Proving implicit function theorem with the methods of series

Qihang Wei; Wenhe Li

doi:10.1117/12.3036732

21 July 2024 Proving implicit function theorem with the methods of series

Qihang Wei, Wenhe Li

Proceedings Volume 13219, Fourth International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2024); 1321911 (2024) https://doi.org/10.1117/12.3036732
Event: 4th International Conference on Applied Mathematics, Modelling and Intelligent Computing (CAMMIC 2024), 2024, Kaifeng, China

Abstract

Implicit function theorem is one of the most classic and important results in modern mathematics[2]. It is used to describe whether one variable of a equation could be written as a function of the other variables, which also has a wide range of application[5]. For the proof of this theorem, one method is to discuss the properties of its partial derivatives in a certain field with the help of given conditions, in order to complete the continuity proof, and prove the differentiability of the function by using the differential mean value theorem. But obviously, such a proof is too complex. Therefore, in this paper, we will introduce some relevant contents of series, including the definition of series convergence and uniform convergence, the necessary conditions of convergent series, the continuity theorem of uniformly convergent function and Weierstrass discriminant, and take this as a starting point, construct a function sequence, and use the method of series to prove the implicit function theorem.

(2024) Published by SPIE. Downloading of the abstract is permitted for personal use only.

Citation Download Citation

Qihang Wei and Wenhe Li "Proving implicit function theorem with the methods of series", Proc. SPIE 13219, Fourth International Conference on Applied Mathematics, Modelling, and Intelligent Computing (CAMMIC 2024), 1321911 (21 July 2024); https://doi.org/10.1117/12.3036732

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