Anonymous
6/21/2025, 11:26:05 PM
No.16704625
How can I prove the equivalence between these two constructions of the differential?
1) [math](\mathrm{d}f)_p: X \mapsto (f\circ c)'(0)[/math] where X is a derivation and c is a curve s.t. c'(0) = X and c(0) = p;
2)[math](\mathrm{d}f)_p:X \mapsto X(f)[/math] where [math]X\in T_pM[/math]
1) [math](\mathrm{d}f)_p: X \mapsto (f\circ c)'(0)[/math] where X is a derivation and c is a curve s.t. c'(0) = X and c(0) = p;
2)[math](\mathrm{d}f)_p:X \mapsto X(f)[/math] where [math]X\in T_pM[/math]