RECENT DEVELOPMENTS IN GEOMETRIC INTEGRATION THEORY

This paper develops the theory of differential forms introduced by E. Cartan. We consider the integral of a completely skew symmetric contravariant tensor T ab. . . s of rank p over a q -dimensional differentiable manifold M in a Riemannian space R n of n dimensions ( n = p + q ). We prove that the integral can be expressed in the form ∫ T ab. . . s χ ab. . . sʼ , where χ ab. . . s = ∂( α 1 , α 2 , . . . , α p )/∂( x a , x b , . . . , x s ) (─1) p , and α 1 , α 2 , . . . , α p are the characteristic functions of p ( n -dimensional) domains A 1 , A 2 , . . . , A p , whose boundaries intersect in M . The integral is taken over the whole of the space R n . The tensor χ ab . . . s is the ‘characteristic tensor’ of M . In the representation by a Grassmann algebra χ ab. . . s is expressed by a ‘characteristic form’ χ = (— 1) p d α 1 ⋀d α 2 ⋀. . . ⋀d α p , and the dual of T ab. . . s by a form U of rank q . If d denotes the exterior derivative, γ the characteristic form of C and ω the characteristic form of Ω ≡ ∂ C , then ω = -d γ . Stokes’s theorem is proved in the form — ∫d γ ⋀ U = ∫ γ ⋀d U . In the case of three-dimensional Euclidean space very simple proofs can be given by these results which form the basic theorems of ‘continuous vector analysis’ as introduced by Weyl (1940) and H. Cartan (1949).

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Trends in Electron Energy Loss Spectroscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100078675 ◽

1977 ◽

Vol 35 ◽

pp. 228-229

Author(s):

C. Colliex ◽

P. Trebbia

Keyword(s):

Energy Loss ◽

Electron Energy ◽

Electron Energy Loss Spectroscopy ◽

Materials Science ◽

Analytical Technique ◽

Recent Developments ◽

Quantitative Results ◽

Electron Microscopes ◽

Electron Energy Loss ◽

Primary Electrons

The physical foundations for the use of electron energy loss spectroscopy towards analytical purposes, seem now rather well established and have been extensively discussed through recent publications. In this brief review we intend only to mention most recent developments in this field, which became available to our knowledge. We derive also some lines of discussion to define more clearly the limits of this analytical technique in materials science problems.The spectral information carried in both low ( 0<ΔE<100eV ) and high ( >100eV ) energy regions of the loss spectrum, is capable to provide quantitative results. Spectrometers have therefore been designed to work with all kinds of electron microscopes and to cover large energy ranges for the detection of inelastically scattered electrons (for instance the L-edge of molybdenum at 2500eV has been measured by van Zuylen with primary electrons of 80 kV). It is rather easy to fix a post-specimen magnetic optics on a STEM, but Crewe has recently underlined that great care should be devoted to optimize the collecting power and the energy resolution of the whole system.

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