Implicit function theorem

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

Etwas präziser, given a system of m equations fi (x1, ..., xn, y1, ..., ym) = 0, i = 1, ..., m (often abbreviated into F(x, j) = 0), der Satz besagt das, 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] Mit anderen Worten, under a mild condition on the partial derivatives, the set of zeros of a system of equations is locally the graph of a function.

Inhalt 1 Geschichte 2 Erstes Beispiel 3 Definitionen 4 Aussage des Theorems 4.1 Higher derivatives 5 Proof for 2D case 6 The circle example 7 Anwendung: change of coordinates 7.1 Beispiel: polar coordinates 8 Verallgemeinerungen 8.1 Banach space version 8.2 Implicit functions from non-differentiable functions 9 Siehe auch 10 Anmerkungen 11 Verweise 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, j) = 1 of the function f(x, j) = x2 + y2. Around point A, y can be expressed as a function y(x). In this example this function can be written explicitly as {Anzeigestil g_{1}(x)={quadrat {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, j) = x2 + y2, then the equation f(x, j) = 1 cuts out the unit circle as the level set {(x, j) | f(x, j) = 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, nämlich {Anzeigestil pm {quadrat {1-x^{2}}}} .

Jedoch, it is possible to represent part of the circle as the graph of a function of one variable. If we let {Anzeigestil g_{1}(x)={quadrat {1-x^{2}}}} for −1 ≤ x ≤ 1, then the graph of y = g1(x) provides the upper half of the circle. Ähnlich, wenn {Anzeigestil g_{2}(x)=-{quadrat {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, j).

Definitions Let {Anzeigestil f:mathbb {R} ^{n+m}zu mathbb {R} ^{m}} be a continuously differentiable function. We think of {Anzeigestil mathbb {R} ^{n+m}} as the Cartesian product {Anzeigestil mathbb {R} ^{n}mal mathbb {R} ^{m},} and we write a point of this product as {Anzeigestil (mathbf {x} ,mathbf {j} )=(x_{1},Punkte ,x_{n},y_{1},ldots y_{m}).} Starting from the given function {Anzeigestil f} , our goal is to construct a function {Anzeigestil g:mathbb {R} ^{n}zu mathbb {R} ^{m}} whose graph {Anzeigestil ({textbf {x}},g({textbf {x}}))} is precisely the set of all {Anzeigestil ({textbf {x}},{textbf {j}})} so dass {Anzeigestil f({textbf {x}},{textbf {j}})={textbf {0}}} .

As noted above, this may not always be possible. We will therefore fix a point {Anzeigestil ({textbf {a}},{textbf {b}})=(a_{1},Punkte ,a_{n},b_{1},Punkte ,b_{m})} which satisfies {Anzeigestil f({textbf {a}},{textbf {b}})={textbf {0}}} , and we will ask for a {Anzeigestil g} that works near the point {Anzeigestil ({textbf {a}},{textbf {b}})} . Mit anderen Worten, we want an open set {displaystyle Usubset mathbb {R} ^{n}} enthält {Anzeigestil {textbf {a}}} , an open set {displaystyle Vsubset mathbb {R} ^{m}} enthält {Anzeigestil {textbf {b}}} , and a function {Anzeigestil g:Uto V} such that the graph of {Anzeigestil g} satisfies the relation {Anzeigestil f={textbf {0}}} an {displaystyle Utimes V} , and that no other points within {displaystyle Utimes V} do so. In Symbolen, {Anzeigestil {(mathbf {x} ,g(mathbf {x} ))mid mathbf {x} in U}={(mathbf {x} ,mathbf {j} )in Utimes Vmid f(mathbf {x} ,mathbf {j} )=mathbf {0} }.} To state the implicit function theorem, we need the Jacobian matrix of {Anzeigestil f} , which is the matrix of the partial derivatives of {Anzeigestil f} . Abbreviating {Anzeigestil (a_{1},Punkte ,a_{n},b_{1},Punkte ,b_{m})} zu {Anzeigestil ({textbf {a}},{textbf {b}})} , the Jacobian matrix is {Anzeigestil (Df)(mathbf {a} ,mathbf {b} )=links[{Start{Matrix}{frac {teilweise f_{1}}{teilweise x_{1}}}(mathbf {a} ,mathbf {b} )&cdots &{frac {teilweise f_{1}}{teilweise x_{n}}}(mathbf {a} ,mathbf {b} )\vdots &ddots &vdots \{frac {teilweise f_{m}}{teilweise x_{1}}}(mathbf {a} ,mathbf {b} )&cdots &{frac {teilweise f_{m}}{teilweise x_{n}}}(mathbf {a} ,mathbf {b} )Ende{Matrix}}Rechts|links.{Start{Matrix}{frac {teilweise f_{1}}{partial y_{1}}}(mathbf {a} ,mathbf {b} )&cdots &{frac {teilweise f_{1}}{partial y_{m}}}(mathbf {a} ,mathbf {b} )\vdots &ddots &vdots \{frac {teilweise f_{m}}{partial y_{1}}}(mathbf {a} ,mathbf {b} )&cdots &{frac {teilweise f_{m}}{partial y_{m}}}(mathbf {a} ,mathbf {b} )\Ende{Matrix}}Rechts]=[X|Y]} wo {Anzeigestil X} is the matrix of partial derivatives in the variables {Anzeigestil x_{ich}} und {Anzeigestil Y} is the matrix of partial derivatives in the variables {Anzeigestil y_{j}} . The implicit function theorem says that if {Anzeigestil Y} is an invertible matrix, dann gibt es {Anzeigestil U} , {Anzeigestil V} , und {Anzeigestil g} wie gewünscht. Writing all the hypotheses together gives the following statement.

Aussage des Theorems Let {Anzeigestil f:mathbb {R} ^{n+m}zu mathbb {R} ^{m}} be a continuously differentiable function, und lass {Anzeigestil mathbb {R} ^{n+m}} have coordinates {Anzeigestil ({textbf {x}},{textbf {j}})} . Fixieren Sie einen Punkt {Anzeigestil ({textbf {a}},{textbf {b}})=(a_{1},Punkte ,a_{n},b_{1},Punkte ,b_{m})} mit {Anzeigestil f({textbf {a}},{textbf {b}})=mathbf {0} } , wo {Anzeigestil mathbf {0} in 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): {Anzeigestil J_{f,mathbf {j} }(mathbf {a} ,mathbf {b} )=links[{frac {teilweise f_{ich}}{partial y_{j}}}(mathbf {a} ,mathbf {b} )Rechts]} is invertible, then there exists an open set {displaystyle Usubset mathbb {R} ^{n}} enthält {Anzeigestil {textbf {a}}} such that there exists a unique continuously differentiable function {Anzeigestil g:Uto mathbb {R} ^{m}} so dass {Anzeigestil g(mathbf {a} )=mathbf {b} } , und {Anzeigestil f(mathbf {x} ,g(mathbf {x} ))=mathbf {0} ~{Text{für alle}}~mathbf {x} in U} . Darüber hinaus, denoting the left-hand panel of the Jacobian matrix shown in the previous section as: {Anzeigestil J_{f,mathbf {x} }(mathbf {a} ,mathbf {b} )=links[{frac {teilweise f_{ich}}{teilweise x_{j}}}(mathbf {a} ,mathbf {b} )Rechts],} the Jacobian matrix of partial derivatives of {Anzeigestil g} in {Anzeigestil U} are given by the matrix product:[3] {Anzeigestil links[{frac {partial g_{ich}}{teilweise x_{j}}}(mathbf {x} )Rechts]_{mtimes n}=-left[J_{f,mathbf {j} }(mathbf {x} ,g(mathbf {x} ))Rechts]_{mtimes m}^{-1},links[J_{f,mathbf {x} }(mathbf {x} ,g(mathbf {x} ))Rechts]_{mtimes n}} Higher derivatives If, Außerdem, {Anzeigestil f} is analytic or continuously differentiable {Anzeigestil k} times in a neighborhood of {Anzeigestil ({textbf {a}},{textbf {b}})} , then one may choose {Anzeigestil U} in order that the same holds true for {Anzeigestil g} Innerhalb {Anzeigestil U} . [4] In the analytic case, this is called the analytic implicit function theorem.

Proof for 2D case Suppose {Anzeigestil F:mathbb {R} ^{2}zu mathbb {R} } is a continuously differentiable function defining a curve {Anzeigestil F(mathbf {r} )=F(x,j)=0} . Lassen {Anzeigestil (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 {teilweise f}{teilweise y}}Rechts|_{(x_{0},y_{0})}neq 0} then for the curve around {Anzeigestil (x_{0},y_{0})} wir können schreiben {displaystyle y=f(x)} , wo {Anzeigestil f} is a real function.

Nachweisen. Since F is differentiable we write the differential of F through partial derivatives: {Anzeigestil mathrm {d} F=operatorname {grad} Fcdot mathrm {d} mathbf {r} ={frac {teilweise f}{teilweise x}}Mathrm {d} x+{frac {teilweise f}{teilweise y}}Mathrm {d} j.} Since we are restricted to movement on the curve {Anzeigestil mathrm {d} F=0} and by assumption {Anzeigestil {tfrac {teilweise f}{teilweise y}}neq 0} around the point {Anzeigestil (x_{0},y_{0})} (seit {Anzeigestil {tfrac {teilweise f}{teilweise y}}} is continuous at {Anzeigestil (x_{0},y_{0})} und {displaystyle left.{tfrac {teilweise f}{teilweise y}}Rechts|_{(x_{0},y_{0})}neq 0} ). Therefore we have a first-order ordinary differential equation: {Anzeigestil teilweise _{x}Fmathrm {d} x+partial _{j}Fmathrm {d} 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 {Anzeigestil (x_{0},y_{0})} wofür, at every point in it, {Anzeigestil teilweise _{j}Fneq 0} . Since F is continuously differentiable and from the assumption we have {Anzeigestil |teilweise _{x}F|

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