| **1. Euler's Identity** | **3. The Fundamental Theorem of Calculus** |
|------------------------|------------------------------------------|
| $e^{i\pi} + 1 = 0$     | $\int_a^b f(x) \, dx = F(b) - F(a)$     |
| Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving. |

| **2. Pythagorean Theorem** | **4. Maxwell's Equations** |
|---------------------------|---------------------------|
| $a^2 + b^2 = c^2$         | $\nabla \cdot E = \frac{\rho}{\varepsilon_0}$ |
| 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. | $\nabla \cdot B = 0$ |
|                          | $\nabla \times E = -\frac{\partial B}{\partial t}$ |
|                          | $\nabla \times B = \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial E}{\partial t}$ |
|                          | 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. |