Editing The Road to Reality Study Notes (section)

=== 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.