Editing The Road to Reality Study Notes (section)

== Chapter 6 Real-number calculus ==

=== 6.1 What makes an honest function? ===

Calculus is built from the two ingredients, [https://en.wikipedia.org/wiki/Derivative differentiation] and [https://en.wikipedia.org/wiki/Integral integration].  Differentiation is a local phenomenon which concerns the rates that things change whereby integration is a more quantity that measures the totality.  Incredibly, these two ingredients are the inverse of one another ([https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus fundamental theorem of calculus]).

These two operate on ‘functions’ and Penrose notes that these can be thought of as ‘mappings’ from some array of numbers (domain) to another (target).  Penrose gives the three examples below which will be used later in the chapter as mapping the real number system to itself:
[[File:Fig 6p2.png|thumb|center]]

=== 6.2 Slopes of functions ===

Differentiation is concerned with and calculates the rates that things change, or ‘slopes’ of these things.  For the curves given in section 6p1 above, two of the three do not have unique slopes at the origin and are said to be not ''differentiable'' at the origin, or not ''smooth'' there.  Further, the curve of theta(x) has a jump at the origin which is to say that it is discontinuous there, whereas <math>|x|</math> and <math>x^2</math> were continuous everywhere.

Taking differentiation a step further, Penrose shows us two plots which look very similar, but are represented by different functions, <math>x^3</math> and <math>x|x|</math>.  Each are differentiable and continuous, but the difference has to do with the curvature ([https://en.wikipedia.org/wiki/Second_derivative second derivative]) at the origin.  <math>x|x|</math> does not have a well-defined curvature here and is said to not be ''twice differentiable''.

=== 6.3 Higher derivatives; <math>C^\infty</math>-smooth functions ===
Looking closer at the concept of two derivatives of the same function (second derivative, or curvature), Penrose shows us the functions from 6p2 and their first and second derivatives.  Note that the first derivative of <math>f(x)</math>, written <math>f’(x)</math>, meets the x-axis at places of local minima or maxima and the second derivative of <math>f(x)</math>, written <math>f’’(x)</math>, meets the x-axis where the curvature goes to <math>0</math> and is said to be a point of inflection.

[[File:Fig 6p5.png|thumb|center]]

In general, a function can be smooth for many derivatives and the mathematical terminology for general smoothness is to say that <math>f(x)</math> is <math>C^n</math>-smooth.  It can be seen that <math>x|x|</math> is <math>C^1</math>-smooth but not <math>C^2</math>-smooth due to the discontinuity at the origin in the derivative. In general <math>x^n|x|</math> is <math>C^n</math>-smooth but not <math>C^{n+1}</math>-smooth.  In fact, a function can be <math>C^\infty</math>-smooth if it is smooth for every positive integer.  Note that negative integers for <math>x^n</math> immediately are not smooth for <math>x^{-1}</math> (discontinuous at the origin).  

Penrose notes that Euler would have required <math>C^\infty</math>-smooth functions to be defined as functions, and then gives the function:
:<math>h(x) = \begin{cases}
  0, & \mbox{if } x \le 0 \\
  e^{-\frac{1}{x}},  & \mbox{if } x > 0 
\end{cases}
</math>
as an example of a <math>C^\infty</math>-smooth function but one that Euler would still not be happy with since it is two functions stuck together.

=== 6.4 The "Eulerian" notion of a function? ===
How, then do we define the notion of a ‘Eulerian’ function?  This can be accomplished in two ways.  The first using complex numbers and is incredibly simple.  If we extend <math>f(x)</math> to <math>f(z)</math> in the complex plane, then all we require is for <math>f(z)</math> to be once differentiable (a kind of <math>C^1</math>-smooth function).  That’s it, magically. We will see that this can be stated with <math>f(x)</math> being an [https://en.wikipedia.org/wiki/Analytic_function analytic function].

The second method involves power series manipulations, and Penrose notes that ‘the fact that complex differentiability turns out to be equivalent to power series expansions is one of the truly great pieces of complex-number magic’.  

For the second method, the power series of <math>f(x)</math> is introduced, <math>f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + …</math> For this series to exist then it must be <math>C^\infty</math>-smooth.  We must take and evaluate derivatives <math>f(x)</math> to find the coefficients, thus an infinite number of derivatives (positive integers) must exist for the power series to exist.  If we evaluate <math>f(x)</math> at the origin, we call this a power series expansion about the origin.  About any other point <math>p</math> would be considered a power series expansion about <math>p</math>. (Maclaurin Series about origin, see also [https://en.wikipedia.org/wiki/Taylor_series Taylor Series] for the general case)

The power series is considered analytic if it encompasses the power series about point <math>p</math>. If it is analytic at all points of its domain, we call it an analytic function or, equivalently, a <math>C^ω</math>-smooth function.  Euler would be pleased with this notion of an analytic function, which is ‘smoothier’ than the set of <math>C^\infty</math>-smooth functions (<math>h(x)</math> from 6p3 is <math>C^\infty</math>-smooth but not <math>C^ω</math>-smooth). 

* Physics in trying to understand reality by approximating it.

=== 6.5 The rules of differentiation ===

* Armed with these few rules (and loads and loads of practice), one can become an "expert at differentiation" without needing to have much in the way of actual understanding of why the rules work!

=== 6.6 Integration ===

As stated in section 6p1, integration is the inverse of differentiation as stated in the fundamental theorem of calculus.  Penrose provides the following visual and explanation.  If we start with the differentiated curve, the area underneath the derivative curve bounded by two points on the x-axis, and the x-axis itself is equal to the difference in heights of the original curve evaluated at the two points.

[[File:Fig6p8 and6p9.png|thumb|center]]

Integration is noted as making the function smoother and smoother, whereas differentiation continues to make things ‘worse’ until some functions reach a discontinuity and become ‘non-differentiable’.

Penrose ends the chapter noting that there are approaches which enable the process of differentiation to be continued indefinitely, even if the function is not differentiable.  One example is the [https://en.wikipedia.org/wiki/Dirac_delta_function Dirac Delta Function] which is of ‘considerable importance in quantum mechanics’.  This extends our notion of <math>C^n</math>-functions into the negative integer space (<math>C^{-1},C^{-2},...</math>) and will be discussed later with complex numbers.  Penrose notes that this leads us further away from the ‘Eulerian’ functions, but complex numbers provide us with an irony that expresses one of their finest magical feats of all.

* If we integrated then differentiate, we get the same answer back. Non-commutative the other way.