Given
(1)
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(2)
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(3)
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if the determinantof the Jacobian
(4)
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then , , and can be solved for in terms of , , and and partial derivatives of , , with respect to , , and can be found by differentiating implicitly.
More generally, let be an open set in and let be a function. Write in the form , where and are elements of and . Suppose that (, ) is a point in such that and the determinant of the matrix whose elements are the derivatives of the component functions of with respect to the variables, written as , evaluated at , is not equal to zero. The latter may be rewritten as
(5)
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Then there exists a neighborhood of in and a unique function such that and for all .