Zero Set
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High Quality Content by WIKIPEDIA articles! In mathematics, the zero set of a real-valued function f : X R (or more generally, a function taking values in some additive group) is the subset f 1(0) of X (the inverse image of 0). In other words, the zero set of the function f is the subset of X on which f(x) = 0. The cozero set of f is the complement of the zero set of f (i.e. the subset of X on which f is nonzero). Zero sets are important in several branches of geometry and topology. In differential geometry, zero sets are frequently used to define manifolds. An important spacial case is the ca...
High Quality Content by WIKIPEDIA articles! In mathematics, the zero set of a real-valued function f : X R (or more generally, a function taking values in some additive group) is the subset f 1(0) of X (the inverse image of 0). In other words, the zero set of the function f is the subset of X on which f(x) = 0. The cozero set of f is the complement of the zero set of f (i.e. the subset of X on which f is nonzero). Zero sets are important in several branches of geometry and topology. In differential geometry, zero sets are frequently used to define manifolds. An important spacial case is the case that f is a smooth function from Rp to Rn. If zero is a regular value of f then the zero-set of f is a smooth manifold of dimension m=p-n by the regular value theorem. For example, the unit m-sphere in Rm+1 is the zero set of the real-valued function f(x) = x 2 - 1.
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