A book worth buying for the preface alone (or how to review a book you haven’t read)’

Everyone in grade school writes a book report on something they haven’t completely read (or read at all). Well, I’m long past, but here’s a book worth buying for the preface alone — Visual Differential Geometry and Forms by Tristan Needham.

His earlier book Visual Complex Analysis (another book I haven’t read completely) is a masterpiece (those parts I’ve managed to read) with all sorts of complex formulas and algebraicism explained visually.  It’s best to be patient, as Needham doesn’t even get to complex differentiation until Chapter 4 (p. 188) The Amplitwist concept.

On page xviii of the preface, Needham describes the third type of calculus Newton invented, and the one he used in the great Principia Mathematica of 1687.

The first one of 1665 was basically the manipulation of power series (which I’ve never heard of.  Have you?)

The second was one we all study in school, dx dy and all that.

Newton called the third method of calculus “the synthetic method of fluxions.”

Bishop Berkeley had a field day with them.  “And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?”

Laugh not at the good Bishop;   Berkeley California is named for him (as long as the woke don’t find out).

Needham gives a completely nontrivial example of the method.  He shows geometrically how the derivative of tan(theta) is 1 + [ tan(theta)]^2,.

The diagram will not be reproduced here but suffice it to say you’ll have to remember a lot of your geometry, e.g. two lines separated by a very small angle are essentially parallel, the radius of a circle is perpendicular to the tangent, what a similar triangle is.

The best thing about it is you can actually visualize the limit process taking place as a triangle shrinks and shrinks and eventually becomes ‘almost’ similar to another.  No deltas and epsilons for Newton (or Needham).  There is some algebra involving the ratios of the sides of similar triangles, but it’s trivial.

Buy the book and enjoy.   You’ll never think of differentiation the same way.  Your life will be better, and you might even meet a better class of people.

A premature book review and a 60 year history with complex variables in 4 acts

“Visual Differential Geometry and Forms” (VDGF) by Tristan Needham is an incredible book.  Here is a premature review having only been through the first 82 pages of 464 pages of text.

Here’s why.

While mathematicians may try to tie down the visual Gulliver with Lilliputian strands of logic, there is always far more information in visual stimuli than logic can appreciate.  There is no such a thing as a pure visual percept (a la Bertrand Russell), as visual processing begins within the 10 layers of the retina and continues on from there.  Remember: half your brain is involved in processing visual information.  Which is a long winded way of saying that Needham’s visual approach to curvature and other visual constructs is an excellent idea.

Needham loves complex variables and geometry and his book is full of pictures (probably on 50% of the pages).

My history with complex variables goes back over 60 years and occurs in 4 acts.

 

Act I:  Complex variable course as an undergraduate. Time late 50s.  Instructor Raymond Smullyan a man who, while in this world, was definitely not of it.  He really wasn’t a bad instructor but he appeared to be thinking about something else most of the time.

 

Act II: Complex variable course at Rocky Mountain College, Billings Montana.  Time early 80s.  The instructor and MIT PhD was excellent.  Unfortunately I can’t remember his name.  I took complex variables again, because I’d been knocked out for probably 30 minutes the previous year and wanted to see if I could still think about the hard stuff.

 

Act III: 1999 The publication of Needham’s first book — Visual Complex Analysis.  Absolutely unique at the time, full of pictures with a glowing recommendation from Roger Penrose, Needham’s PhD advisor.  I read parts of it, but really didn’t appreciate it.

 

Act IV 2021 the publication of Needham’s second book, and the subject of this partial review.  Just what I wanted after studying differential geometry with a view to really understanding general relativity, so I could read a classmate’s book on the subject.  Just like VCA, and I got through 82 pages or so, before I realized I should go back and go through the relevant parts (several hundred pages) of VCA again, which is where I am now.  Euclid is all you need for the geometry of VCA, but any extra math you know won’t hurt.

 

I can’t recommend both strongly enough, particularly if you’ve been studying differential geometry and physics.  There really is a reason for saying “I see it” when you understand something.

 

Both books are engagingly and informally written, and I can’t recommend them enough (well at least the first 82 pages of VDGF).