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As far as I know, the tangent space of a manifold at a point is often defined abstractly as follows. First, we define the germ of smooth real valued functions at a point a, which is the set of equivalence classes of smooth functions where two functions are considered equivalent if they are the same on some neighborhood of a. Pointwise addition, multiplication and scalar multiplication makes this into an algebra.

Then we define a derivation on this algebra. This is where I'm a little unclear. This is something like a linear functional on the algebra which satisfies:

[tex] \partial(fg) = \partial(f) g(a) + f(a) \partial(g) [/tex]

My question is, in the more general abstract setting, what are f(a) and g(a)? It seems like you need another linear functional e given by evaluation at a, and then the above condition is:

[tex] \partial(fg) = \partial(f) e(g) + e(f) \partial(g) [/tex]

Is this the correct general definition of a derivation, so that you must also specify a linear functional on the algebra? According to wikipedia, a derivation on an algebra satisfies:

[tex] \partial(fg) = \partial(f) g + f \partial(g) [/tex]

So that it takes values in the algebra. This gets around needing a linear functional, but it seems like this space of derivations would be much bigger.

Then we define a derivation on this algebra. This is where I'm a little unclear. This is something like a linear functional on the algebra which satisfies:

[tex] \partial(fg) = \partial(f) g(a) + f(a) \partial(g) [/tex]

My question is, in the more general abstract setting, what are f(a) and g(a)? It seems like you need another linear functional e given by evaluation at a, and then the above condition is:

[tex] \partial(fg) = \partial(f) e(g) + e(f) \partial(g) [/tex]

Is this the correct general definition of a derivation, so that you must also specify a linear functional on the algebra? According to wikipedia, a derivation on an algebra satisfies:

[tex] \partial(fg) = \partial(f) g + f \partial(g) [/tex]

So that it takes values in the algebra. This gets around needing a linear functional, but it seems like this space of derivations would be much bigger.

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