Potentials for tangent tensor fields on spheroids

In three-dimensional Euclidean space let S be a closed simply connected, smooth surface (spheroid). Let \(\hat n\) be the outward unit normal to S, ▽ _S the surface gradient on S, I _S the metric tensor on S, g_ij the four covariant components of I _S (i,j = 1, 2), h _ij the four covariant components of -\(\hat n\)xI _S , and D _i covariant differentiation on S. It is well known that for any tangent vector field u on S there exist scalars ? and ψ on S, unique to within additive constants, such that \(u = \nabla _s \varphi - \hat n \times \nabla _s \psi \); the covariant components of u are \(u_i = D_i \varphi + h_i^j D_j \psi \). This theorem is very useful in the study of vector fields in spherical coordinates. The present paper gives an analogous theorem for real second-order tangent tensor fields F on S: for any such F there exist scalar fields H, L, M, N such that the covariant components of F are

$$F_{ij} = H h{}_{ij} + Lg_{ij} + E_{ij} (M,N),$$