Proof of the implicit function theorem for a surjective derivative

Builds on Implicit function theorem Inverse function theorem Jacobian matrix Rank of a matrix Linear independence Chain rule in several variables

Proof

We choose $m$ coordinates in which the derivative is invertible, and apply the implicit function theorem.

Let $U$ , $m \le n$ , $F$ and $p$ be as in the statement. Let $J = J_F(p)$ be the $m \times n$ Jacobian matrix, so $DF(p) v = J v$ .
The image of $DF(p)$ is the column space of $J$ . Since $DF(p)$ is surjective, the column space is $\R^m$ , so the rank of $J$ is $m$ .
A spanning set of $\R^m$ contains a basis. So some $m$ columns of $J$ , with indices $i_1 < \dots < i_m$ , are linearly independent. The $m \times m$ matrix $C$ of these columns is invertible.
Case $m = n$ . Then $C = J$ , and $DF(p)$ is invertible. By the inverse function theorem, $F$ is injective on an open neighbourhood $W$ of $p$ . So $\set{x \in W : F(x) = 0} = \set{p}$ . This is the graph of the constant function of the empty list of remaining coordinates.
Case $m < n$ . For $x \in \R^n$ , let $x'' = (x_{i_1}, \dots, x_{i_m}) \in \R^m$ , and let $x' \in \R^{n-m}$ list the other coordinates in their order. Let $P(x) = (x', x'')$ .
The map $P$ is a linear bijection that only reorders coordinates, so it preserves norms. Hence $\Omega = P(U)$ is open, and $G = F \circ P^{-1} \colon \Omega \to \R^m$ is of class $C^1$ .
Write $P(p) = (p', p'')$ . By the chain rule, the partial derivatives of $G$ with respect to the coordinates of $x''$ at $(p', p'')$ are the columns $i_1, \dots, i_m$ of $J$ . So $D_{x''} G(p', p'') = C$ , which is invertible.
Also $G(p', p'') = F(p) = 0$ . By the implicit function theorem, there are open sets $U' \ni p'$ in $\R^{n-m}$ and $V' \ni p''$ in $\R^m$ and a $C^1$ function $\varphi \colon U' \to V'$ with this property. For $(x', x'') \in U' \times V'$ , $G(x', x'') = 0$ if and only if $x'' = \varphi(x')$ .
The function $G$ is evaluated on $U' \times V'$ in this statement, so $U' \times V' \subseteq \Omega$ . Let $W = P^{-1}(U' \times V')$ . It is open, since $P$ is continuous, and $p \in W \subseteq U$ .
Let $x \in W$ . Then $(x', x'') = P(x) \in U' \times V'$ and $F(x) = G(x', x'')$ . So $F(x) = 0$ if and only if $x'' = \varphi(x')$ .
Hence $\set{x \in W : F(x) = 0}$ is the set of $x \in \R^n$ with $x' \in U'$ and $x'' = \varphi(x')$ . This is the graph of the $C^1$ function $\varphi$ of the $n - m$ coordinates in $x'$ .
So the coordinates $x_{i_1}, \dots, x_{i_m}$ and the neighbourhood $W$ have the property in the statement.

Idea

Surjectivity of $DF(p)$ means that the Jacobian matrix has $m$ independent columns. The coordinates of these columns play the role of $y$ in the implicit function theorem, and the remaining coordinates play the role of $x$ . After this renumbering, the statement is the implicit function theorem.

Reading path

To understand this page from the ground up, read these 453 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of Rolle's theorem, Proof of the derivative of a map into R^n by components, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the decomposition of a permutation into transpositions, Proof of the implicit function theorem from the inverse function theorem, Proof that the symmetric group is generated by transpositions, Proof of the Laplace expansion along a column by transposition, Proof that open sets are the sets that are neighbourhoods of all their points, Proof that closed balls are closed, Proof of the diameter characterisation of Cauchy sequences, Proof of the commutativity of addition by induction, Proof of the Lebesgue number lemma by a finite cover of balls, Proof of the epsilon-delta form of continuity, Proof that equivalence classes form a partition, Proof of the triangle inequality for norms, Proof of the sum of a finite geometric series, Proof that a complex number times its conjugate is the square of its modulus, Proof of the Cauchy–Schwarz inequality, Proof of the Heine–Borel theorem by sequential compactness, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule in several variables, Proof that the determinant is multiplicative by expanding the rows, Proof that permuting rows multiplies the determinant by the sign, Proof that closed subsets of complete spaces are complete, Proof of the derivative as a little-o approximation, Proof that disjoint cycles commute, Proof of the equality of complex numbers, Proof that a matrix is invertible iff its determinant is a unit, Proof that a linear map between normed spaces is continuous if and only if it is bounded, Proof of the determinant of a block triangular matrix by the Leibniz formula, Proof that a function is bijective iff it has a two-sided inverse, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of continuity via closed sets, Proof that a matrix with two equal rows has determinant zero, Proof of the double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof that compact subsets of a metric space are bounded, Proof of the Laplace expansion along a row by the Leibniz formula, Proof that compactness and sequential compactness agree in metric spaces, Proof that subsequences of a convergent sequence converge to the same limit, Proof of the uniqueness of limits, Proof that the coordinate map is a linear isomorphism, Proof of strong induction from ordinary induction, Proof that linear maps on finite-dimensional normed spaces are continuous, Proof that the determinant of a transpose equals the determinant, Proof of the laws of exponents in a group, Proof of the limit of a product of sequences, Proof of the inverse in terms of the adjugate, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the existence of square roots of nonnegative reals, Proof of the chain rule for a composition with a curve, Proof of the linearity of differentiation, Proof that a continuous image of a compact set is compact, Proof that compact subsets of a Hausdorff space are closed, Proof that the composition of continuous maps is continuous, Proof of epsilon-delta continuity at a point in metric spaces, Proof of the sequential characterization of limits, Proof of the description of the open sets of a metric space, Proof of the Banach fixed point theorem by iteration, Proof of the triangle inequality for real numbers, Proof that an intersection of subgroups is a subgroup, Proof of the epsilon-N definition of the limit of a sequence, Proof that compactness of a subset can be tested with open sets of the ambient space, Proof that the sign is a homomorphism, Proof of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof of the determinant of the inverse, Proof that every spanning set contains a basis, by Zorn's lemma, Proof that the determinant of the identity matrix is one, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the inverse function theorem by the contraction mapping principle, Proof that closed sets contain their limits, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that the derivative gives the directional derivatives, Proof that differentiable functions are continuous, Proof of the decomposition of a permutation into disjoint cycles, Proof of the product of a matrix and its adjugate by Laplace expansion, Proof of the sequence characterisation of uniform continuity, Proof of the epsilon-delta form of the limit of a function, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the division algorithm, Proof of the Archimedean property in an Archimedean ordered field, Proof of the sign of a cycle, Proof that finite-dimensional normed spaces are complete, Proof that a continuous real function on a compact set attains its bounds, Proof of the Leibniz formula from the defining properties of the determinant, Proof of the uniqueness of inverses, Proof of the derivative of a map into a product, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof that the determinant is multilinear in the rows, Proof that open balls are open, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that Lipschitz maps are uniformly continuous, Proof that C^k functions are characterised by continuous derivatives, Proof that continuous partial derivatives imply differentiability, Proof that a set is open if and only if its complement is closed, Proof of the subgroup criterion, Proof of the mean value theorem from Rolle's theorem, Proof of continuity via open sets in metric spaces, Proof of the mean value inequality, Proof of the constant multiple rule.

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Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
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Equivalence classes form a partition statement
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Division algorithm statement
Epsilon characterisation of Cauchy sequences statement
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
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Proof that the derivative gives the directional derivatives proof
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The reals form an ordered field statement
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Decomposition of a permutation into disjoint cycles statement
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Proof that the coordinate map is a linear isomorphism proof
Absolute value definition
Proof that every spanning set contains a basis, by Zorn's lemma proof
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Proof of the existence of the infimum by negation proof
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Every spanning set contains a basis statement
Proof of the epsilon-N definition of the limit of a sequence proof
Isomorphism with the coordinate space statement
Existence of square roots of nonnegative reals statement
Approximation property of the supremum statement
Existence of the infimum statement
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Proof of continuity via closed sets proof
Decomposition of a permutation into transpositions statement
Closed sets contain their limits statement
Triangle inequality statement
Absolute value is multiplicative statement
Closed balls are closed statement
Open balls are open statement
Square root definition
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Sign of a permutation definition
Continuity via closed sets statement
Modulus of a complex number definition
Compactness via open covers statement
Proof of the limit of a sum of sequences proof
Proof that closed subsets of complete spaces are complete proof
Proof of the uniqueness of limits proof
Proof that the symmetric group is generated by transpositions proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of the squeeze theorem for sequences proof
Proof of continuity via open sets in metric spaces proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Proof that metric spaces are Hausdorff proof
Proof that convergent sequences are bounded proof
The reals are Archimedean statement
Proof that subsequences of a convergent sequence converge to the same limit proof
Proof of the monotone convergence theorem for sequences proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a complex number times its conjugate is the square of its modulus proof
Proof that a continuous image of a compact set is compact proof
The symmetric group is generated by transpositions statement
Proof that the sign is a homomorphism proof
Closed subsets of complete spaces are complete statement
Proof of the sign of a cycle proof
Lebesgue number lemma statement
Epsilon-delta continuity in metric spaces statement
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Differentiable functions are continuous statement
Limit of a sum of sequences statement
Proof of the sequence characterisation of uniform continuity proof
Uniqueness of limits statement
Proof of the triangle inequality for norms proof
Limits preserve non-strict inequalities statement
Convergent sequences are bounded statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Squeeze theorem statement
Subsequences of a convergent sequence converge to the same limit statement
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Compact subsets of a Hausdorff space are closed statement
Sign of a cycle statement
Continuous image of a compact set is compact statement
Sign is a homomorphism statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Compact subsets of a metric space are bounded statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
Proof that permuting rows multiplies the determinant by the sign proof
There is a natural number n with 1/n below any positive epsilon statement
Proof that a decreasing sequence bounded below converges proof
Triangle inequality for norms statement
The sequence r^n tends to zero for |r| < 1 example
Proof that a linear map between normed spaces is continuous if and only if it is bounded proof
Permuting rows multiplies the determinant by the sign statement
Proof of the Banach fixed point theorem by iteration proof
Proof that Lipschitz maps are uniformly continuous proof
Proof of the sequential characterization of limits proof
Proof that continuity is equivalent to sequential continuity proof
Proof of the Cauchy–Schwarz inequality proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Banach fixed point theorem statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Sequential characterization of limits statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Leibniz formula for determinants statement
Proof of the linearity of differentiation proof
Bolzano–Weierstrass theorem statement
Proof of the derivative of a map into a product proof
Proof of the determinant of a block triangular matrix by the Leibniz formula proof
Linearity of differentiation statement
Proof of the Heine–Borel theorem by sequential compactness proof
Operator norm definition
Proof of the Laplace expansion along a row by the Leibniz formula proof
Proof that the determinant is multiplicative by expanding the rows proof
Proof that the determinant of a transpose equals the determinant proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Completeness of the real numbers statement
Derivative of a map into a product statement
Proof of the constant multiple rule proof
Heine–Borel theorem statement
Boundedness theorem statement
Determinant of a block triangular matrix statement
Determinant is multiplicative statement
Laplace expansion along a row statement
Determinant of a transpose statement
Constant multiple rule statement
Derivative of a map into R^n by components statement
Proof of the determinant of the inverse proof
Proof of the Laplace expansion along a column by transposition proof
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Proof that a continuous real function on a compact set attains its bounds proof
Extreme value theorem statement
Determinant of the inverse statement
Continuous real function on a compact set attains its bounds statement
Laplace expansion along a column statement
Proof of Fermat's theorem on stationary points proof
Proof of the equivalence of norms in finite dimension by compactness of the unit sphere proof
Proof of the product of a matrix and its adjugate by Laplace expansion proof
Fermat's theorem on stationary points statement
Equivalence of norms in finite dimension statement
Product of a matrix and its adjugate statement
Proof of Rolle's theorem proof
Rolle's theorem statement
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Proof that finite-dimensional normed spaces are complete proof
Proof of the inverse in terms of the adjugate proof
Linear maps on finite-dimensional spaces are continuous statement
Finite-dimensional normed spaces are complete statement
Proof of the mean value theorem from Rolle's theorem proof
Inverse in terms of the adjugate statement
Mean value theorem statement
Proof that a matrix is invertible iff its determinant is a unit proof
Proof of the chain rule in several variables proof
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
A matrix is invertible iff its determinant is a unit statement
Derivative of the inverse function in several variables statement
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Chain rule for a composition with a curve statement
Proof that C^k functions are characterised by continuous derivatives proof
C^k function characterised by continuous derivatives statement
Proof of the mean value inequality proof
Mean value inequality statement
Proof of the inverse function theorem by the contraction mapping principle proof
Inverse function theorem statement
Proof of the implicit function theorem from the inverse function theorem proof
Implicit function theorem statement
Proof of the implicit function theorem for a surjective derivative proof

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Implicit function theorem for a surjective derivative

Tag	`2MCN92CX58`
URI	`m2://2MCN92CX58`
Kind	proof
Writing	W0 stub
MSC	26B10
Proves	Implicit function theorem for a surjective derivative
Uses	Implicit function theorem, Inverse function theorem, Jacobian matrix, Rank of a matrix, Linear independence, Chain rule in several variables
Changed	4 November 2026

Proof of the implicit function theorem for a surjective derivative

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