Equation examples
The Quadratic Formula
\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
Cauchy’s Integral Formula
\(f(a) = \frac{1}{2\pi i} \oint\frac{f(z)}{z-a}dz\)
Angle Sum Formula for Cosines
\(\cos(\theta+\phi)=\cos(\theta)\cos(\phi)−\sin(\theta)\sin(\phi)\)
Gauss’ Divergence Theorem
\(\int_D ({\nabla\cdot} F)dV=\int_{\partial D} F\cdot ndS\)
Curl of a Vector Field
\(\vec{\nabla} \times \vec{F} =
\left( \frac{\partial F_z}{\partial y} – \frac{\partial F_y}{\partial z} \right) \mathbf{i}
+ \left( \frac{\partial F_x}{\partial z} – \frac{\partial F_z}{\partial x} \right) \mathbf{j}
+ \left( \frac{\partial F_y}{\partial x} – \frac{\partial F_x}{\partial y} \right) \mathbf{k}\)
\left( \frac{\partial F_z}{\partial y} – \frac{\partial F_y}{\partial z} \right) \mathbf{i}
+ \left( \frac{\partial F_x}{\partial z} – \frac{\partial F_z}{\partial x} \right) \mathbf{j}
+ \left( \frac{\partial F_y}{\partial x} – \frac{\partial F_x}{\partial y} \right) \mathbf{k}\)
Standard Deviation
\(\sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^N (x_i -\mu)^2}\)
Definition of Christoffel Symbols
\((\nabla_X Y)^k = X^i (\nabla_i Y)^k =
X^i \left( \frac{\partial Y^k}{\partial x^i} + \Gamma_{im}^k Y^m \right)\)
X^i \left( \frac{\partial Y^k}{\partial x^i} + \Gamma_{im}^k Y^m \right)\)