Classical gallery

Rendered with Purple Pi.

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