Undergrad Differential of the coordinate functions

[Rico1990]

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

 

[fresh_42]

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 $D\varphi = L_a\, : \,x \longmapsto L_a(x)=a\cdot x$.