Softcover reprint of the hardcover 1st edition 1969Library of Congress Catalog Card Number 68 26689 All rights reserved. No part of this work covered by the copyright hereon may be reproduced or used in any form or by any meansgraphic, electronic, or mechanical, including photocopying, recording, taping, or information storage and retrieval systems-without written permission of the publisher.
Abstract.By separating the algebraic and analytic aspect of Frobenius' theorem on involutive distributions, we are able to give a simplified proof.The Frobenius theorem may be stated as follows [B, p. 161Theorem. An r-distribution A on an m-manifold M is involutive if and only if A is completely integrable. Clearly, if A is completely integrable then A is involutive. Thus we assume that A is involutive.The following proof separates the algebraic and analytic aspects of the theorem. The algebraic part is completely elementary; we give a reference for the analytic part. where the functions aj are C°° on V . Since the Y, are independent, the rxm matrix A = (aj) is of rank r, and without loss of generality, we may assume
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