= Pushforward of a vector field
{title2=$f_*X=df(X)$}
For a <vector field> $X$ on $M$, its pushforward by a <smooth map between manifolds> $f:M\to N$ is the <vector field along a map> defined by $(f_*X)_p=df_p(X_p)$. Acting on a function $h$ on $N$, it satisfies $(f_*X)_p(h)=X_p(h\circ f)$. For a <diffeomorphism> this defines a vector field on $N$ by evaluation at $p=f^{-1}(q)$. For a general map there need not be a single vector at each image point: $f(x)=x^2$, $X=\partial_x$ gives $2x\partial_y$, with opposite values over $y>0$.
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