ISSN:
1432-0673

Source:
Springer Online Journal Archives 1860-2000

Topics:
Mathematics
,
Physics

Notes:
Abstract 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, gij 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 ϕ 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 $$F_{ij} = H h{}_{ij} + Lg_{ij} + E_{ij} (M,N),$$ where $$E_{ij} (M,N) = ( - \nabla _s ^2 M)g_{ij} + 2D_i D_j M + (h_i ^k D_j + h_j ^k D_i )D_k N$$ It is shown that H and L are uniquely determined by F but that M and N are not. The set of complex scalar fields ℳ′ = M′+iN′ such that E ij (M′, N′)=0 is shown to constitute a four-dimensional complex linear space $$\mathfrak{W}$$ . The scalars M and N which help to generate a given F are uniquely determined by F and the condition that, for every ℳ′ in $$\mathfrak{W}$$ , $$\mathop \smallint \limits_s (M - iN)\mathcal{M}\prime dS = 0$$ The real linear space of second-order tangent tensor fields on S which have simultaneously the form E(M, 0) and the form E(0, N) is shown to have dimension zero on a sphere, dimension four on a non-spherical, intrinsically axisymmetric spheroid (a spheroid whose isometries form a compact, one parameter group), and dimension six on a spheroid which is not intrinsically axisymmetric. Applications of the representation theorem to tensor problems in spherical coordinates are briefly discussed.

Type of Medium:
Electronic Resource

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