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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
by 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 agoI 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 agoI 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 agoYes, you are right. My apologies.
- chombier 8y agoOf 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.