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--- 17_equations [2024-05-05 11:08] – nik
+++ 17_equations [2024-08-12 11:04] (current) – nik
@@ Line 4: / Line 4: @@
-==== Pythagorean Theorem ====
+=== Pythagorean Theorem ===
 $$a^2 + b^2 = c^2$$
-==== Logarithms ====
+----
+=== Logarithms ===
 $$log{xy} = log{x} + log{y}$$
-==== Calculus ====
+----
+=== Calculus ===
-$$\frac{\partial f}{\partial t} = \lim_{x\to\infty} =  \frac{f{(t+h)}- f{(t)}}{h}$$
+$$\frac{\partial f}{\partial t} = \lim_{h\to\infty} =  \frac{f{(t+h)}- f{(t)}}{h}$$
-==== Newton’s Law of Gravity ====
+----
+=== Newton’s Law of Gravity ===
 $${F}_\text{gravity}=G\frac{m_{1}m_{2}}{r^{2}}$$
-==== Complex Numbers ====
+----
+=== Complex Numbers ===
 $$i^2=-1$$
-==== Euler’s Formula for Polyhedra ====
+----
+=== Euler’s Formula for Polyhedra ===
-==== Normal Distribution ====
+$$V-E+F=2$$
+----
+=== Normal Distribution ===
-==== Wave Equation ====
+$$\Phi(x)= \frac{1}{\sqrt{2\pi\rho}} e^{\frac{(x-\mu)^2}{2\rho^2}}$$
-==== Fourier Transform ====
+----
+=== Wave Equation ===
-==== Navier-Stokes Equation ====
+$$\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$$
-==== Maxwell’s Equations ====
+----
+=== Fourier Transform ===
-$$\begin{aligned}
+$$f(\omega) = \int_{\infty}^{\infty}f(x)e^{-2\pi i x \omega} \text{d}x$$
-&\nabla\cdot\mathcal{E} = 0
+----
+=== Navier-Stokes Equation ===
+$$\rho\left(\frac{d\text{v}}{dt} + \text{v} \cdot \text{v}\nabla \right) = -\nabla p + \nabla \cdot \text{T} + \text{f}$$
+----
+=== Maxwell’s Equations ===
+$$\begin{aligned}
+&\nabla\cdot\mathcal{E} = 0
 &\nabla\cdot\mathcal{H} = 0
 \end{aligned}$$
 $$\begin{aligned}
 &\nabla\times\mathcal{E} = - \frac{1}{c}\frac{\partial\mathcal{H}}{\partial t}
 &\nabla\times\mathcal{H} = - \frac{1}{c}\frac{\partial\mathcal{E}}{\partial t}
 \end{aligned}$$
-==== Second Law of Thermodynamics ====
+----
+=== Second Law of Thermodynamics ===
 $$dS\geq0$$
-==== Relativity ====
+----
+=== Relativity ===
 $$E=mc^2$$
-==== Schrodinger’s Equation ====
+----
+=== Schrödinger’s Equation ===
+$$ih \frac{\delta}{\delta t}\Psi = H\Psi$$
-==== Information Theory ====
+----
+=== Information Theory ===
 $$H=-\sum p(x) + log{p(x)}$$
-==== Chaos Theory ====
+----
+=== Chaos Theory ===
 $$x_{t+1} = kx_t(1-x_i)$$
-==== Black-Scholes Equation ====
+----
+=== Black-Scholes Equation ===
+$$\frac{1}{2}\sigma^2S^2 \frac{\delta^2 V}{\delta S^2} + rS \frac{\delta V}{\delta S} + \frac{\delta V}{\delta t} - rV = 0$$
+----

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