```math gallery

Euler's theorem

If $n$ and $a$ are coprime positive integers, then $a$ raised to the power of the totient of $n$ is congruent to one, modulo $n$, or:

$$a^{\varphi (n)} \equiv 1 \pmod{n}$$

where $\varphi(n)$ is Euler's totient function.

Divergence theorem

Suppose $V$ is a subset of $\mathbb{R}^{n}$ (in the case of $n = 3$, $V$ represents a volume in three-dimensional space) which is compact and has a piecewise smooth boundary $S$ (also indicated with $\partial V = S$). If $F$ is a continuously differentiable vector field defined on a neighborhood of $V$, then

$$\iiint_V \left(\mathbf{\nabla} \cdot \mathbf{F} \right)\,dV = \oint_S (\mathbf {F} \cdot \mathbf{\hat{n}})\,dS.$$

The left side is a volume integral over the volume $V$, the right side is the surface integral over the boundary of the volume $V$. The closed manifold $\partial V$ is oriented by outward-pointing normals, and $n$ is the outward pointing unit normal at each point on the boundary $\partial V$.

Cauchy's integral formula

Let $U$ be an open subset of the complex plane $\mathbb{C}$, and suppose the closed disk $D$ defined as

$$D = \bigl\{z:|z-z_{0}|\leq r\bigr\}$$

is completely contained in $U$. Let $f: U\to\mathbb{C}$ be a holomorphic function, and let $\gamma$ be the circle, oriented counterclockwise, forming the boundary of $D$. Then for every $a$ in the interior of $D$,

$$f(a) = \frac{1}{2\pi i} \oint _{\gamma}\frac{f(z)}{z-a} dz.$$