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In page Pushforward (differential):
"Any vector field Y on N defines a pullback section φ∗Y of φ∗TN with (φ∗Y)x = Yφ(x). A vector field X on M and a vector field Y on N are said to be φ-related if φ∗X = φ∗Y as vector fields along φ. In other words, for all x in M, dφx(X) = Yφ(x).
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