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A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as x n, while the surface area, being (n − 1)-dimensional, scales as x n−1.
An example of dimensional analysis can be found for the case of the mechanics of a thin, solid and parallel-sided rotating disc. There are five variables involved which reduce to two non-dimensional groups. The relationship between these can be determined by numerical experiment using, for example, the finite element method. [10]
A Fermi problem (or Fermi quiz, Fermi question, Fermi estimate), also known as an order-of-magnitude problem (or order-of-magnitude estimate, order estimation), is an estimation problem in physics or engineering education, designed to teach dimensional analysis or approximation of extreme scientific calculations.
One example is the surface of the 6-sphere, S 6. This is the set of all points in seven-dimensional space (Euclidean) R 7 {\displaystyle \mathbb {R} ^{7}} that are a fixed distance from the origin. This constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions.
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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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