Nontrivial

In mathematics, the adjective trivial is frequently used for objects (for example, groups or topological spaces) that have a very simple structure. The noun triviality usually refers to a simple technical aspect of some proof or definition. The origin of the term in mathematical language comes from the medieval trivium curriculum. The antonym nontrivial is commonly used by engineers and mathematicians to indicate a statement or theorem that is not obvious or easy to prove.

In mathematics, the term trivial is frequently used for objects (for examples, groups or topological spaces) that have a very simple structure.

Examples are as follows:

Trivial can also be used to describe solutions to an equation that have a very simple structure, but for the sake of completeness cannot be omitted. These solutions are called the trivial solutions. For example, consider the differential equation

where y = f(x) is a function whose derivative is y′. The trivial solution is

while a nontrivial solution is

The differential equation ${\displaystyle f''(x)=-\lambda f(x)}$ with boundary conditions ${\displaystyle f(0)=f(L)=0}$ is important in math and physics, for example describing a particle in a box in quantum mechanics, or standing waves on a string. It always has the solution ${\displaystyle f(x)=0}$. This solution is considered obvious and is called the "trivial" solution. In some cases, there may be other solutions (sinusoids), which are called "nontrivial".

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