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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
by 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 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.