Rank (differential topology)

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In mathematics, the rank of a differentiable map ${\displaystyle f:M\to N}$ between differentiable manifolds at a point ${\displaystyle p\in M}$ is the rank of the derivative of ${\displaystyle f}$ at ${\displaystyle p}$. Recall that the derivative of ${\displaystyle f}$ at ${\displaystyle p}$ is a linear map

${\displaystyle d_{p}f:T_{p}M\to T_{f(p)}N\,}$

from the tangent space at p to the tangent space at f(p). As a linear map between vector spaces it has a well-defined rank, which is just the dimension of the image in T_f(p)N:

${\displaystyle \operatorname {rank} (f)_{p}=\dim(\operatorname {im} (d_{p}f)).}$