $\alpha$ gives the `connection' of the metric (or Christoffel symbol [7]). This means that while derivatives of scalars obey the usual Cartesian rules, derivatives of vectors (or their local components) introduce extra terms. This key observation was omitted in the work of VS [195,194]. For a general vector field ${\mathbf b}$ we have,

$$\partial_n b_n \equiv \partial_n ({\mathbf n}\cdot{\mathbf b}) \ = \ {\mathbf n}\cdot\partial_n{\mathbf b}$$ (I.6)

$$\partial_n b_t \equiv \partial_n ({\mathbf t}\cdot{\mathbf b}) \ = \ {\mathbf t}\cdot\partial_n{\mathbf b}$$ (I.7)

$$\partial_t b_n \equiv \partial_t ({\mathbf n}\cdot{\mathbf b}) \ = \ {\mathbf n}\cdot\partial_t{\mathbf b} + \alpha b_t$$ (I.8)

$$\partial_t b_t \equiv \partial_t ({\mathbf t}\cdot{\mathbf b}) \ = \ {\mathbf t}\cdot\partial_t{\mathbf b} - \alpha b_n .$$ (I.9)

The four derivative terms on the RHS are the components of the covariant derivative tensor. If we apply the above to the choice ${\mathbf b}={\mathbf a}$ which we take to be a constant (translation) field according to the prescription of the previous section, then all the derivatives of ${\mathbf a}$ vanish leaving

$$\partial_n a_n = \partial_n a_t \ = \ 0,$$ (I.10)

$$\partial_t a_n = \alpha a_t,$$ (I.11)

$$\partial_t a_t = -\alpha a_n,$$ (I.12)

which we will use later. We will also need the divergence of a general vector field ${\mathbf b}$, which is not simply $\partial_n b_n + \partial_t b_t$, but rather

$$\nabla \cdot {\mathbf b} \ = \ {\mathbf n}\cdot\partial_n {... ...mathbf b} \ = \ \partial_n b_n + \partial_t b_t + \alpha b_n,$$ (I.13)

the trace of the covariant derivative tensor. Choosing ${\mathbf b} = \nabla \psi$ finally gives the Laplacian

$$\nabla^2 \psi \ = \ \nabla\cdot\nabla \psi \ = \ \psi^{nn} + \psi^{tt} + \alpha \psi^n,$$ (I.14)

equivalent to the form in locally-cylindrical coordinates.