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I'm trying to get a hold of tensor analysis on manifolds and the idea of vectors and tangent spaces are just starting to click, but I don't really get how the differential of a function can be viewed as a dual or covarient(is there any explicit difference or is it just a matter of context and convention between covarient vectors and dual vectors) vectors.

I'm coming from physics, so I have a lot of holes in my formal mathematical knowledge, please feel free to be somewhat pedantic to help me refine my language when talking about this stuff. So I am intentionally stretching my understanding so that it can be corrected. It boils down to this:

  1. In GR, we want to be able to talk about vector fields on manifolds to describe, for instance, the electric and magnetic fields in space. But the standard definition of vectors in flat space comes in terms of a coordinate system.

    • First, is this a reasonable statement, and why is this a big problem? Why can't we just define vectors in terms of their components in coordinate system provided by some chart from the manifold to $R^n$ and then use our transition maps to see how that vector would be represented in other coordinate systems. What am I missing, why go through all this trouble and say we must talk about vectors in terms of directional derivatives?
  2. For whatever reason, we need to find some way to parameterize the tangent space of some point on a manifold. We do this by considering the fact that any curve on the manifold, $M$, is a map from $R \to M$. Then we can say that the set of all derivative of an arbitrary field at a point $P$ on the manifold are a basis for the tangent space at $P$, and we can label these derivatives by their direction in terms of which curve, out of the set of all (smooth?) curves through $P$ they follow.

    • This is where I start to feel like I have no idea what I'm talking about. First off, don't curves have two directions we could follow. What is the space of all curves on the manifold? The space of all $C^{\infty}$ maps from $R \to M\to R^n$? Doesn't this bring in the coordinate charts once again if I am looking for smoothness of the curve in $M$, do we not care for smooth curves, or should I not be thinking of curves as maps from $R \to M$ and instead think of them as an intangible continuum of points with no other properties?
  3. If we choose to use the directional derivatives in terms of coordinates as our basis for $T_p(M)$, then dual vectors are differentials of functions.

    • How on earth do differentials act on partial derivative operators? Maybe I'm just letting the notation dictate my thinking too much, but $df \frac{\partial}{\partial x^u} = \frac{\partial f}{\partial x^u}$ bothers me so much. In all the other math I've learned and hopefully understand the idea of taking the derivative of a differential is ill defined at best, but now we have this whole new can of worms, I don't know I just think I need some more convincing that this is a natural or at least useful definition. Are n-forms meant to generalize some other notion that I'm familiar with from more elementary mathematics? I'm particularly troubled because it seems in some sense we are saying that you can take the partial derivative of the gradient of a function, and that's the same as just the dot product of the gradient some vector (in terms of more classical vector constructions)

Okay, I think that's long enough, thanks for the help in advance.

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    $\begingroup$ A brief answer to (1) and (2): there are lots of ways to define tangent vectors; representing them as directional derivative operators are just one way to do so. Representing them as a curve that pass through a point in the correct direction and rate is another. A gadget that produces a coordinate vector when you select a coordinate chart is a third. $\endgroup$ – Hurkyl Feb 24 '14 at 0:29
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    $\begingroup$ And for (3): yes, they do generalize something familiar: integrands. "Differential 1-form" roughly means the same thing as "integrand of path integrals". "Differential 2-form" roughly means "integrand of surface integrals" and so forth. The point of the construction mentioned is that you can represent a differential $1$-form as a cotangent vector field, which gives you a way to work with a differential $1$-form in terms of its values at individual points. $\endgroup$ – Hurkyl Feb 24 '14 at 0:34
  • $\begingroup$ In terms of physics terminology, a tangent vector is like an "infinitesimal displacement", and a $df$ represents how much $f$ varies in different directions. When you put the two together, you get a number. Since you've already defined tangent vector, it's easiest to use this product to let you define cotangent vector space as the dual of the tangent vector space. In other fields (e.g. algebraic geometry) it's actually the other way around: it's easiest to define the cotangent vector space, and then you define the tangent vector space as the dual vectors. $\endgroup$ – Hurkyl Feb 24 '14 at 0:37
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GR can be a hard place to start for someone beginning their journey into differential geometry. Some concepts are easier to imagine in terms of manifolds embedded in ambient spaces, but GR is all very intrinsic--you're just plopping a metric on top of a space(time).

To be honest, I'm not sure what you're saying with (1). Stuff about directional derivatives is very common, in the sense that directional derivatives obey all the algebraic properties we would want of vectors, but I think it's an unnecessary level of abstraction. You will inevitably read some literature that rigidly adheres to this correspondence, saying (for instance) that the action of a vector field on a scalar field is the directional derivative and so on. Outside of such marginal cases, though, there's no real issue with thinking about vectors and vector fields the same way you're used to from vector calculus in 3d or from electrodynamics.

For (2), again this is something that's more easily imagined in terms of embedded manifolds. Consider a surface embedded in 3d space. That surface could be parameterized by 2 coordinate functions $u^1, u^2$. If $p(u^1, u^2)$ is a position vector on the manifold, then $\partial p/\partial u^1$ and $\partial p/\partial u^2$ are vector fields on the manifold. Holding $u^2$ constant and allowing $u^1$ to vary generates a curve on the manifold, whose tangent vector is $\partial p/\partial u^1$. These two vector fields span a 2d vector space, the tangent space at that particular point.

Again, the reason I say that GR is a poor first example to think about is that you don't usually think of spacetime as embedded in anything, so it's not immediately obvious what the significance of the tangent space is there, except to say that any tangent vector must necessarily lie in it.

For (3), this is a time where it does help to get more abstract. All a differential geometer would need to imagine is that there exists some map that takes in vectors (whether they're viewed as partial derivative operators or something else) and spits out scalars. This is one case where you can neatly duck the problem because in GR, you have a metric, and the distinction between tangent vectors and cotangent vectors is rather arbitrary, since you can always use the metric to convert from one to another.

Still, you're going to hear about this because cotangent vectors arise from any notion of differentiation with respect to position. Differential forms people will call this $d$, the exterior derivative, and there is a whole algebra of differential forms generated by $d$ and successively higher dual-vector fields. Any $n$-form-field gets taken to an $n+1$-form-field thanks to $d$. Just as a scalar field gets taken to a 1-form field when you hit it with a gradient--this is one of the simplest guises of $d$ to consider. Again, in GR as well as ordinary vector calculus, the distinction between 1-form fields and vector fields isn't always maintained in the strictest sense because the metric allows you to freely convert from one to the other.

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