## **The 20 Most Important Mathematical Equations**

A journey through the most elegant and influential formulas in mathematics

**1. Euler's Identity**  $\text{Contextsts } \text{five } \text{fundamental} \text{ constraints (e, i, n, 1, 0), } \text{revezling the} $ profound relationship between exponential functions and trigonometry. **3. The Fundamental Theorem of Calculus** Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral  $\text{еquals } \mathsf{F}(\mathsf{b}) \text{ - } \mathsf{F}(\mathsf{a}). \text{ Веvolutionized паўнепайсаl то problem-so/wing.} $ **2. Pythagorean Theorem 4. Maxwell's Equations** *e* + *iπ* 1 = 0 ∫ *f*(*x*) *dx* = *a b F*(*b*) − *F*(*a*)  $a^2 + b^2 = c^2 $ <sup>2</sup> <sup>∇</sup> <sup>⋅</sup> **<sup>E</sup>** <sup>=</sup> *ρ*

In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.

$$
\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon\_0}
$$

$$
\nabla \cdot \mathbf{B} = 0
$$

$$
\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
$$

$$
\nabla \times \mathbf{B} = \mu\_0 \mathbf{J} + \mu\_0 \varepsilon\_0 \frac{\partial \mathbf{E}}{\partial t}
$$

Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.