Implicit function theorem

Implicit function theorem In mathematics, more specifically in multivariable calculus, the implicit function theorem[un] 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.

Plus précisément, given a system of m equations fi (x1, ..., xn, y1, ..., ym) = 0, i = 1, ..., m (often abbreviated into F(X, y) = 0), le théorème dit que, 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.[1] Autrement dit, under a mild condition on the partial derivatives, the set of zeros of a system of equations is locally the graph of a function.

Contenu 1 Histoire 2 Premier exemple 3 Définitions 4 Énoncé du théorème 4.1 Higher derivatives 5 Proof for 2D case 6 The circle example 7 Application: change of coordinates 7.1 Exemple: polar coordinates 8 Généralisations 8.1 Banach space version 8.2 Implicit functions from non-differentiable functions 9 Voir également 10 Remarques 11 Références 12 Further reading History 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.[2] First example The unit circle can be specified as the level curve f(X, y) = 1 of the function f(X, y) = x2 + y2. Around point A, y can be expressed as a function y(X). In this example this function can be written explicitly as {style d'affichage g_{1}(X)={sqrt {1-x^{2}}};} in many cases no such explicit expression exists, but one can still refer to the implicit function y(X). No such function exists around point B.

If we define the function f(X, y) = x2 + y2, 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, à savoir {displaystyle pm {sqrt {1-x^{2}}}} .

Cependant, it is possible to represent part of the circle as the graph of a function of one variable. If we let {style d'affichage g_{1}(X)={sqrt {1-x^{2}}}} for −1 ≤ x ≤ 1, then the graph of y = g1(X) provides the upper half of the circle. De la même manière, si {style d'affichage g_{2}(X)=-{sqrt {1-x^{2}}}} , then the graph of y = g2(X) gives the lower half of the circle.

The purpose of the implicit function theorem is to tell us the existence of functions like g1(X) and g2(X), even in situations where we cannot write down explicit formulas. It guarantees that g1(X) and g2(X) are differentiable, and it even works in situations where we do not have a formula for f(X, y).

Definitions Let {style d'affichage f:mathbb {R} ^{n+m}à mathbb {R} ^{m}} be a continuously differentiable function. We think of {style d'affichage mathbb {R} ^{n+m}} as the Cartesian product {style d'affichage mathbb {R} ^{n}fois mathbb {R} ^{m},} and we write a point of this product as {style d'affichage (mathbf {X} ,mathbf {y} )=(X_{1},ldots ,X_{n},y_{1},ldots y_{m}).} Starting from the given function {style d'affichage f} , our goal is to construct a function {style d'affichage g:mathbb {R} ^{n}à mathbb {R} ^{m}} whose graph {style d'affichage ({textbf {X}},g({textbf {X}}))} is precisely the set of all {style d'affichage ({textbf {X}},{textbf {y}})} tel que {style d'affichage f({textbf {X}},{textbf {y}})={textbf {0}}} .

As noted above, this may not always be possible. We will therefore fix a point {style d'affichage ({textbf {un}},{textbf {b}})=(un_{1},des points ,un_{n},b_{1},des points ,b_{m})} which satisfies {style d'affichage f({textbf {un}},{textbf {b}})={textbf {0}}} , and we will ask for a {style d'affichage g} that works near the point {style d'affichage ({textbf {un}},{textbf {b}})} . Autrement dit, we want an open set {displaystyle Usubset mathbb {R} ^{n}} contenant {style d'affichage {textbf {un}}} , an open set {displaystyle Vsubset mathbb {R} ^{m}} contenant {style d'affichage {textbf {b}}} , and a function {style d'affichage g:Uto V} such that the graph of {style d'affichage g} satisfies the relation {style d'affichage f={textbf {0}}} sur {displaystyle Utimes V} , and that no other points within {displaystyle Utimes V} do so. En symboles, {style d'affichage {(mathbf {X} ,g(mathbf {X} ))mid mathbf {X} in U}={(mathbf {X} ,mathbf {y} )in Utimes Vmid f(mathbf {X} ,mathbf {y} )= mathbf {0} }.} To state the implicit function theorem, we need the Jacobian matrix of {style d'affichage f} , which is the matrix of the partial derivatives of {style d'affichage f} . Abbreviating {style d'affichage (un_{1},des points ,un_{n},b_{1},des points ,b_{m})} à {style d'affichage ({textbf {un}},{textbf {b}})} , the Jacobian matrix is {style d'affichage (Df)(mathbf {un} ,mathbf {b} )=gauche[{commencer{matrice}{frac {partial f_{1}}{partiel x_{1}}}(mathbf {un} ,mathbf {b} )&cdots &{frac {partial f_{1}}{partiel x_{n}}}(mathbf {un} ,mathbf {b} )\vdots &ddots &vdots \{frac {partial f_{m}}{partiel x_{1}}}(mathbf {un} ,mathbf {b} )&cdots &{frac {partial f_{m}}{partiel x_{n}}}(mathbf {un} ,mathbf {b} )fin{matrice}}droit|la gauche.{commencer{matrice}{frac {partial f_{1}}{partial y_{1}}}(mathbf {un} ,mathbf {b} )&cdots &{frac {partial f_{1}}{partial y_{m}}}(mathbf {un} ,mathbf {b} )\vdots &ddots &vdots \{frac {partial f_{m}}{partial y_{1}}}(mathbf {un} ,mathbf {b} )&cdots &{frac {partial f_{m}}{partial y_{m}}}(mathbf {un} ,mathbf {b} )\fin{matrice}}droit]=[X|Oui]} où {style d'affichage X} is the matrix of partial derivatives in the variables {style d'affichage x_{je}} et {style d'affichage Y} is the matrix of partial derivatives in the variables {style d'affichage y_{j}} . The implicit function theorem says that if {style d'affichage Y} is an invertible matrix, alors il y a {style d'affichage U} , {style d'affichage V} , et {style d'affichage g} comme voulu. Writing all the hypotheses together gives the following statement.

Énoncé du théorème Soit {style d'affichage f:mathbb {R} ^{n+m}à mathbb {R} ^{m}} be a continuously differentiable function, et laissez {style d'affichage mathbb {R} ^{n+m}} have coordinates {style d'affichage ({textbf {X}},{textbf {y}})} . Fixer un point {style d'affichage ({textbf {un}},{textbf {b}})=(un_{1},des points ,un_{n},b_{1},des points ,b_{m})} avec {style d'affichage f({textbf {un}},{textbf {b}})= mathbf {0} } , où {style d'affichage mathbf {0} en mathbb {R} ^{m}} is the zero vector. If the Jacobian matrix (this is the right-hand panel of the Jacobian matrix shown in the previous section): {displaystyle J_{F,mathbf {y} }(mathbf {un} ,mathbf {b} )=gauche[{frac {partial f_{je}}{partial y_{j}}}(mathbf {un} ,mathbf {b} )droit]} is invertible, then there exists an open set {displaystyle Usubset mathbb {R} ^{n}} contenant {style d'affichage {textbf {un}}} such that there exists a unique continuously differentiable function {style d'affichage g:Uto mathbb {R} ^{m}} tel que {style d'affichage g(mathbf {un} )= mathbf {b} } , et {style d'affichage f(mathbf {X} ,g(mathbf {X} ))= mathbf {0} ~{texte{pour tous}}~mathbf {X} in U} . En outre, denoting the left-hand panel of the Jacobian matrix shown in the previous section as: {displaystyle J_{F,mathbf {X} }(mathbf {un} ,mathbf {b} )=gauche[{frac {partial f_{je}}{partiel x_{j}}}(mathbf {un} ,mathbf {b} )droit],} the Jacobian matrix of partial derivatives of {style d'affichage g} dans {style d'affichage U} are given by the matrix product:[3] {style d'affichage à gauche[{frac {partial g_{je}}{partiel x_{j}}}(mathbf {X} )droit]_{mtimes n}=-left[J_{F,mathbf {y} }(mathbf {X} ,g(mathbf {X} ))droit]_{mtimes m}^{-1},la gauche[J_{F,mathbf {X} }(mathbf {X} ,g(mathbf {X} ))droit]_{mtimes n}} Higher derivatives If, moreover, {style d'affichage f} is analytic or continuously differentiable {style d'affichage k} times in a neighborhood of {style d'affichage ({textbf {un}},{textbf {b}})} , then one may choose {style d'affichage U} in order that the same holds true for {style d'affichage g} à l'intérieur {style d'affichage U} . [4] In the analytic case, this is called the analytic implicit function theorem.

Proof for 2D case Suppose {style d'affichage F:mathbb {R} ^{2}à mathbb {R} } is a continuously differentiable function defining a curve {style d'affichage F(mathbf {r} )=F(X,y)=0} . Laisser {style d'affichage (X_{0},y_{0})} be a point on the curve. The statement of the theorem above can be rewritten for this simple case as follows: Theorem — If {displaystyle left.{frac {Fa partiel}{y partiel}}droit|_{(X_{0},y_{0})}neq 0} then for the curve around {style d'affichage (X_{0},y_{0})} nous pouvons écrire {displaystyle y=f(X)} , où {style d'affichage f} is a real function.

Preuve. Since F is differentiable we write the differential of F through partial derivatives: {style d'affichage mathrm {ré} F=operatorname {grad} Fcdot mathrm {ré} mathbf {r} ={frac {Fa partiel}{partiel x}}mathrm {ré} x+{frac {Fa partiel}{y partiel}}mathrm {ré} y.} Since we are restricted to movement on the curve {style d'affichage mathrm {ré} F=0} and by assumption {style d'affichage {tfrac {Fa partiel}{y partiel}}neq 0} around the point {style d'affichage (X_{0},y_{0})} (puisque {style d'affichage {tfrac {Fa partiel}{y partiel}}} is continuous at {style d'affichage (X_{0},y_{0})} et {displaystyle left.{tfrac {Fa partiel}{y partiel}}droit|_{(X_{0},y_{0})}neq 0} ). Therefore we have a first-order ordinary differential equation: {style d'affichage partiel _{X}Fmathrm {ré} x+partial _{y}Fmathrm {ré} y=0,quad y(X_{0})=y_{0}} Now we are looking for a solution to this ODE in an open interval around the point {style d'affichage (X_{0},y_{0})} Pour qui, at every point in it, {style d'affichage partiel _{y}Fneq 0} . Since F is continuously differentiable and from the assumption we have {style d'affichage |partiel _{X}F|

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