I am a bit confused with the way Jack Lee (and my teacher in class too) defines vector fields. For instance, in Introduction to Smooth Manifolds, page 182, Example 8.17, Lee defines a smooth vector field on $\mathbb{R}^{2}$ by $$Y = x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}.$$ Is this just shorthand for $$\forall (x,y) \in \mathbb{R}^{2} \, \colon \: Y_{(x,y)} = x \frac{\partial}{\partial y}\Big\vert_{(x,y)} - y \frac{\partial}{\partial x}\Big\vert_{(x,y)},$$ or am I missing something, i.e., should the $x$ and $y$ multiplying the vector fields $\partial / \partial y$ and $\partial / \partial x$ be interpreted as functions here?

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    $\begingroup$ yes and yes. This is not the last time you will be annoyed by (necessarily) terse notation in differential geometry :) $\endgroup$ – Tim kinsella Jan 5 '17 at 10:02
  • $\begingroup$ Quick question : why is there a $-$ sign in the definition of the vector field $Y$ ? $\endgroup$ – Odile Jan 5 '17 at 10:20
  • $\begingroup$ @Odile $Y$ is just an example of a smooth vector field on $\mathbb{R}^{2}$ $\endgroup$ – MK7 Jan 5 '17 at 10:25
  • $\begingroup$ @Odile: The $-$ sign is there so that this vector field will be tangent to circles centered at the origin. This is a geometrically pleasing and important vector field. $\endgroup$ – Ted Shifrin Jan 9 '17 at 1:31

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