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No. 1223:

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Transcribed painstakingly from the charred remains of the original notes by: unstattedCommoner

The author writes:

One of the dangers in regarding a two-dimensional vector field as a function from the plane to itself is that one is liable to forget that for an arbitrary two-dimensional smooth manifold one does not necessarily have at every point a natural diffeomorphism between the tangent space of that point and the manifold. (As, for example, in the case of the product of the half-line and the circle.)