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Integration in Euclidean spaces and manifolds
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Type
Thesis
Author
Lu, Jitan
Supervisor
Lee, P. Y. (Peng Yee)
Abstract
The main objective of this thesis is to develop the theories of nonabsolute integrals in Euclidean spaces and Reimannian manifolds.
In the first part, we develop further the theory of Kurzweil-Henstock integral in the Euclidean space. The Kurzweil-Henstock integral in the Euclidean space has been well defined and developed by many mathematicians. In this part we give new results this well-known integral. In Chapter one, we state the definition of the kurzweil-Henstock integral in the Euclidean space and some properties of the integral, which will be used later in this part. In Chapter two, we give some complete characterizations of the primitives of Kurzweil-Henstock integral functions. As a simple application of the results in this chapter, we give a new proof of a theorem in two dimensional integration by Henstock. In Chapter three, we give more applications.using the characterization results obtained in Chapter two, we prove a new controlled convergence theorem. Based on this controlled convergence theorem, in Chapter four, we give Riesz-type definition of the Kurzweil-Henstock integral in the Euclidean space.
In the second part, we define and develop a new integral on Riemannian manifolds. First, we give a brief description of Reimannian manifolds in Chapter five for easy reference later. Then, in Chapter six, we define a new integral, called the S-integral, which is not only a generalization of the usual integral defined on Riemannian manifolds, but also a nonabsolute integral. Finally, we give some properties of this integral.
Note that the main part of Chapter two has been accepted for publication in Bulletin of the London Mathematical Society and the main part of Chapter seven has been accepted for publication in Mathematica Japonica.
In the first part, we develop further the theory of Kurzweil-Henstock integral in the Euclidean space. The Kurzweil-Henstock integral in the Euclidean space has been well defined and developed by many mathematicians. In this part we give new results this well-known integral. In Chapter one, we state the definition of the kurzweil-Henstock integral in the Euclidean space and some properties of the integral, which will be used later in this part. In Chapter two, we give some complete characterizations of the primitives of Kurzweil-Henstock integral functions. As a simple application of the results in this chapter, we give a new proof of a theorem in two dimensional integration by Henstock. In Chapter three, we give more applications.using the characterization results obtained in Chapter two, we prove a new controlled convergence theorem. Based on this controlled convergence theorem, in Chapter four, we give Riesz-type definition of the Kurzweil-Henstock integral in the Euclidean space.
In the second part, we define and develop a new integral on Riemannian manifolds. First, we give a brief description of Reimannian manifolds in Chapter five for easy reference later. Then, in Chapter six, we define a new integral, called the S-integral, which is not only a generalization of the usual integral defined on Riemannian manifolds, but also a nonabsolute integral. Finally, we give some properties of this integral.
Note that the main part of Chapter two has been accepted for publication in Bulletin of the London Mathematical Society and the main part of Chapter seven has been accepted for publication in Mathematica Japonica.
Date Issued
1999
Call Number
QA308 Lu
Date Submitted
1999