Implicit function theorem

5 Implicit Function Theorem Given a function fix,y) such that 

Given a bounded function y(x) defined on a bounded interval [a,b], the definite Riemann integral of fix) is defined as 

The term/x) is known as the integrand a and b are the limits of integration. The interpretation of a definite integral is the area bounded by the function/(x) and the x-axis between the lines x = a and x = b. 

The fundamental theorem of calculus Given a continuous function/(x) on the bounded interval [0,x], then 

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Hence, by the implicit functions theorem, one can assert that for a sufficiently small y there exists only one solution /3X = (/ ,), j8a = fl2 y) which vanishes with y and, besides, this solution is analytic in y. This means that the solution x(t) is of the form ... 

Let us emphasize one typical inaccuracy met in the description of the quasi-stationarity hypothesis for chemical systems. It is suggested that the rate of changing the amount of intermediate particles (fast sub-system) tends to or even equals zero. But this is not true since it is not difficult to obtain an expression for y by differentiating the relationship g(x, y) = 0 and using an implicit function theorem... 

We refer to this as the local equation. Since L range L -> range L is invertible, it follows from the implicit function theorem that the local equation [Eq. (20)] with the constraint given by Eq. (15) can be solved uniquely for c in terms of (c). Substitution of this in Eq. (19) gives the reduced or averaged model. 

A good explanation of the theory can be found in Smoller [Smo] (see also [MM]). All forms of the basic result are essentially equivalent. Bifurcation theory is not restricted to differential equations but is actually concerned with mappings or functions. A principal tool in developing the theory is the implicit function theorem. When the theory is used in infinitedimensional spaces, quite sophisticated mathematics is required. However, the problem here can be dealt with in a finite-dimensional setting. 

Fortunately, although the implicit function theorem would appear to be inapplicable as a tool to discover the structure of solutions of (6.1) in a neighborhood of a bifurcation point, it can be successfully applied once... 

In this section, we compile some results from nonlinear analysis that are used in the text. The implicit function theorem and Sard s theorem are stated. A brief overview of degree theory is given and applied to prove some results stated in Chapters 5 and 6. The section ends with an outline of the construction of a Poincare map for a periodic solution of an autonomous system of ordinary differential equations and the calculation of its Jacobian (Lemma 6.2 of Chapter 3 is proved). 

Implicit Function Theorem. Suppose that F U xU" K " has continuous first partial derivatives and satisfies F(0,0) = 0. If the Jacobian matrix of F x,y) with respect to x satisfies... 

Techniques based on the implicit function theorem have been used to predict the existence of multiple solutions in a CSTR (Chang and Calo, 1979). An extension of catastrophe theory known as singularity theory has also been effectively used to determine the conditions for the existence of multiple solutions in a CSTR and a tubular reactor (Balakotaiah and Luss, 1981, 1982 Witmer et al., 1986). In this subsection, the technique of singularity to find the maximum number of solutions of a single mathematical equation and its application to analysis of the multiplicity of a CSTR are presented (Luss, 1986 Balakotaiah at al., 1985). The details of singularity theory can be found in Golubitsky and Schaeffer (1985). 

In the last section, we have seen that many familiar concepts involving differentials can be transferred from differentials and analytic geometry to algebraic geometry. But one very important theorem in the differential and analytic situations is false in the algebraic case - the implicit function theorem. This asserts that if we are given k differentiable (resp. analytic) functions /i,..., / near a point x in Rn+k (resp. Cn+fc) such that... 

M. ARTIN.- The implicit function theorem in Algebraic Geometry. Proc. Bombay Colloquium on Algebraic Geometry. Tata Institute (1969). 

Implicit function theorem Triple product rules... 

Thermodynamic descriptions of systems are often reformulated by exploiting connections among properties. One reformulation among differential properties can be obtained by applying the implicit function theorem. Consider some function... 

In thermodynamics the implicit function theorem is usually written in the form of a triple product rule ... 

Equation (8.29). Prom the Implicit Function Theorem (Section 9.16, p. 277), Equations (8.32) are valid in an open region around /3 = 0 and have solutions... 

In principle such an equation can be solved (locally at least) for y as a function of X due to the implicit function theorem. Hence one may write y = if(x). Reinsert this into the first differential equation to get a reduced equation in just one dependent variable ... 

Krantz, S.G., Parks, H.R. (2003) The Implicit Function Theorem History. Theory, and Applications, Birkhauser, Boston. 

Then, using the implicit function theorem, we obtain the following... 

Therefore, the implicit function theorem and the continuity of imply... 

In the context of ARRs it is sufficient to consider a simplified version of the implicit function theorem. 

Krantz, S. G., Parks, H. R. (2013). The implicit function theorem—History, theory, and applications. Berlin Springer. ISBN 978-1-4614-5980-4. doilO.1007/978-1-4614-5981-1. (Modem Birkhauser Classics, Boston, 2003, second printing). 

Although it is now clear why increasing best responses ensure existence of an equilibrium, it is not immediately obvious why Definition 3 provides a suificient condition, given that it only concerns the sign of the second-order cross-partial derivative. To see this connection, consider separately the continuous and the dis-continuous parts of the best response x (xj). When the best response is continuous, we can apply the Implicit Function Theorem to find its slope as follows... 

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