The Implicit Function Theorem History, Theory, and Applications
The implicit function theorem is part of the bedrock of mathematics analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into power
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The Implicit Function Theorem History, Theory, and Applications
Springer Science+Business Media, LLC
Steven G. Krantz Department of Mathematics Washington Vniversity St. Louis, MO 63130-4899 V.S.A.
Harold R. Parks Department of Mathematics Oregon State Vniversity Corvallis, OR 97331-4605 V.S.A.
Library of Congress Cataloglng-In-PubUcatioD Data Krantz, Steven G. (Steven George), 1951The implicit function theorem : history, theory, and applications / Steven G. Krantz and Harold R. Parks. p.cm. Includes bibliographical references and index. ISBN 978-1-4612-6593-1 ISBN 978-1-4612-0059-8 (eBook) DOI 10.1007/978-1-4612-0059-8 1. Implicit functions. 1. Parks, Harold R., 1949- II. Title. QA331.5.K71362002 515' .8-
-00
and
0,
so F(y) is strictly increasing as y increases. By the intermediate value theorem, we see that F(y) attains the value 0 for a unique value of y. That value of y is the value of I(x) for the fixed value of x under consideration. 0 Note that it is not clear from (1.4) by itself that y is a function of x. Only by doing the extra work in the example can we be certain that y really is uniquely defined as a function of x. Because it is not immediately clear from the defining equation that a function has been given, we say that the function is defined implicitly by (1.4). In contrast, when we see (1.5)
y = I(x)
written, we then take it as a hypothesis that I (x) is a function of x; no additional verification is required, even when in the right-hand side the function is simply a symbolic representation as in (1.5) rather than a formula as in (1.1), (1.2), and (1.3). To distinguish them from implicitly defined functions, the functions in (1.1), (1.2), (1.3), and (1.5) are called (in this book) explicit functions.
1.2
An Informal Version of the Implicit Function Theorem
Thinking heuristically, one usually expects that one equation in one variable
F(x)
= c,
c a constant, will be sufficient to determine the value of x (though the existence of more than one, but only finitely many, solutions would come as no surprise).1 When there are two variables, one expects that it will take two simultaneous equations
F(x,y)
=
c,
G(x, y)
=
d,
1 What we are doing is informally describing the notion of "degrees of freedom" that is commonly used in physics.
4
1. Introduction to the Implicit Function Theorem
and d constants, to determine the values of both x and y. In general, one expects that a system of m equations in m variables
C
FI (XI, X2, ... ,xm ) F2(Xt, X2, ... ,xm )
= =
C2,
Fm(XI, X2, ... , x m )
=
Cm ,
CI,
(1.6)
constants, will be just the right number of equations to determine the values of the variables. But of course we must beware of redundancies among the equations. That is, we must check that the system is nondegenerate-in the sense that a certain determinant does not vanish. In case the equations in (1.6) are all linear, we can appeal to linear algebra to make our heuristic thinking precise (see any linear algebra textbook): A necessary and sufficient condition to guarantee tha
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