Differential of the coordinate functions

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SUMMARY

The discussion centers on the differential of coordinate functions within the context of differential geometry. Rico inquires about the assertion that the differential of the i-th projection is itself the i-th projection. The response clarifies that the differential of a linear map remains a linear map, specifically illustrating this with the example of the linear function φ: x ↦ a·x, where the derivative d/dx φ(x) equals a, confirming that Dφ = La: x ↦ La(x) = a·x.

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Rico1990
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Hello folks,
I'm glad that I discovered this forum. :) You might save me.
I'm hearing right now differential geometry and am having some problems with the subject.
May you explain me the follwoing. We had the special case of the i-th projection. My lecturer now posited that the differential of the i-th projection is again the i-th projection. Can you explain me how to see that?

Greetings
Rico
 
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Without any further context, it's simply the following: The differential of a linear map is again the linear map.
##\varphi\, : \,x\longmapsto a\cdot x## gets ##\dfrac{d}{dx}\varphi(x) = a## which is ##D\varphi = L_a\, : \,x \longmapsto L_a(x)=a\cdot x##.
 

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