Implicit function theorem
Encyclopedia
In multivariable calculus
, the implicit function theorem is a tool which allows relations to be converted to function
s. 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
(an implicit function
) satisfies some mild conditions on its partial derivative
s, then one can in principle solve this equation for
, at least over some small interval
. Geometrically, the locus
defined by
will overlap locally
with the graph of a function
(an explicit function, see article on implicit function
s).
If we define the function 
, then the equation 
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 
because for each choice of 
there are two choices of 
, namely 
.
However, it is possible to represent part of the circle as the graph of a function of one variable. If we let
for 
, 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 
.
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, an open set V of Rm, and a function g : U → V such that the graph of g satisfies the relation f = 0 on U × V. In symbols,
To state the implicit function theorem, we need the Jacobian, also called the differential or total derivative, of
. This is the matrix of partial derivative
s of
. Abbreviating (a1, ..., an, b1, ..., bm) to (a, b), the Jacobian matrix is
where
is the matrix of partial derivatives in the 
's and 
is the matrix of partial derivatives in the 
's. The implicit function theorem says that if 
is an invertible matrix, then there are 
, 
, and 
as desired. Writing all the hypotheses together gives the following statement.
.
Similarly, if f is analytic
inside U×V, then the same holds true for the explicit function g inside U. This generalization is called the analytic implicit function theorem.
. In this case
and 
. The matrix of partial derivatives is just a 1×2 matrix, given by
Thus, here,
is just a number; the linear map defined by it is invertible iff
. By the implicit function theorem we see that we can write the circle in the form 
for all points where 
. For 
and 
we run into trouble, as noted before.
. We can introduce a new coordinate system 
by supplying m functions 
. These functions allow to calculate the new coordinates 
of a point, given the point's old coordinates 
using 
. One might want to verify if the opposite is possible: given coordinates 
, can we 'go back' and calculate the same point's original coordinates 
? The implicit function theorem will provide an answer to this question. The (new and old) coordinates 
are related by 
, with
Now the Jacobian matrix of f at a certain point
[ where 
] is given by
where
denotes the 
identity matrix
, and J is the
matrix of partial derivatives, evaluated at 
. (In the above, these blocks were denoted by X and Y. As it happens, in this particular application of the theorem, neither matrix depends on 
.) The implicit function theorem now states that we can locally express 
as a function of 
if J is invertible. Demanding J is invertible is equivalent to 
, thus we see that we can go back from the primed to the unprimed coordinates if the determinant of the Jacobian J is non-zero. This statement is also known as the inverse function theorem
.
. We can go to a new coordinate system (cartesian coordinates) by defining functions 
and
. This makes it possible given any point 
to find corresponding cartesian coordinates 
. When can we go back and convert cartesian into polar coordinates? By the previous example, we need 
, with
Since
, the conversion back to polar coordinates is only possible if 
. This is a consequence of the fact that at the origin, polar coordinates don't exist: the value of 
is not well-defined.
in Banach space
s, it is possible to extend the implicit function theorem to Banach space valued mappings.
Let
, 
, 
be Banach space
s. Let the mapping
be continuously Fréchet differentiable. If 
, 
, and 
is a Banach space isomorphism from 
onto 
, then there exist neighbourhoods 
of 
and 
of 
and a Frechet differentiable function 
such that 
and 
if and only if 
, for all 
.
is not differentiable. It is standard that it holds in one dimension. The following more general form was proven by Kumagai based on an observation by Jittorntrum.
Consider a continuous function
such that 
. If there exist open neighbourhoods 
and 
of 
and 
, respectively, such that, for all 
, 
is locally one-to-one then there exist open neighbourhoods 
and 
of 
and 
,
such that, for all
, the equation
has a unique solution
,
where
is a continuous function from 
into 
.
Multivariable calculus
Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable: the differentiated and integrated functions involve multiple variables, rather than just one....
, the implicit function theorem is a tool which allows relations to be converted to function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s. It does this by representing the relation as the graph of a function
Graph of a function
In mathematics, the graph of a function f is the collection of all ordered pairs . In particular, if x is a real number, graph means the graphical representation of this collection, in the form of a curve on a Cartesian plane, together with Cartesian axes, etc. Graphing on a Cartesian plane is...
. 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
Domain (mathematics)
In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...
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
(an implicit functionImplicit function
The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...
) satisfies some mild conditions on its partial derivative
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
s, then one can in principle solve this equation for
, at least over some small intervalInterval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
. Geometrically, the locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
defined by
will overlap locallyLocal property
In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points.-Properties of a single space:...
with the graph of a function
(an explicit function, see article on implicit functionImplicit function
The implicit function theorem provides a link between implicit and explicit functions. It states that if the equation R = 0 satisfies some mild conditions on its partial derivatives, then one can in principle solve this equation for y, at least over some small interval...
s).
First example
, then the equation cuts out the unit circleUnit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
as the level set
Level set
In mathematics, a level set of a real-valued function f of n variables is a set of the formthat is, a set where the function takes on a given constant value c....
. There is no way to represent the unit circle as the graph of a function of one variable because for each choice of there are two choices of , namely .However, it is possible to represent part of the circle as the graph of a function of one variable. If we let
for , 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 .Statement of the theorem
Let f : Rn+m → Rm be a continuously differentiable function. We think of Rn+m as the Cartesian productCartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
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
Open set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
U of Rn, an open set V of Rm, and a function g : U → V such that the graph of g satisfies the relation f = 0 on U × V. In symbols,
To state the implicit function theorem, we need the Jacobian, also called the differential or total derivative, of
. This is the matrix of partial derivativePartial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
s of
. Abbreviating (a1, ..., an, b1, ..., bm) to (a, b), the Jacobian matrix iswhere
is the matrix of partial derivatives in the 's and is the matrix of partial derivatives in the 's. The implicit function theorem says that if is an invertible matrix, then there are , , and as desired. Writing all the hypotheses together gives the following statement.- Let f : Rn+m → Rm be a continuously differentiable function, and let Rn+m have coordinates (x, y). Fix a point (a1,...,an,b1,...,bm) = (a,b) with f(a,b)=c, where c∈ Rm. If the matrix [(∂fi/∂yj)(a,b)] is invertible, then there exists an open set U containing a, an open set V containing b, and a unique continuously differentiable function g:U → V such that
Regularity
It can be proven that whenever we have the additional hypothesis that f is continuously differentiable up to k times inside U×V, then the same holds true for the explicit function g inside U and.Similarly, if f is analytic
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
inside U×V, then the same holds true for the explicit function g inside U. This generalization is called the analytic implicit function theorem.
The circle example
Let us go back to the example of the unit circleUnit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
. In this case
and . The matrix of partial derivatives is just a 1×2 matrix, given byThus, here,
is just a number; the linear map defined by it is invertible iffIFF
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...
. By the implicit function theorem we see that we can write the circle in the form for all points where . For and we run into trouble, as noted before.Application: change of coordinates
Suppose we have an m-dimensional space, parametrised by a set of coordinates. We can introduce a new coordinate system by supplying m functions . These functions allow to calculate the new coordinates of a point, given the point's old coordinates using . One might want to verify if the opposite is possible: given coordinates , can we 'go back' and calculate the same point's original coordinates ? The implicit function theorem will provide an answer to this question. The (new and old) coordinates are related by , withNow the Jacobian matrix of f at a certain point
[ where ] is given bywhere
denotes the identity matrixIdentity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
, and J is the
matrix of partial derivatives, evaluated at . (In the above, these blocks were denoted by X and Y. As it happens, in this particular application of the theorem, neither matrix depends on .) The implicit function theorem now states that we can locally express as a function of if J is invertible. Demanding J is invertible is equivalent to , thus we see that we can go back from the primed to the unprimed coordinates if the determinant of the Jacobian J is non-zero. This statement is also known as the inverse function theoremInverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
.
Example: polar coordinates
As a simple application of the above, consider the plane, parametrised by polar coordinates. We can go to a new coordinate system (cartesian coordinates) by defining functions and. This makes it possible given any point to find corresponding cartesian coordinates . When can we go back and convert cartesian into polar coordinates? By the previous example, we need , withSince
, the conversion back to polar coordinates is only possible if . This is a consequence of the fact that at the origin, polar coordinates don't exist: the value of is not well-defined.
Banach space version
Based on the inverse function theoremInverse function theorem
In mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
in Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s, it is possible to extend the implicit function theorem to Banach space valued mappings.
Let
, , be Banach spaceBanach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s. Let the mapping
be continuously Fréchet differentiable. If , , and is a Banach space isomorphism from onto , then there exist neighbourhoods of and of and a Frechet differentiable function such that and if and only if , for all .Implicit functions from non-differentiable functions
Various forms of the implicit function theorem exist for the case when the function is not differentiable. It is standard that it holds in one dimension. The following more general form was proven by Kumagai based on an observation by Jittorntrum.Consider a continuous function
such that . If there exist open neighbourhoods and of and , respectively, such that, for all , is locally one-to-one then there exist open neighbourhoods and of and ,such that, for all
, the equationhas a unique solution
,where
is a continuous function from into .See also
- Constant rank theorem: Both the implicit function theorem and the Inverse function theoremInverse function theoremIn mathematics, specifically differential calculus, the inverse function theorem gives sufficient conditions for a function to be invertible in a neighborhood of a point in its domain...
can be seen as special cases of the constant rank theorem.