Help with some questions from spivak calc on manifolds

[osnarf]

Hi everyone, been away for a while I got bogged down with my classes so didn't have time to work on this book and haven't been on the forums much. Was getting caught back up to where I was before in here and I ran into a problem that I can't figure out the notation on.

I am only looking for help understanding what he is asking, please no hints to the problem (yet, lol).

In my book, it is on page 33 (problem 2-29).

Homework Statement

Let f: Rⁿ ---> R. For x in Rⁿ, the limit

limit as t-->0 of [ ( f(a+tx) - f(a) ) / t ]

if it exists, is denoted D_xf(a), and called the directional derivative of fa at a, in the direction x.

(part a) Show that D_eif(A) = D_if(a) <----edit: kind of hard to see, the first subscript is e_i

(part b) Show that D_txf(a) = tD_xf(a)

(part c) If f is differentiable at a, show that D_xf(a) = Df(a)(x), and therefore D_x+yf(a) = D_xf(a) + D_yf(a).

Homework Equations

The Attempt at a Solution

Again, haven't really tried to solve it yet, as I'm not sure what it's asking. (I'll probably start working on part b immediately after posting this, because I'm pretty sure I get what he's saying there). So, my questions:

(part a) ? what is e_i? I looked earlier in the book, and I can't see anything close to what this might mean (he denotes second partial derivatives D_i,jf(a), so that's not it). I would assume the i^th element in the e vector, but that doesn't really seem to make sense here...

(part b) Just confirm for me that he means t $$\in$$ R, and we're all set.

(part c) I'm pretty sure I get this, I've been slightly unsure of what he means when he uses notation like this: Df(a)(x). Is this the same as saying f'(a)*(x), where x is the jacobian? Or is it f'(a)*a*x?

He mentions it earlier on in the book (Referencing page 20, theorem 2-3, part 5), for instance:

If p: R²-->R is defined by p(x,y) = x*y, then
Dp(a,b)(x,y) = bx + ay

I'm a little confused about the notation here, as well. In fact this has kind of had me confused the whole book.

In the quoted theorem,

p'(a,b) = (b a)

(b a) * (x y)^transform = 1x1 matrix (bx + ay)

Does this mean that Dp(a,b) is the matrix of the linear transformation (Dp(a,b) = p'(a,b) ?) and he is multiplying it by the vector (x y)^transform? Earlier on, he refers to Df(a) as the linear transform, so I'm assuming (x,y) are the inputs to be 'plugged in' to the linear transform Df(a) (like x and y are the variables to be plugged into the function f(x,y)). But I still feel like it should be written Dp(a,b)((x,y)^transform) if this is the case (or written vertically to begin with), so I can't feel sure about it.

I wish he was a little more descriptive with his notations, as those are the only problems I have had with this book, his explanations of concepts are great, as in his single variable book.

Thanks for reading, I appreciate your help.
PS: Is there a way to input matrices, or at least vertical vectors so you don't have to put (x, y, z)^t? I'm not very handy with tex yet.

 

[osnarf]

If you don't feel like reading it all, more than anything else I want to know what he means in part a.

 

[Sethric]

Usually when I see $$e_i$$ I think of a basis vector, if that helps.

 

[osnarf]

That makes perfect sense actually, and its pretty easy to see why it's true after that. Don't know how I didn't get that thanks, been buggin me all night.

edit:
On further review (page 3 of the book), he defines the usual basis vectors as e₁, e₂, etc... didn't go back far enough i guess :( Fail on my part.

Still could use clarification on part C.