“Differential geometry is the study of properties that are invariant under change of notion” — Preface p. vii of “Introduction to Smooth Manifolds” J. M. Lee second edition. Lee says this is “funny primarily because it is so close to the truth”.

Having ascended to a substantial base camp for the assault on the Einstein Field equations (e.g. understanding the Riemann curvature tensor), I thought I’d take a break and follow Needham’s advice about another math book “Elementary Differential Geometry” 2nd edition (revised 2006) by Barrett O’Neill. “First published in 1966, this trail-blazing text pioneered the use of Forms at the undergraduate level. Today, more than a half-century later, O’Neill’s work remains, in my view, the single most clear-eyed, elegant and (ironically) modern treatment of the subject available — present company excepted! — at the undergraduate level.”

Anyone taking calculus has seen plenty of dx’s — in derivatives, in integrals etc. etc.. They’re rarely explained. O’Neill will get you there in just the first 24 pages. One more page and you’ll understand

df = (partial f/partial x) * dx + (partial f/partial y) * dy + (partial f/partial z) * dz

which you’ve doubtless seen before primarily as (heiroglyphics) before you moved on.

Is it easy? No, not unless you read definitions and internalize them immediately. The definitions are very clearly explained.

His definition of vector is a bit different — two points in Euclidean 3-space (written R^3 which is the only space he talks about in the first 25 pages). His 3-space is actually a vector space in which point can be added and scalar multiplied.

You’ll need to bone up on the chain rule from calculus 101.

A few more definitions — natural coordinate functions, tangent space to R^3 at p, vector field on R^3, pointwise principle, natural frame field, Euclidean coordinate function (written x_i, where i is in { 1, 2, 3 } ), derivative of a function in direction of vector v (e.g. directional derivative), operation of a vector on a function, tangent vector, 1-form, dual space. I had to write them down to keep them straight as they’re mixed in paragraphs containing explanatory text.

and

at long last,

differential of x_i (written dx_i)

All is not perfect. On p. 28 you are introduced to the alternation rule

dx_i ^ dx_j = – dx_j ^ dx_i with no justification whatsoever

On p. 30 you are given the formula for the exterior derivative of a one form again with no justification. So its back to mumbling incantations and plug and chug