The mathematics of infinitesimals is an area of calculus and mathematical analysis that involves the study of quantities that are exceedingly small, often approaching zero but not necessarily reaching it. Infinitesimals provide a framework for dealing with concepts such as limits, derivatives, and integrals in a rigorous way. The usage of infinitesimals can be traced back to the work of mathematicians like John Wallis and Isaac Newton in the 17th century.

Recreational mathematics is a branch of mathematics that is primarily concerned with mathematical games, puzzles, and interesting problems that are enjoyed for their entertainment value rather than for practical applications. It often involves creative thinking, problem-solving skills, and exploration of mathematical concepts in a fun and engaging way. Some common themes in recreational mathematics include: 1. **Puzzles and Games**: This includes everything from logic puzzles and Sudoku to strategy games like chess and checkers.

In differential geometry, the term "structures on manifolds" refers to various mathematical frameworks and properties that can be defined on smooth manifolds. A manifold is a topological space that locally resembles Euclidean space and supports differentiable structures.

Nonlinear algebra is a branch of mathematics that deals with systems of equations that are not linear. While linear algebra focuses on linear systems, characterized by linear equations (which can be expressed in the form \(Ax = b\), where \(A\) is a matrix, \(x\) is a vector of variables, and \(b\) is a constant vector), nonlinear algebra involves the study of equations where the relationships between variables are nonlinear.