Suppose we have an "m"-dimensional space, parametrised by a set of coordinates formula_47. We can introduce a new coordinate system formula_48 by supplying m functions formula_49 each being continuously differentiable. These functions allow us to calculate the new coordinates formula_48 of a point, given the point's old coordinates formula_47 using formula_52. One might want to verify if the opposite is possible: given coordinates formula_48, can we 'go back' and calculate the same point's original coordinates formula_47? The implicit function theorem will provide an answer to this question. The (new and old) coordinates formula_55 are related by "f" = 0, with