## Motivation
In the study of Partial Differential Equations and manifold theory, we often encounter a "local-to-global" problem. We possess powerful tools to solve problems locally (on small balls or coordinate charts)—for example, we can smooth a function locally using [mollifiers](/page/Standard%20Mollifier), or flatten a manifold locally into Euclidean space.
However, gluing these local solutions into a single global solution is non-trivial. Simply adding them up fails because the sum would exceed the desired values where the local patches overlap.
A **Partition of Unity** provides the mathematical "glue" to solve this. It is a family of smooth [functions](/page/Function) $\{\zeta_i\}$ that "weights" the domain. Each function $\zeta_i$ is supported entirely within a specific local patch $U_i$, and crucially, at every point in the domain, the sum of these weights is exactly $1$. This allows us to decompose any global object $u$ into localized pieces $\zeta_i u$, manipulate them, and sum them back together perfectly.
## Formal Definition
[definition: Partition Of Unity]
Let $U \subset \mathbb{R}^n$ be an [open set](/page/Open%20Set), and let $\mathcal{V} = \{V_i\}_{i \in I}$ be an open cover of $U$ (i.e., $U = \bigcup_{i \in I} V_i$).
A **partition of unity subordinate to the cover $\mathcal{V}$** is a collection of smooth functions $\{\zeta_i\}_{i \in I}$ defined on $U$ satisfying the following properties:
1. **Smoothness:** $\zeta_i \in C^\infty(U)$ for all $i \in I$.
2. **Range:** $0 \le \zeta_i(x) \le 1$ for all $x \in U$.
3. **Support:** For each $i$, the support of $\zeta_i$ is contained in the cover [set](/page/Set) $V_i$:
\begin{align*}
\operatorname{supp}(\zeta_i) \subset V_i.
\end{align*}
4. **Locally Finite:** For any compact subset $K \subset U$, $\operatorname{supp}(\zeta_i) \cap K \neq \emptyset$ for only finitely many indices $i$.
5. **Unity:** For every $x \in U$:
\begin{align*}
\sum_{i \in I} \zeta_i(x) = 1.
\end{align*}
[/definition]
## Auxiliary Tool: Bump Functions
Before proving existence, we define the standard building block for smooth cutoffs.
[example:StandardBumpFunction]
We construct a smooth function with compact support on the unit ball.
First, define $f: \mathbb{R} \to \mathbb{R}$ by:
\begin{align*}
f(t) := \begin{cases} \exp\left(-\frac{1}{t}\right) & t > 0 \\ 0 & t \le 0 \end{cases}
\end{align*}
This function is $C^\infty$ on $\mathbb{R}$. Next, define $g: \mathbb{R} \to \mathbb{R}$ by:
\begin{align*}
g(t) := \frac{f(t)}{f(t) + f(1-t)}.
\end{align*}
Then $g$ is smooth, $g(t) = 0$ for $t \le 0$, and $g(t) = 1$ for $t \ge 1$.
Finally, to create a bump on a ball $B(y, r)$, we define $\psi_{y,r}: \mathbb{R}^n \to \mathbb{R}$:
\begin{align*}
\psi_{y,r}(x) := g\left( 1 - \frac{|x-y|^2}{r^2} \right).
\end{align*}
This function is smooth, identically $1$ at the center $y$, and vanishes identically outside $B(y, r)$.
[/example]
## Existence Theorem
The power of this tool comes from the fact that such partitions always exist for any open cover of a subset of $\mathbb{R}^n$ (or more generally, any paracompact manifold).
[quotetheorem:57]
## Concrete Example
[problem]
Construct a partition of unity for $\mathbb{R}$ subordinate to the cover of overlapping intervals $V_n = (n-1, n+1)$ for $n \in \mathbb{Z}$.
[/problem]
[solution]
**Step 1: The Bump Function**
We use a simpler 1D bump function based on the exponential construction. Let $h: \mathbb{R} \to \mathbb{R}$ be:
\begin{align*}
h(t) := \begin{cases} \exp\left(\frac{1}{t^2 - 1}\right) & |t| < 1 \\ 0 & |t| \ge 1 \end{cases}
\end{align*}
This function is smooth and supported on $(-1, 1)$.
**Step 2: Translation**
For each $n \in \mathbb{Z}$, define $\psi_n(x) := h(x-n)$.
The support of $\psi_n$ is $(n-1, n+1)$, which is exactly $V_n$.
**Step 3: Normalization**
Form the sum $\Psi(x) = \sum_{m \in \mathbb{Z}} \psi_m(x)$.
For any $x$, at most two intervals $(m-1, m+1)$ contain $x$, so the sum is finite and positive everywhere.
Define:
\begin{align*}
\zeta_n(x) := \frac{\psi_n(x)}{\sum_{m \in \mathbb{Z}} \psi_m(x)}.
\end{align*}
Then $\{\zeta_n\}_{n \in \mathbb{Z}}$ is a partition of unity subordinate to $\{(n-1, n+1)\}$.
[/solution]
## References
* Evans, L. C. (2010). *Partial Differential Equations*. American Mathematical Society. (Appendix C).
* Tu, L. W. (2011). *An Introduction to Manifolds*. Springer.
