Why write vector fields on manifolds as directional differential

>>9059180
>Why write vector fields on manifolds as
>directional differential operators?
To scare away brainlets.

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*blocks your path integral*

We say both. It's very easy to compute things once calculus is involved though. Some identities or theorems are best learned by direct computation, then proving that your computation was independent of coordinate choice, rather than some abstract logic approach.
One of those methods is better (the second one) but one is more helpful as a learning procedure. For example, in my undergrad math/physics class we took the time to exactly calculate things like the curl of a gradient of some function in a series of different coordinates. It was very instructive at the time to see how calculus could be used to prove important identities. But years later, when learning some differential forms and Hodge theory and what-not, I learned how all of that can be written down without using calculus at all. It's more enlightening, but I couldn't have learned it that way from the start.

>>9059180
>but why not say that a vector field is just a map from the manifold to the tangent bundle where the image of every point is an element of the corresponding tangent space
but this IS the definition. vector field = section of the tangent bundle.

>>9059388

why then do all my differential geometry books call a vector field something of the form [math]v^{\mu}\partial_{\mu}[/math]

I thought one of the first things you learnt in theoretical physics is that it doesn't matter how you choose to represent the vector, so long as you get the corresponding operations and identities right.

So pick your favorite definition, learn to convert between them as and when needed, and leave the philosophising to the people with too much free time on their hands.

>>9059444
share the names of said books

>>9059444
That definition of a vector field only works in a coordinate patch (you generally won't have global
coordinates [math]x_\mu[/math] which induce the tangent vectors [math]\partial_\mu=\partial_{ x_\mu}[/math]), but locally it shows how the geometry
of manifolds is like [math]\mathbb{R}^n[/math].

As to why they are written as differential operators: note that under a coordinate change [math]x\mapsto x'[/math],
the operator transforms as
[math]\partial_{x_\mu}\mapsto \sum_\nu
\frac{\partial x_\mu}{\partial x_\nu'}\partial_{x_\nu'}[/math].
So writing them as partial differential operators makes the calculus of the tangent vectors look intuitive.

Further in your course you will learn how to associate a one-form (a section of the co-tangent bundle)
[math]df[/math] to a function [math]f[/math]. You can apply such a one-form to a vector field that locally has the form [math]v^\mu\partial_\mu[/math],
to get a function that has the value
[math]df[v^\mu\partial_\mu] = v^\mu\frac{\partial f}{\partial x_\mu}[/math]
on your chart. So in a way, a vectorfield can be used as a differential operator.