3 ms·

Well this is the whole point of derivatives (i.e. tangent maps): to be linear approximations of functions. So yes, a linear approximation of a linear function

Well this is the whole point of derivatives (i.e. tangent maps): to be linear approximations of functions. So yes, a linear approximation of a linear function is the function itself.

giomasce 8y ago
You are right, but what you are saying has nothing to do with the author's point: what he is saying is that the differentiation operator itself is linear, which is a meaningful and true fact even in spaces where you have no idea of what a linear function is.
- hgibbs 8y ago
  I think you'll find that derivatives are overwhelming defined as operators on vector spaces. It is definitely correct to talk about a linear function on a vector space.
  - giomasce 8y ago
    I agree with you, but this is not what GGP is saying. GGP is saying another true fact (i.e., that derivative of a linear function is that same function), which a different thing than stating, as the article says, that the differentiation operator is linear. On a manifold there is no concept of a linear function, so you cannot say that the derivative of a linear function is the same function, but the differentiation operator is still defined and linear.
    - hgibbs 8y ago
      Yes, you are right. My apologies.
    - chombier 8y ago
      Of course you are correct, even though one could argue the linearity of differentiation is a property you can obtain by differentiating in coordinate charts, where the reasoning is still valid. In any case, it is probably a good thing to get a good intuition of what differentiation and derivatives are in the vector space setting before digging into differential geometry.
      - 4ad 8y ago
        Not all vector spaces are manifolds (in fact most aren't), and you don't need charts to define differential operators (just see functional analysis).
        chombier 8y ago
        I most definitely agree with you, however this brings us even further away from the author's setting.
thaumasiotes 8y ago
If a function f passes through some point (a,b), then the tangent to f through that point is given by (y-b) = f'(a)·(x-a) and that function is affine but usually not linear. (For the tangent curve to be a linear function, you would need a·f'(a) = b, so that the tangent goes through the point (0,0).) It's not at all obvious to me that this means that the function d(f) = df/dx is linear. It is linear, but I don't see how the tangent curve demonstrates it.
- giomasce 8y ago
  It is common in this context to say "linear" to also mean "affine", because after all affine functions are not much more complicated than linear functions.
- chombier 8y ago
  The sum operator is linear, so its derivative is itself: the derivative of the sum is the sum of derivatives. Same goes for multiplication by a scalar. From this you obtain that the derivative of any linear combination is the linear combination of the derivatives: differentiation is linear.
- amelius 8y ago
  The derivative is only the direction of the tangent line. The affine part does not come into play.
geoalchimista 8y ago
That is my feeling after reading this article. For people who have learned a little bit of Analysis, it is like saying "the earth is round". There is nothing new. The Wikipedia page of Derivative (https://en.wikipedia.org/wiki/Derivative https://en.wikipedia.org/wiki/Derivative) has a detailed description on the linearity. (Well, I do appreciate the author's way of presenting the idea, but I don't think it deserves an in-depth discussion on Hacker News.)

Well this is the whole point of derivatives (i.e. tangent maps): to be linear approximations of functions. So yes, a linear approximation of a linear function is the function itself.

giomasce 8y ago

You are right, but what you are saying has nothing to do with the author's point: what he is saying is that the differentiation operator itself is linear, which is a meaningful and true fact even in spaces where you have no idea of what a linear function is.

hgibbs 8y ago
I think you'll find that derivatives are overwhelming defined as operators on vector spaces. It is definitely correct to talk about a linear function on a vector space.
- giomasce 8y ago
  I agree with you, but this is not what GGP is saying. GGP is saying another true fact (i.e., that derivative of a linear function is that same function), which a different thing than stating, as the article says, that the differentiation operator is linear. On a manifold there is no concept of a linear function, so you cannot say that the derivative of a linear function is the same function, but the differentiation operator is still defined and linear.
  - hgibbs 8y ago
    Yes, you are right. My apologies.
  - chombier 8y ago
    Of course you are correct, even though one could argue the linearity of differentiation is a property you can obtain by differentiating in coordinate charts, where the reasoning is still valid. In any case, it is probably a good thing to get a good intuition of what differentiation and derivatives are in the vector space setting before digging into differential geometry.
    - 4ad 8y ago
      Not all vector spaces are manifolds (in fact most aren't), and you don't need charts to define differential operators (just see functional analysis).
      - chombier 8y ago
        I most definitely agree with you, however this brings us even further away from the author's setting.

thaumasiotes 8y ago

If a function f passes through some point (a,b), then the tangent to f through that point is given by (y-b) = f'(a)·(x-a) and that function is affine but usually not linear. (For the tangent curve to be a linear function, you would need a·f'(a) = b, so that the tangent goes through the point (0,0).) It's not at all obvious to me that this means that the function d(f) = df/dx is linear. It is linear, but I don't see how the tangent curve demonstrates it.

giomasce 8y ago
It is common in this context to say "linear" to also mean "affine", because after all affine functions are not much more complicated than linear functions.
chombier 8y ago
The sum operator is linear, so its derivative is itself: the derivative of the sum is the sum of derivatives. Same goes for multiplication by a scalar. From this you obtain that the derivative of any linear combination is the linear combination of the derivatives: differentiation is linear.
amelius 8y ago
The derivative is only the direction of the tangent line. The affine part does not come into play.

geoalchimista 8y ago

That is my feeling after reading this article. For people who have learned a little bit of Analysis, it is like saying "the earth is round". There is nothing new. The Wikipedia page of Derivative (https://en.wikipedia.org/wiki/Derivative https://en.wikipedia.org/wiki/Derivative) has a detailed description on the linearity. (Well, I do appreciate the author's way of presenting the idea, but I don't think it deserves an in-depth discussion on Hacker News.)