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## Substance

*A summary of Penrose's summary of physics and math.*

Thinking of functions as maps allows us to define non-smooth functions as well as those pieced together from other functions like:
\[\theta(x) = \frac{|x|- x}{2x}\] ie. the Heaviside Step Function ## C^n-smoothness
\(f(x) = x^{n}|x|\) with \(n\) a positive integer is \(C^{n}\)-smooth because it can only be differentiated to a smooth function n times.
Now consider \(h(x)=\begin{cases}0 & x\leq 0\\ e^{-1/x} & x > 0 \\ \end{cases}\) which is \(\bullet\) \(C^{\infty}\)-smooth over \(\mathbb{R}\) \(\bullet\) can think of as 2 functions glued together Sticking to real number functions, let the power series of \(f(x)\) be \[f(x) = a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4} + ...\] To have an expansion in this form, \(f(x)\) must be \(C^{\infty}\)-smooth so that one may differentiate \(f(x)\) infinitely to obtain an approximation of the terms of the expansion which are of course infinite. ## analytic functions
In \(h(x)\), the coefficients of the Maclauren series at the origin \(x = 0\), ie. \(a_{n} = f^{n}(0)/n!\) are all zero on both sides. Thus, \(h(x)\) is NOT analytic at the origin. You cannot gain any information about \(h(x)\) from its expansion about the origin.
However, a power series about some other point is given by \[f(x) = a_{0} + a_{1}(x-p) + a_{2}(x-p)^{2} + a_{3}(x-p)^{3} + ...\] with \(a_{n} = f^{n}(p)/n!\). For some \(x\) infinitesimally close to \(p\) st. we can essentially say \(x=p\), if the values of \(a_{n}\) give more information than before (ie. are not always zero at p), then the function is called analytic. \(\bullet\) analytic/\(C^{\infty}\)-smooth function = analytic at all \(p\) in its domain ## Differentiation Rules
\[\frac{d(x^{n})}{dx} = nx^{n-1}\] \[d[f(x) + g(x)] = df(x) + dg(x)\] \[d{af(x)} = adf(x)\] for \(a\) a constant
the "Leibniz Law": \[d\{f(x)g(x)\} = f(x)dg(x) + g(x)df(x)\] \[d\{f(g(x))\} = f'(g(x))g'(x)dx\] \[d\left(\frac{f(x)}{g(x)}\right) = \frac{g(x)df(x) - f(x)dg(x)}{g(x)^{2}}\] ## Integration
## Other Notes:
- Penrose refers to \(C^{n}\) as the differentiability class, ie. functions that can be differentiated n times
-defines the Heaviside step function \(\theta(x)\) as a \(C^{-1}\) function and the Dirac delta function (its derivative) as a \(C^{-2}\) function
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