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 .
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 .
i believe *you* are misunderstanding the question. V is a vector field, not a curve in R^{3}.
i think calling the coordinate functions of V, "y^{2}" and "x" is a bad idea...if i understand this correctly what we really have is the following:
V(x,y,z) = y^{2}U_{1}(x,y,z) - xU_{3}(x,y,z)
= y^{2}(1,0,0)_{(x,y,z)} - x(0,0,1)_{(x,y,z)}
= (y^{2},0,-x)_{(x,y,z)}.
in that case, i concur that V[f](x,y,z) = y^{2}(y) + 0(x) + (-x)(0) = y^{3}. 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:
V_{1}(x,y,z) = y^{2}
V_{2}(x,y,z) = 0
V_{3}(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:
V_{1}(p_{1},p_{2},p_{3}) = (p_{2})^{2}
V_{2}(p_{1},p_{2},p_{3}) = 0
V_{3}(p_{1},p_{2},p_{3}) = -p_{1}
understanding what function is INTENDED by V_{1},V_{2}, and V_{3} 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.