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Why do we try to define $dx$ as a linear functional from the tangent space to the real numbers when we use a similar notation for the local coords on the tangent space?

Example, let $M$ be a smooth manifold and $T_pM$ be its tangent space at $p$. Then the linear functional $dx: T_pM \to R$ maps $t \in T_pM$ to $t^i$, $dx^i(t) = t^i$. But when describing the basis vectors on $T_pM$ we write the local coordinates as $\left ( d/dx_i\right)$, so we use $dx_i$ in the bottom too.

Lemon
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  • Shouldn't you really be using $\frac{\partial}{\partial x_i}$? –  Nov 23 '16 at 14:23
  • @Bye_World, sure but that doesn't answer what $\partial x_i$ now. – Lemon Nov 23 '16 at 14:25
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    I'm not seeing the problem. $dx^i$ and $\frac{\partial}{\partial x_i}$ are distinct enough notations that no one is going to confuse one for the other. As to why we use those particular notations, it's just because they're convenient. Look around the site a bit and you'll see some explanations. –  Nov 23 '16 at 14:28
  • @Hawk: To clarify, writing a tangent vector with respect to $d/dx_{i}$ requires you to fix a local coordinate system; this isn't an intrinsic construction on the tangent space of a manifold. Are you asking why we use differential operator notation to denote vector fields in coordinates, and why we use differential notation to denote cotangent vectors...? – Andrew D. Hwang Nov 23 '16 at 14:29
  • @AndrewD.Hwang, yeah. We are essentially using the same thing, but we are just 'hiding' it by cosmetically changing the notations. – Lemon Nov 23 '16 at 14:37
  • Not sure I understand your comment, but the "differential" notations come from classical (pre-20th Century) calculus, and are motivated by the chain rule. Spivak (in both Calculus on Manifolds and A Comprehensive Introduction to Differential Geometry Vol. I) explains this perspective in detail, see also Dual space and covectors: force, work and energy. – Andrew D. Hwang Nov 23 '16 at 14:49

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