Award Date


Degree Type


Degree Name

Master of Science in Mathematical Science


Mathematical Sciences

First Committee Member

Angel S. Muleshkov

Second Committee Member

Zhonghai Ding

Third Committee Member

Dieudonne Phanord

Fourth Committee Member

Stephen Lepp

Number of Pages



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 (not perturbation) method, is introduced 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.


Coordinates, Polar; Jacobians; N-dimensional Sphere; Sphere





Included in

Mathematics Commons