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 $\displaystyle V = y^2 U_1 - x U_3 $ and let $\displaystyle f = xy $ [See attachment page 9 for definition of $\displaystyle U_1, U_2, U_3 $, the natural frame field on$\displaystyle \mathbb{R}^3 $

O'Neil uses the notation $\displaystyle v_p [f] $ for the derivative of f with respect to $\displaystyle v_p $ and derives the formula

$\displaystyle v_p [f] = \sum v_i \frac{\partial f}{\partial x_i} (p) $ [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 $\displaystyle V[f] $ 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:

$\displaystyle V(p) = (V_1, V-2, V_3) = V_1 U_1 + V_2 U_2 + V_3 U_3$

Also I assume that, analogous to $\displaystyle v_p [f] = \sum v_i \frac{\partial f}{\partial x_i} (p) $ we can write

$\displaystyle V_p [f] = \sum V_i \frac{\partial f}{\partial x_i} (p) $

Thus using the field and function in the exercise above i.e. $\displaystyle V = y^2 U_1 - x U_3 $ and $\displaystyle f = xy $ we have

$\displaystyle V = y^2.U_1 + 0.U_2 + (-x).U_3 $

i.e. $\displaystyle V_1 = y^2 , V_2 = 0 , V_3 = -x $

Also $\displaystyle \frac{\partial f}{\partial x} = y $ , $\displaystyle \frac{\partial f}{\partial y} = x $ and $\displaystyle \frac{\partial f}{\partial z} = 0 $

Thus $\displaystyle V(p)[f] = V_1 \frac{\partial f}{\partial x} + V_2 \frac{\partial f}{\partial y} + V_3 \frac{\partial f}{\partial z} $

so

$\displaystyle V(p)[f] = y^2 . y + 0.x + (-x).0 = y^3 $

However, at the back of the book, O'Neill gives the answer as $\displaystyle y^2$

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

Peter