I am reading O'Neill: Elementary Differential Geonetry Ch 1. I am having a problem with Exercise 3 of Exercises 1.3 page 15, which I suspect may be due to me misuderstanding O'Neill's notation. [So that others may follow his terminology I have attached a copy of O'Neill Ch1 pages 6 to 15.]

Exercise 3 reads as follows:

Let and let [See attachment page 9 for definition of , the natural frame field on

O'Neil uses the notation for the derivative of f with respect to and derives the formula

[see the attachment pages 11 and 12]

O'Neill then defines the operation of a vector field V on a function f. The result is the real-valued function whose value at each point p is the number V(p)[f], that is the derivative of f with respect to the tangent vector V(p) at p.

Thus following this (and the notation and definitions in O'Neil - see attachment) that we can (correctly) write the following:

Also I assume that, analogous to we can write

Thus using the field and function in the exercise above i.e. and we have

i.e.

Also , and

Thus

so

However, at the back of the book, O'Neill gives the answer as

This is really disconcerting ... can anyone locate my error? Could you also confirm that I am using the terminology properly.

Peter