Implicit function theorem

In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent 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. In mathematics, more specifically in multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent 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. More precisely, given a system of m equations fi (x1, ..., xn, y1, ..., ym) = 0, i = 1, ..., m (often abbreviated into F(x, y) = 0), the theorem states that, under a mild condition on the partial derivatives (with respect to the yis) at a point, the m variables yi are differentiable functions of the xj in some neighborhood of the point. As these functions can generally not be expressed in closed form, they are implicitly defined by the equations, and this motivated the name of the theorem. In other words, under a mild condition on the partial derivatives, the set of zeros of a system of equations is locally the graph of a function. Augustin-Louis Cauchy (1789–1857) is credited with the first rigorous form of the implicit function theorem. Ulisse Dini (1845–1918) generalized the real-variable version of the implicit function theorem to the context of functions of any number of real variables. If we define the function f ( x , y ) = x 2 + y 2 {displaystyle f(x,y)=x^{2}+y^{2}} , then the equation f(x, y) = 1 cuts out the unit circle as the level set {(x, y)| f(x, y) = 1}. 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 ± 1 − x 2 {displaystyle pm {sqrt {1-x^{2}}}} . However, it is possible to represent part of the circle as the graph of a function of one variable. If we let g 1 ( x ) = 1 − x 2 {displaystyle g_{1}(x)={sqrt {1-x^{2}}}} for −1 ≤ x ≤ 1, then the graph of y = g 1 ( x ) {displaystyle y=g_{1}(x)} provides the upper half of the circle. Similarly, if g 2 ( x ) = − 1 − x 2 {displaystyle g_{2}(x)=-{sqrt {1-x^{2}}}} , then the graph of y = g 2 ( x ) {displaystyle y=g_{2}(x)} gives the lower half of the circle. The purpose of the implicit function theorem is to tell us the existence of functions like g 1 ( x ) {displaystyle g_{1}(x)} and g 2 ( x ) {displaystyle g_{2}(x)} , even in situations where we cannot write down explicit formulas. It guarantees that g 1 ( x ) {displaystyle g_{1}(x)} and g 2 ( x ) {displaystyle g_{2}(x)} are differentiable, and it even works in situations where we do not have a formula for f(x, y). Let f : Rn+m → Rm be a continuously differentiable function. We think of Rn+m as the Cartesian product Rn × Rm, and we write a point of this product as (x, y) = (x1, ..., xn, y1, ..., ym). Starting from the given function f, our goal is to construct a function g: Rn → Rm whose graph (x, g(x)) is precisely the set of all (x, y) such that f(x, y) = 0. As noted above, this may not always be possible. We will therefore fix a point (a, b) = (a1, ..., an, b1, ..., bm) which satisfies f(a, b) = 0, and we will ask for a g that works near the point (a, b). In other words, we want an open set U of Rn containing a, an open set V of Rm containing b, and a function g : U → V such that the graph of g satisfies the relation f = 0 on U × V, and that no other points within U × V do so. In symbols,