====== September 1: Differentiation and Integration ======


===== Differentiation =====
What is differentiation? it is measuring the ratio of how the output change versus how the input changes. It is a linear map from the vector space of small change of input, to the vector space of small changes of output. 

==== Chain Rule ====
There is also the chain rules, which says, if quantity x affect y,and y  affect z, then x affects z. If $y=2x$, $z=3y$, then $z = 6x$. 

==== Partial Derivative ====
If you have a function $f(x,y)$ that depends on two input variables, you can ask how sensitive the output is on each of them, say
$$ \frac{\d f}{\d x}(x_0, y_0) = \lim_{\epsilon \to 0} \frac{f(x_0 + \epsilon, y_0) - f(x_0, y_0)}{\epsilon} $$


===== Integration =====

What is integration? It is a process of collecting stuff / contributions along the way.For example, the integral (when $f(x)$ is continuous) is the limit of the following approximations
$$ \int_a^b f(x) dx = \lim_{N\to \infty} f(x_{N,i}) \Delta_N x, \quad x_{N,i} = a + \frac{b-a}{N} i, \Delta_N x = \frac{b-a}{N}. $$

The fundamental theorem of calculus says
$$ \int_a^b f'(x) dx = f(b) - f(a).$$

  * line integral
  * Green's theorem.