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There are other commonly used definitions for derivatives in multivariable spaces. For topological vector spaces, the most familiar is the Fréchet derivative, which makes use of a norm. In the case of matrix spaces, there are several matrix norms available, all of which are equivalent since the space is finite-dimensional. For example, it is possible for a map to have all partial derivatives exist at a point, and yet not be continuous in the topology of the space. See for example Hartogs' theorem. The matrix derivative is not a special case of the Fréchet derivative for matrix spaces, but rather a convenient notation for keeping track of many partial derivatives for doing calculations, though in the case that a function is Fréchet differentiable, the two derivatives will agree.