ece205: add periodicity

docs/2a/ece205.md

@@ -214,6 +214,38 @@ Thus we also have:
 
 $$\mathcal L\{\delta (t-a)\}=e^{-as}\implies\mathcal L^{-1}\{1\}=\delta(t)$$
 
+## Heat flow
+
+The temperature of a tube from $x=0$ to $x=L$ can be represented by the following DE:
+
+$$\text{temp}=u(x,t)=\boxed{u_t=a^2u_{xx}},0<x<L,y>0$$
+
+Two boundary conditions are requred to solve the problem for all $t>0$ — that at $t=0$ and at $x=0,x=L$.
+
+- $u(x,0)=f(x),0\leq x\leq L$
+- e.g., $u(0,t)=u(L,t)=0,t>0$
+
+### Periodicity
+
+The **period** of a function is an increment that always returns the same value: $f(x+T)=f(x)$, and its **fundamental period** of a function is the smallest possible period.
+
+!!! example
+    The fundamental period of $\sin x$ is $2\pi$, but any $2\pi K,k\in\mathbb N$ is a period.
+    
+    The fundamental periods of $\sin \omega x$ and $\cos\omega x$ are both $\frac{2\pi}{\omega}$.
+
+If functions $f$ and $g$ have a period $T$, then both $af+bg$ and $fg$ also must have period $T$.
+
+#### Manipulating polarity
+
+- even: $\int^L_{-L}f(x)dx=2\int^L_0f(x)dx$
+- odd: $\int^L_{-L}f(x)dx=0$
+
+- even × even = even
+- odd × odd = even
+- even × odd = odd
+
+
 ## Resources
 
 - [Laplace Table](/resources/ece/laplace.pdf)