- #1

orion

- 93

- 2

In some books, the differential is defined by:

(a) ##df(v) = v(f) : T_pM → \mathbb {R}##

while in other books, a more abstract definition is made:

(b) Given two manifolds ##M,N##, ##v \in T_pM## and a map ##F :M → N## then ##dF## is defined

##dF(v)(f) = v(f\circ F) : T_pM → T_{F(p)}N##

where ##f \in C^∞(N)##.My questions:

(1) How are these two definitions equivalent? And if they are not equivalent how to reconcile them?

(2) I need help understanding (b). Why are there ##F## and ##f##? Also, how in what space does ##v(F \circ f)## operate since F is a map from ##M## to ##N## and ##v## belongs to ##T_pM## and ##f \in C^∞(N)##?

(3) It would seem that definition (a) is more in line with the idea that a differential is a linear functional on ##T_pM## and hence a covector in a space dual to ##T_pM##. How to understand (2) in this context?

(a) ##df(v) = v(f) : T_pM → \mathbb {R}##

while in other books, a more abstract definition is made:

(b) Given two manifolds ##M,N##, ##v \in T_pM## and a map ##F :M → N## then ##dF## is defined

##dF(v)(f) = v(f\circ F) : T_pM → T_{F(p)}N##

where ##f \in C^∞(N)##.My questions:

(1) How are these two definitions equivalent? And if they are not equivalent how to reconcile them?

(2) I need help understanding (b). Why are there ##F## and ##f##? Also, how in what space does ##v(F \circ f)## operate since F is a map from ##M## to ##N## and ##v## belongs to ##T_pM## and ##f \in C^∞(N)##?

(3) It would seem that definition (a) is more in line with the idea that a differential is a linear functional on ##T_pM## and hence a covector in a space dual to ##T_pM##. How to understand (2) in this context?

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