N-Dimensional Quasipolar Coordinates - Theory and Application

Abstract

N-Dimensional Quasipolar Coordinates Theory and Application By Tan Nguyen Dr. Angel Muleshkov, Examination Committee Chair Associate Professor of Mathematics University of Nevada, Las Vegas In this thesis, various generalizations to the n-dimension of the polar coordinates and spherical coordinates are introduced and compared with each other and the existent ones in the literature. The proof of the Jacobian of these coordinates is very often wrongfully claimed. Currently, prior to our proof, there are only two complete proofs of the Jacobian of these coordinates known to us. A friendlier definition of these coordinates is introduced and an original, direct, short, and elementary proof of the Jacobian of these coordinates is given. A method, which we call a perturbative method, is introduced and applied so that the approach in the general case is also valid in all special cases. After the proof, the definitions of the n-dimensional quasiballs (hyperballs for n ≥ 4) and the n-dimensional quasispheres (hyperspheres for n ≥ 4) are given. The Jacobian is used to calculate the n-dimensional quasivolume of the n-dimensional quasiball and the n-dimensional quasi-surface area of the n-dimensional quasisphere directly. The formulas obtained afterwards are free of any special functions and could be introduced without any advanced mathematical knowledge. Numerical results are provided in a table followed by interpretations of these results. iii ACKNOWLEDGEMENTS I would like to thank Dr. Lepp, Dr. Ding, Dr. Phanord for taking their time to read and review my thesis. I would like to thank Dr. Ding and Dr. Phanord for the educations and recommendations that I receive from them. Two people that I would like to thank specially are Dr. Muleshkov and his wife, Mrs. Sonya Muleshkov. I have received so much love and caring from them during the last seven years that in some sense, I was like their own child. I am very much thankful for that. My mathematical knowledge and ability were special gifts that were either received from or developed by Dr. Muleshkov since I was a young student at the age of 18. They have shown me what it means to have real passions; not only for mathematics but for the things and people around me. I am truly blessed to know them.