|
In multivariable calculus, the implicit function theorem, also known, especially in Italy, as Dini's theorem, is a tool that allows relations to be converted to functions of several real variables. It does this by representing the relation as the graph of a function. There may not be a single function whose graph is the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function. The theorem states that if the equation ''F''(''x1'', ..., ''xn'', ''y1'', ..., ''ym'') = ''F''(x, y) = 0 satisfies some mild conditions on its partial derivatives, then one can in principle (though not necessarily with an analytic expression) express the ''m'' variables ''yi'' in terms of the ''n'' variables ''xj'' as ''yi'' = ''fi ''(x), at least in some disk. Then each of these implicit functions ''fi ''(x),〔Chiang, Alpha C. ''Fundamental Methods of Mathematical Economics'', McGraw-Hill, third edition, 1984〕 implied by ''F''(x, y) = 0, is such that geometrically the locus defined by ''F''(x, y) = 0 will coincide locally (that is in that disk) with the hypersurface given by y = f(x). == First example == If we define the function , then the equation ''f''(''x'', ''y'') = 1 cuts out the unit circle as the level set . There is no way to represent the unit circle as the graph of a function of one variable ''y'' = ''g''(''x'') because for each choice of ''x'' ∈ (−1, 1), there are two choices of ''y'', namely . However, it is possible to represent part of the circle as the graph of a function of one variable. If we let for −1 < ''x'' < 1, then the graph of provides the upper half of the circle. Similarly, if , then the graph of gives the lower half of the circle. The purpose of the implicit function theorem is to tell us the existence of functions like and , even in situations where we cannot write down explicit formulas. It guarantees that and are differentiable, and it even works in situations where we do not have a formula for ''f''(''x'', ''y''). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「implicit function theorem」の詳細全文を読む スポンサード リンク
|