How do we interpret the differential of a function on a smooth manifold?

[panzervlad]

I have done a diagram of pushforwards/pullbacks and I am stuck as to how to graphically interpret the differential of a function.

Let M be a smooth manifold, and let p be a point in M.

Do they lie in the dual tangent space of a point p, like any other covector? (makes sense since its coordinates are expressed with the dual basis dx, but at the same time it does not because it carries with it the partial derivative symbol (coord. of tangent space to p), its 'action' on a function f (since its a linear map)

As shown here,
http://upload.wikimedia.org/math/9/0/5/905edada8d9cc7f0f6982b7ec8583844.png

So there seems to be some connection with the derivations at a point in p(denoted X) which lie in the tangent space of p, yet differential are covectors.

I can't seem to glue together differentials, dual tangent space of p, and derivations of p, in addition to the fact that it acts on a function f !

I understand the intepretation of differentials in the euclidean 3 space as the gradient of f, I understand its properties, properties under pullbacks, but I just can't figure in where they lie in.

Thanks in advance for any help on the topic!

 

[mathwonk]

the differential of f is df, and its value df(p) at each p linves in Tp* the dual tangent space at p. If ∂/∂xj(p) is a tangent vector at p, the value of df(p) on ∂/∂xj(p), is ∂f/∂xj(p).

a derivation on functions induces a linear map on differentials.

 

[lavinia]

I have done a diagram of pushforwards/pullbacks and I am stuck as to how to graphically interpret the differential of a function.

Let M be a smooth manifold, and let p be a point in M.

Do they lie in the dual tangent space of a point p, like any other covector? (makes sense since its coordinates are expressed with the dual basis dx, but at the same time it does not because it carries with it the partial derivative symbol (coord. of tangent space to p), its 'action' on a function f (since its a linear map)

As shown here,
http://upload.wikimedia.org/math/9/0/5/905edada8d9cc7f0f6982b7ec8583844.png

So there seems to be some connection with the derivations at a point in p(denoted X) which lie in the tangent space of p, yet differential are covectors.

I can't seem to glue together differentials, dual tangent space of p, and derivations of p, in addition to the fact that it acts on a function f !

I understand the intepretation of differentials in the euclidean 3 space as the gradient of f, I understand its properties, properties under pullbacks, but I just can't figure in where they lie in.

Thanks in advance for any help on the topic!

I look at df as a covector because it is a linear map on tangent vectors.

It maps the tangent vector, X, to X.f the derivative of f with respect to X.

In notation, df(x) = X.f.

It is easy to see that df is a linear map.

df(aX + bY) = (aX + bY).f = aX.f + bY.f = adf(X) + bdf(Y)

More generally if f is a map between manifolds then df is a linear map between their tangent spaces.

Derivations are vectors.