Provider: JSTOR http://www.jstor.org Database: JSTOR Content: text/plain; charset="us-ascii" TY - CHAP TI - Local Weierstrass Integrals A2 - Morse, Marston AB -

§3. Weierstrass preintegrands. Weierstrass integrals, as we shall define them on our Reimannian manifold Mn, are generalizations of R-length on Mn. With Weierstrass, was the (x,y)-plane, so that no presentation theory was required. However, general differentiable manifolds presuppose a set of compatible presentations. It is accordingly necessary to begin with a presentation (ϕ,U) in${\cal {D}\text{M}_{\text {n}}}$and replace the R-preintegrands$\caption {(3.0)} (\text{u},\text{r})\to {f}(\text{u},\text{r})={{({{\text{a}}_{\text{ij}}}(\text{u})\ {{\text{r}}^{\text{i}}}{{\text{r}}^{\text{j}}})}^{\frac{1}{2}}}\quad \ ((\text{u},\text{r})\in \text{U}\times {{\dot{\text {R}}}^{\text{n}}})$by more general preintegrands.

To this end there is associated with each presentation$(\phi ,\text{U})\in {\cal {D}\text{M}_{\text {n}}}$a unique mapping$\caption {(3.1)}^* (\text{u},\text{r})\to \text{F}(\text{u},\text{r}):\text{U}\times {\dot{\text{{R}}}^{\text{n}}}\to \text{R}$of class C, subject to the homogeneity condition,

(3.2) F(u,kr) = kF(u,r),

valid for each pair (u,r) in the domain of F

EP - 33 PB - Princeton University Press PY - 1976 SN - null SP - 16 T2 - Global Variational Analysis: Weierstrass Integrals on a Riemannian Manifold. (MN-16) UR - http://www.jstor.org/stable/j.ctt130hkgk.6 Y2 - 2021/09/26/ ER -