By Washek F. Pfeffer
This booklet provides a close and ordinarily user-friendly exposition of the generalized Riemann-Stieltjes integrals came upon by way of Henstock, Kurzweil, and McShane. in addition to the classical effects, it comprises a few contemporary advancements hooked up with lipeomorphic swap of variables and the divergence theorem for discontinuously differentiable vector fields.
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Additional resources for The Riemann approach to integration: local geometric theory
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Let A(G),A(H),A(K) be the corresponding Lie algebras, and choose some ad(H)-invariant complement P of A(H) in A(G), some ad(K)-invariant complement Q of A(K) in A(H). Then any ad(H)-invariant scalar product on P gives rise to a G-invariant metric h on G/H; similarly any ad(K)-invariant scalar product on Q gives rise to a H-invariant metric on H/K. Now if g is the G-invariant metric on G/K corresponding to the scalar product on P @ Q given by the two preceding scalar products on P and Q and the condition that P and Q are orthogonal, then~rg is a Riemannian submersion from (G/K,g) onto (G/H,h) with totally geodesic fibres isometric to (H/K,k).
The Riemann approach to integration: local geometric theory by Washek F. Pfeffer