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--- 17_equations [2024-05-05 10:45] – [Calculus] nik
+++ 17_equations [2024-05-05 11:08] (current) – nik
@@ Line 6: / Line 6: @@
 ==== Pythagorean Theorem ====
-$a^2 + b^2 = c^2$
+$$a^2 + b^2 = c^2$$
 ==== Logarithms ====
-$log{xy} = log{x} + log{y}$
+$$log{xy} = log{x} + log{y}$$
 ==== Calculus ====
 $$\frac{\partial f}{\partial t} = \lim_{x\to\infty} =  \frac{f{(t+h)}- f{(t)}}{h}$$
 ==== Newton’s Law of Gravity ====
-${F}_\text{gravity}=G\frac{m_{1}m_{2}}{r^{2}}$
+$${F}_\text{gravity}=G\frac{m_{1}m_{2}}{r^{2}}$$
 ==== Complex Numbers ====
-$i^2=-1$
+$$i^2=-1$$
 ==== Euler’s Formula for Polyhedra ====
@@ Line 33: / Line 36: @@
 ==== 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 ====
-$dS\geq0$
+$$dS\geq0$$
 ==== Relativity ====
-$E=mc^2$
+$$E=mc^2$$
 ==== Schrodinger’s Equation ====
@@ Line 47: / Line 60: @@
 ==== Information Theory ====
-$H=-\sum p(x) + log{p(x)}$
+$$H=-\sum p(x) + log{p(x)}$$
 ==== Chaos Theory ====
-$x_{t+1} = kx_t(1-x_i)$
+$$x_{t+1} = kx_t(1-x_i)$$
 ==== Black-Scholes Equation ====