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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the rank of a differentiable map f : M N between differentiable manifolds at a point p M is the rank of the derivative of f at p. Recall that the derivative of f at p is a linear map Df_p : 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 Tf(p)N: operatorname{rank}(f)_p = dim(operatorname{im}(Df_p)).A differentiable map f : M N is said to have constant rank if the rank of f is the same for all p in M. Constant rank maps have a number of nice properties and are an important concept in differential topology.