Editing Partition of unity (section)

== Example ==
Let <math>p</math> and <math>q</math> be antipodal points on the circle <math>S^1</math>. We can construct a partition of unity on <math>S^1</math> by looking at a chart on the complement of the point <math>p \in S^1</math> that sends <math>S^1 -\{p\}</math> to <math>\mathbb{R}</math> with center <math>q \in S^1</math>. Now let <math>\Phi</math> be a [[bump function]] on <math>\mathbb{R}</math> defined by <math display="block">\Phi(x) = \begin{cases}
\exp\left(\frac{1}{x^2-1}\right) & x \in (-1,1) \\
0 & \text{otherwise}
\end{cases}</math> then, both this function and <math>1 - \Phi</math> can be extended uniquely onto <math>S^1</math> by setting <math>\Phi(p) = 0</math>. Then, the pair of functions <math>\{ (S^1 - \{p\}, \Phi), (S^1 - \{q\}, 1-\Phi) \}</math> forms a partition of unity over <math>S^1</math>.