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Mathematical Preliminaries
Published in R. Ravi, Chemical Engineering Thermodynamics, 2020
In section 0.1, we introduce the notation regarding functions and variables. Section 0.2 discusses the inverse function theorem for a function of a single variable. In section 0.3, the differential of a function of one or many variables is introduced and the various misconceptions associated with the term are addressed. Section 0.4 extends the ideas in section 0.2 to a function of many variables. In section 0.5, the implicit function theorem is introduced and it is pointed out how the inverse function theorem may be regarded as a consequence of it. Section 0.6 contains formulae for the chain rule of differentiation in various forms. The interrelationship between the material in sections 0.4–0.6 is brought out. While a procedure for obtaining the formulae in these sections is outlined, the emphasis is on developing an intuitive ability to write these formulae down by inspection. This is on account of the repeated occurrence of instances in thermodynamics where these formulae are required.
Vector analysis
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Consider an open set Ω in ℝn × ℝk and a continuously differentiable function f : Ω → ℝk. We write the points in Ω as pairs (x, y), where x ∈ ℝn and y ∈ ℝk. Suppose that f(x0, y0) = 0. The implicit function theorem provides circumstances in which we can solve the equation f(x, y) = 0 for y as a function of x.
Traveling waves of nonlocal delayed disease models: critical wave speed and propagation speed
Published in Applicable Analysis, 2023
Hongying Shu, Xuejun Pan, Bruce Wade, Xiang-Sheng Wang
Let . We have and . Now we treat as an independent variable, while and are dependent variables. By implicit function theorem, the partial derivative of with respect to equals to . Since and , it follows that is an increasing function of . This completes the proof.
Weak rigidity of entropy spectra
Published in Dynamical Systems, 2021
For , we denote by the characteristic polynomial of A. Since the Perron root has algebraic multiplicity one, the fraction is non-zero at , and hence, Thus, by the implicit function theorem, there exist a neighbourhood of and a real analytic map such that, for , is an eigenvalue of A and . It follows from (10) that . We conclude that (i) holds.
Contrast enhanced tomographic reconstruction of vascular blood flow based on the Navier-Stokes equation.
Published in Inverse Problems in Science and Engineering, 2020
B. Sixou, M. Sigovan, L. Boussel
The methodology of the adjoint method is explained in [14,34]. The results of the former section show that the implicit function theorem can be applied and that the mapping is Fréchet differentiable in a neighbourhood of with Lipschitz continuous derivatives.