Mathematical formula

1. NavierStokes Equations

Formula:

rho left( frac{partial mathbf{u}}{partial t} + (mathbf{u} cdot nabla)mathbf{u} right) = -nabla p + mu nabla^2 mathbf{u} + mathbf{f}

– Description: These are nonlinear partial differential equations that describe the motion of viscous fluids. They are one of the Millennium Prize Problems, with a general solution still elusive.

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2. Einstein Field Equations (General Theory of Relativity)

Formula:

R_{mu nu} – frac{1}{2} R g_{mu nu} + Lambda g_{mu nu} = frac{8 pi G}{c^4} T_{mu nu}

– Description: These equations explain gravity as the curvature of spacetime caused by mass and energy, involving advanced concepts like tensors and differential geometry.

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3. Schrdinger Equation

Formula:

ihbar frac{partial}{partial t} Psi(mathbf{r}, t) = hat{H} Psi(mathbf{r}, t)

– Description: The foundation of quantum mechanics, it describes the behavior of subatomic particles through the concept of wave funct

ions.

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4. Riemann Hypothesis (Related to Riemann Zeta Function)

Formula:

zeta(s) = sum_{n=1}^{infty} frac{1}{n^s} = prod_{p text{ prime}} frac{1}{1 – p^{-s}}

– Description: This conjecture is about the distribution of prime numbers and remains unproven. It states that all non-trivial zeros of the Riemann zeta function lie on the critical line text{Re}(s) = frac{1}{2}

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5. Feynman Path Integral

Formula:

langle q_f, t_f | q_i, t_i rangle = int mathcal{D}[q(t)] e^{frac{i}{hbar} S[q(t)]}

– Description: Used in advanced quantum mechanics to calculate the transition amplitude between two states of a system, considering all possible paths the system could take.

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