By Phillip Griffiths, Daniel Grossman, Visit Amazon's Robert L. Bryant Page, search results, Learn about Author Central, Robert L. Bryant,

ISBN-10: 0226077942

ISBN-13: 9780226077949

*Exterior Differential Systems*, the authors current the result of their ongoing improvement of a concept of the geometry of differential equations, focusing specifically on Lagrangians and Poincaré-Cartan kinds. additionally they conceal yes elements of the idea of external differential platforms, which gives the language and strategies for the whole examine. since it performs a primary position in uncovering geometric homes of differential equations, the strategy of equivalence is especially emphasised. additionally, the authors speak about conformally invariant platforms at size, together with effects at the category and alertness of symmetries and conservation legislation. The booklet additionally covers the second one edition, Euler-Lagrange PDE platforms, and higher-order conservation laws.

This well timed synthesis of partial differential equations and differential geometry should be of basic significance to either scholars and skilled researchers operating in geometric analysis.

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**Sample text**

12^ Remark: The latter argument apphes to an arbitrary (say, compact for safety) metric space X admitting a topological fibration p : X -^ B, all of whose fibers are circles of length < 27r. , / i = fo ^ P- Now let us suppose that the fibers p~^{b) C X are /c-dimensional and that their /c-dimensional Hausdorfl^ measures are strictly smaller than that of the unit sphere S^. Here, one wants to homotope each map f : X -^ S^ with dil(/) < 1 to / i , sending each fiber p~^{b) onto a subset of topological (or Hausdorff") dimension < fc in 5^.

20. The requirements for a map to be an isometry or even local isometry are too stringent to provide a sufficiently rich class of morphisms for path metric spaces. For example, any Riemannian n-manifold that is locally isometric to R^ must be flat. A more flexible notion is the following. 21. Definition: An arc-wise isometry of path metric spaces X,Y is a map f:X-^Y such that i{f o c) = £{c) for each Lipschitz curve c: I -^ X oiX. Examples: (1) Every closed, piecewise C^ curve admits an arc-wise isometric mapping into R.

The proportionality constant equals the degree of f defined above. (See [Massey] and [Godb], p. 212). 7. Remark: If M is a compact, oriented Riemannian manifold, then its volume form is defined as the unique differential n-form VM , which for every x e M has value 1 on each oriented, orthonormal frame in T^M. The volume of M equals the integral vol(M) = J^^ UM- If N is another compact, oriented Riemannian manifold with volume form z/^v, and / : M -^ iV is smooth, then deg(/) = {YO\{N))~^ J^j f^u^.

### Differential Systems & Euler-Lagrange by Phillip Griffiths, Daniel Grossman, Visit Amazon's Robert L. Bryant Page, search results, Learn about Author Central, Robert L. Bryant,

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