Directional Derivatives - O'Neil - Elementary Differential Geonetry - Ch 1

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

I believe you are misunderstanding "V". If a curve is given by (which I would have called ) then V is its tangent vector at each point, . Of course, that would still give as the derivative, not .

Quote:

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Originally Posted by HallsofIvy

I believe you are misunderstanding "V". If a curve is given by (which I would have called ) then V is its tangent vector at each point, . Of course, that would still give as the derivative, not .

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i believe *you* are misunderstanding the question. V is a vector field, not a curve in R³.

i think calling the coordinate functions of V, "y²" and "x" is a bad idea...if i understand this correctly what we really have is the following:

V(x,y,z) = y²U₁(x,y,z) - xU₃(x,y,z)

= y²(1,0,0)_(x,y,z) - x(0,0,1)_(x,y,z)

= (y²,0,-x)_(x,y,z).

in that case, i concur that V[f](x,y,z) = y²(y) + 0(x) + (-x)(0) = y³. the answer in the back appears to be incorrect.

EDIT: take a look here: http://www.calpoly.edu/~jborzell/Cou...ff_Geo_2ed.pdf

and see if the answer in question is on page 451.

Deveno,

Thanks for the help - and a thanks to HallsofIvy as well!

You write "i think calling the coordinate functions of V, "y2" and "x" is a bad idea"

I am trying to understand why you say this, but I think you are not happy with my statement:



I am seeking to understand what is wrong with breaking up a vector field in into three coordinate functions?

Can you please clarify?

[By the way, you are right, O'Neill has picked up the error - or someone has for him]

Peter

no, it's not your statement, you're fine. i think what is better is to say:

V₁(x,y,z) = y²

V₂(x,y,z) = 0

V₃(x,y,z) = -x

notice is all 3 of these statements, x,y and z are "dummy variables" we get the exact same functions if we write:

V₁(p₁,p₂,p₃) = (p₂)²
V₂(p₁,p₂,p₃) = 0
V₃(p₁,p₂,p₃) = -p₁

understanding what function is INTENDED by V₁,V₂, and V₃ depends on recognizing (x,y,z) as a "typical point", which is fine, but O'Neill uses "p"!

in other words: "-x" is isn't a function, it's the VALUE of a function (namely the function that takes the negative of the third coordinate of a triple of real numbers) when we take the first coordinate of p to be x. perhaps this seems like needless nit-picking, but i have to wonder: why be so careful about the definition of the "standard frame", and then so sloppy about defining the "coordinate functions"? especially since the point p in the definition of the vector field (for the tangent space) is just along for the ride, we actually USE the coordinate functions, and then we just tag the p as a subscript.

it's like if i say: "what is the derivative of a?" is "a" a number, the constant function f(x) = a, for all x in R, or the identity function f(a) = a, for all a in R? i haven't said, i've just written down a letter. it's little better if i say, "what is the derivative of y at a?" now there's two ambiguities, and you don't know whether you're trying to find y'(a) or evaluating the constant function 1 at a, or if (even *worse*) you have to use the chain rule:

y'(a(t))(a'(t))

in which case do i want the whole thing, or just the first factor?

perhaps i am way out in left field, here, but mathematical communication should possess clarity.

Thanks for that! Most helpful to me

I totally agree with you on the need for precision and clarity

Peter