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osnarf
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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).
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 Dxf(a), and called the directional derivative of fa at a, in the direction x.
(part a) Show that Deif(A) = Dif(a) <----edit: kind of hard to see, the first subscript is ei
(part b) Show that Dtxf(a) = tDxf(a)
(part c) If f is differentiable at a, show that Dxf(a) = Df(a)(x), and therefore Dx+yf(a) = Dxf(a) + Dyf(a).
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 ei? I looked earlier in the book, and I can't see anything close to what this might mean (he denotes second partial derivatives Di,jf(a), so that's not it). I would assume the ith 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 [tex]\in[/tex] 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:
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.
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 Dxf(a), and called the directional derivative of fa at a, in the direction x.
(part a) Show that Deif(A) = Dif(a) <----edit: kind of hard to see, the first subscript is ei
(part b) Show that Dtxf(a) = tDxf(a)
(part c) If f is differentiable at a, show that Dxf(a) = Df(a)(x), and therefore Dx+yf(a) = Dxf(a) + Dyf(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 ei? I looked earlier in the book, and I can't see anything close to what this might mean (he denotes second partial derivatives Di,jf(a), so that's not it). I would assume the ith 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 [tex]\in[/tex] 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.
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