Euler's identity is often called the most beautiful equation in mathematics. It relates five fundamental mathematical constants: \(e, \pi, i \) (the imaginary unit), 1, and 0, using addition, multiplication, exponentiation, and equality.
$$e^{i\pi} + 1 = 0$$When \(a \ne 0\), there are two solutions to \(ax^2 + bx + c = 0\) and they are:
$$x = -b \pm \sqrt {b^2-4ac} \over 2a$$The Navier-Stokes equation is a set of nonlinear partial differential equations that describe the motion of fluid substances. It's used extensively in fluid dynamics to model phenomena such as airflow around an airplane wing or ocean currents.
$$ \frac{\partial \mathbf{v}} {\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} = -\frac{\nabla p}{\rho} + \nu \nabla^2 \mathbf{v}$$The standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. It represents the average distance of each data point from the mean of the dataset. In other words, it measures how spread out the values in a dataset are around the mean.
$$\sigma = \sqrt{ \frac{1}{N} \sum_{i=1}^N (x_i -\mu)^2}$$The Black-Scholes formula is a mathematical model used for pricing options contracts in finance. It gives the theoretical price of European-style options over time.
$$ C(S,t) = N(d_1)S - N(d_2)Ke^{-r(T-t)}$$The Taylor series is a mathematical representation of a function as an infinite sum of terms, each term being a polynomial function evaluated at a particular point. It provides a way to approximate a wide variety of functions using polynomials, enabling the approximation of complex functions by simpler ones. The Taylor series expansion is centered around a specific point and expresses the function as a sum of terms involving the function's derivatives evaluated at that point. It is a fundamental tool in calculus and mathematical analysis, used in fields such as physics, engineering, and computer science for numerical approximation and analysis.
$$ P(x) = \color{teal}{f(0)} + \color{blue} \frac{df}{dx}(0)\frac{x^1}{1!}+ \color{green} \frac{d^2f}{dx^2}(0) \frac{x^2}{2!}+ \color{red} \frac{d^3f}{dx^3}(0)\frac{x^3}{3!}+ \color{black}… $$