Proof of Liouville's formula

Builds on Leibniz formula for determinants Determinant of a matrix with two equal rows Trace of a matrix Product rule Linearity of differentiation The integral of a continuous function is an antiderivative Chain rule The exponential function is its own derivative A function with zero derivative is constant Exponential of a sum

Proof

We show that $w = \det \Phi$ satisfies $w' = (\tr A)\, w$ and then solve this equation.

Write $\Phi = (\varphi_{ij})$ and $A = (A_{ij})$ . Each $\varphi_{ij}$ is differentiable on $I$ . By the Leibniz formula, $\det \Phi = \sum_{\sigma} \operatorname{sgn}(\sigma) \prod_{i=1}^{n} \varphi_{i, \sigma(i)}$ , summed over all permutations $\sigma$ .
By induction on the number of factors, the product rule gives $(f_1 \cdots f_n)' = \sum_{k} f_1 \cdots f_k' \cdots f_n$ for differentiable $f_1, \dots, f_n$ .
With linearity of differentiation, $(\det \Phi)' = \sum_{k=1}^{n} \det \Phi_k$ . Here $\Phi_k$ is $\Phi$ with row $k$ replaced by row $k$ of $\Phi'$ .
From $\Phi' = A \Phi$ , row $k$ of $\Phi'$ has entries $\varphi_{kj}' = \sum_{l} A_{kl} \varphi_{lj}$ . So it is $\sum_l A_{kl}$ times row $l$ of $\Phi$ .
Each term of the Leibniz formula contains exactly one entry of row $k$ . So the determinant is linear in row $k$ , and $\det \Phi_k = \sum_l A_{kl} \det \Phi_{k, l}$ . Here $\Phi_{k, l}$ is $\Phi$ with row $k$ replaced by row $l$ of $\Phi$ .
For $l \ne k$ , the matrix $\Phi_{k, l}$ has two equal rows, so its determinant is $0$ . For $l = k$ , $\Phi_{k, k} = \Phi$ . Hence $\det \Phi_k = A_{kk} \det \Phi$ .
Summing over $k$ and using the definition of the trace, $w' = (\tr A)\, w$ on $I$ , where $w = \det \Phi$ .
The function $\tr A$ is continuous, as a sum of continuous entries. Let $T(t) = \int_{t_0}^{t} \tr A(s)\, ds$ . By the antiderivative theorem, $T' = \tr A$ , and $T(t_0) = 0$ .
Let $g = w\, e^{-T}$ . By the product rule, the chain rule and the derivative of the exponential, $g' = (\tr A)\, w\, e^{-T} - w\, (\tr A)\, e^{-T} = 0$ .
So $g$ is constant on $I$ , and $g(t) = g(t_0) = \det \Phi(t_0)$ .
By the exponential of a sum, $e^{T(t)} e^{-T(t)} = 1$ . Multiplying $g(t)$ by $e^{T(t)}$ gives $\det \Phi(t) = \det \Phi(t_0) \exp\left(\int_{t_0}^{t} \tr A(s)\, ds\right)$ .

Idea

Differentiating a determinant row by row gives a sum of $n$ determinants. In the $k$ -th one, the new row $k$ is a combination of the rows of $\Phi$ , and only the multiple $A_{kk}$ of row $k$ survives. So $\det \Phi$ solves a scalar linear equation with coefficient $\tr A$ , whose solutions are exponentials.

Reading path

To understand this page from the ground up, read these 333 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of the Cauchy–Hadamard theorem, Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of Rolle's theorem, 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 that the symmetric group is generated by transpositions, Proof of the formula for the derivative of a polynomial, Proof of the commutativity of addition by induction, Proof of term-by-term differentiation, Proof of the epsilon-delta form of continuity, Proof that equivalence classes form a partition, Proof of the power rule by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof that there is a natural number n with 1/n below any positive epsilon, Proof of the chain rule with an auxiliary difference quotient, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that permuting rows multiplies the determinant by the sign, Proof that the exponential series has infinite radius of convergence, Proof that disjoint cycles commute, Proof that continuous functions on a closed bounded interval are uniformly continuous, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the root test, Proof that a matrix with two equal rows has determinant zero, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof that the integral of a continuous function is an antiderivative, Proof of strong induction from ordinary induction, Proof that power series are differentiable inside the radius of convergence, Proof of the laws of exponents in a group, Proof of the limit of a product of sequences, Proof that uniform convergence of derivatives allows term-by-term differentiation, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of Pascal's rule from the recursive definition, Proof of the linearity of differentiation, Proof of the mean value theorem for integrals, Proof of Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof that Cauchy sequences are bounded, 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 of the first fundamental theorem of calculus, Proof of the comparison test, 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 linearity of the integral, Proof of the factorial formula for binomial coefficients by induction, Proof of the intermediate value theorem by the supremum, Proof of the binomial theorem by induction, Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms, Proof that differentiable functions are continuous, Proof of the decomposition of a permutation into disjoint cycles, 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 Leibniz formula from the defining properties of the determinant, Proof of the uniqueness of inverses, Proof that a composition of continuous functions is continuous, Proof of the exponential of a sum by the Cauchy product, Proof of the additivity of the integral over intervals, Proof that a decreasing sequence bounded below converges, Proof of the term test, Proof that the exponential function is its own derivative, by term-by-term differentiation, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the product rule, 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 a function with zero derivative on an open interval is constant, Proof of the monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof of the ratio formula for the radius of convergence, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof of the subgroup criterion, Proof of the mean value theorem from Rolle's theorem, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product, Proof that continuous functions are Riemann integrable, Proof of the constant multiple rule.

Binary operation definition
Binomial coefficient definition
Construction of the reals from Cauchy sequences concept
Cycle notation concept
Definition by recursion concept
Divisibility definition
Element definition
Factorial definition
Group concept
Identity matrix object
Integers object
Limit point of a set definition
Local maximum definition
Mathematical induction concept
Matrix definition
Multilinear map concept
Natural numbers object
Ordered field definition
Partition of an interval definition
Peano axioms concept
Permutation concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Transposition concept
Vector space definition
Identity element definition
Proof of Pascal's rule from the recursive definition proof
Set definition
Ring definition
Subgroup definition
Symmetric group definition
Ring of integers statement
Limit of a function definition
Local minimum definition
Square matrix definition
Archimedean ordered field definition
Successor function definition
Mesh of a partition definition
Refinement of a partition definition
Riemann sum definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Linear map definition
Inverse element definition
Pascal's rule statement
Empty set object
Family of sets definition
Ordered pair definition
Commutative ring definition
Transposition definition
Proof that an intersection of subgroups is a subgroup proof
Continuity at a point definition
Derivative definition
One-sided limit definition
Intersection definition
Union definition
Trace of a matrix definition
Addition of natural numbers definition
Set-builder notation concept
Riemann integrable function definition
Proof of the Archimedean property in an Archimedean ordered field proof
The rationals form a field statement
The successor function is injective statement
Zero is not a successor statement
Biconditional definition
Truth table concept
Modus ponens concept
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Sequence tending to infinity definition
Series definition
Completeness axiom statement
Multilinear map definition
Proof of the factorial formula for binomial coefficients by induction proof
Cartesian product definition
Proof of the uniqueness of inverses proof
Disjoint sets definition
Continuous function definition
Antiderivative definition
Differentiable function definition
One-sided derivative definition
Subset definition
Proof of the commutativity of addition by induction proof
Union of a family of sets definition
Archimedean property statement
Axiom of extensionality statement
Order on the natural numbers definition
Riemann integral definition
The rationals form an ordered field statement
Propositional logic concept
Upper and lower Darboux sums definition
Mathematical proof concept
Uniqueness quantifier definition
Limit superior definition
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Sequential continuity definition
The sequence r^n tends to infinity for r > 1 example
Cauchy product definition
Partial sum definition
Power series definition
Alternating multilinear map definition
Factorial formula for binomial coefficients statement
Uniqueness of inverses statement
Binary relation definition
Proof that a composition of continuous functions is continuous proof
Partition of a set definition
Order on the reals definition
Commutativity of addition statement
Definite integral definition
Proof of strong induction from ordinary induction proof
Determinant definition
Upper and lower Darboux integrals definition
Tautology definition
Uniform convergence definition
Sum of a finite geometric series statement
Convergent series definition
Radius of convergence definition
Proof of the binomial theorem by induction proof
Antisymmetric relation definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Proof of the subgroup criterion proof
Proof of the laws of exponents in a group proof
Interval definition
Proof that a matrix with two equal rows has determinant zero proof
Proof that the determinant of the identity matrix is one proof
Function definition
Strong induction statement
Proof that the determinant is multilinear in the rows proof
Uniform convergence on a set definition
Logical equivalence definition
Sum of a series definition
Series of functions definition
Absolute convergence definition
Binomial theorem statement
Equivalence relation definition
Preorder definition
Subgroup criterion statement
Laws of exponents in a group statement
Proof of the epsilon-delta form of the limit of a function proof
Proof of the epsilon-delta form of continuity proof
Determinant of a matrix with two equal rows statement
Determinant of the identity matrix statement
Injective function definition
Determinant is multilinear in the rows statement
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Intersection of subgroups is a subgroup statement
Proof that disjoint cycles commute proof
Epsilon-delta definition of a limit statement
Epsilon-delta definition of continuity statement
Equivalence class definition
Well-ordering principle statement
Double negation law statement
Total order definition
Subgroup generated by a subset definition
Disjoint cycles commute statement
Composition of continuous functions is continuous statement
Uniform continuity definition
Proof that equivalence classes form a partition proof
Proof of the division algorithm proof
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Uniform continuity on a set definition
Equivalence classes form a partition statement
Division algorithm statement
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Proof of the decomposition of a permutation into disjoint cycles proof
The reals form an ordered field statement
Absolute value definition
Decomposition of a permutation into disjoint cycles statement
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the decomposition of a permutation into transpositions proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Absolute value is multiplicative statement
Decomposition of a permutation into transpositions statement
Proof of the intermediate value theorem by the supremum proof
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Sign of a permutation definition
Intermediate value theorem statement
Proof that the symmetric group is generated by transpositions proof
Proof of the uniqueness of limits proof
Proof that limits preserve non-strict inequalities proof
Proof that differentiable functions are continuous proof
Proof of the linearity of the integral proof
The reals are Archimedean statement
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The sequence 1/n tends to zero example
Proof of the term test proof
Proof that Cauchy sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
The symmetric group is generated by transpositions statement
Differentiable functions are continuous statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Linearity of the integral statement
Limit of a sum of sequences statement
Uniqueness of limits statement
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Proof of the ratio formula for the radius of convergence proof
Squeeze theorem statement
Proof that Darboux and Riemann integrals agree proof
Convergent sequences are bounded statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Proof of the limit of a product of sequences proof
There is a natural number n with 1/n below any positive epsilon statement
Proof that permuting rows multiplies the determinant by the sign proof
Monotonicity of the integral statement
Darboux and Riemann integrals agree statement
Term test statement
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Permuting rows multiplies the determinant by the sign statement
Proof of the sequential characterization of limits proof
Decreasing sequence bounded below converges statement
Proof that continuity is equivalent to sequential continuity proof
Limit of a product of sequences statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Proof of Riemann's criterion for integrability proof
Geometric series example
Continuity is equivalent to sequential continuity statement
Proof of the Leibniz formula from the defining properties of the determinant proof
Riemann's criterion for integrability statement
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Sequential characterization of limits statement
Proof of the root test proof
Proof of the Cauchy–Hadamard theorem proof
Proof that exponentials dominate powers in sequences, by the ratio of consecutive terms proof
Proof of the ratio test proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Proof of the chain rule with an auxiliary difference quotient proof
Leibniz formula for determinants statement
Proof of the product rule proof
Proof of the linearity of differentiation proof
Comparison test statement
Proof of the additivity of the integral over intervals proof
Cauchy product of absolutely convergent series statement
Bolzano–Weierstrass theorem statement
Exponentials dominate powers in sequences statement
Chain rule statement
Product rule statement
Linearity of differentiation statement
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Additivity of the integral over intervals statement
Ratio test statement
Root test statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Proof of the constant multiple rule proof
Continuous functions on a closed bounded interval are uniformly continuous statement
Proof of the power rule by induction proof
Boundedness theorem statement
Proof of the first fundamental theorem of calculus proof
Ratio formula for the radius of convergence statement
Cauchy–Hadamard theorem statement
Cauchy criterion for sequences statement
Constant multiple rule statement
Power rule statement
Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Extreme value theorem statement
Proof of the formula for the derivative of a polynomial proof
Proof of Fermat's theorem on stationary points proof
Cauchy criterion for series statement
Derivative of a polynomial statement
Proof that continuous functions are Riemann integrable proof
Proof that absolutely convergent series converge proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable statement
Proof of Rolle's theorem proof
Absolutely convergent series converge statement
Rolle's theorem statement
Proof of the mean value theorem for integrals proof
Proof of the Weierstrass M-test proof
Mean value theorem for integrals statement
Proof of the mean value theorem from Rolle's theorem proof
Weierstrass M-test statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Proof that a function with zero derivative on an open interval is constant proof
Proof that the integral of a continuous function is an antiderivative proof
Proof that uniform convergence of derivatives allows term-by-term differentiation proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
A function with zero derivative is constant statement
The integral of a continuous function is an antiderivative statement
Uniform convergence of derivatives allows term-by-term differentiation statement
Proof that the exponential series has infinite radius of convergence proof
The exponential series has infinite radius of convergence statement
Proof of term-by-term differentiation proof
Term-by-term differentiation statement
Exponential function definition
Proof of the exponential of a sum by the Cauchy product proof
Proof that power series are differentiable inside the radius of convergence proof
Exponential of a sum statement
Power series are differentiable inside the radius of convergence statement
Proof that the exponential function is its own derivative, by term-by-term differentiation proof
The exponential function is its own derivative statement
Proof of Liouville's formula proof

Where this leads

2 pages build on this one.

Tag	`A4TJRMRSKB`
URI	`m2://A4TJRMRSKB`
Kind	proof
Writing	W0 stub
MSC	34A30
Proves	Liouville's formula
Uses	Leibniz formula for determinants, Determinant of a matrix with two equal rows, Trace of a matrix, Product rule, Linearity of differentiation, The integral of a continuous function is an antiderivative, Chain rule, The exponential function is its own derivative, A function with zero derivative is constant, Exponential of a sum
Changed	4 November 2026

Proof of Liouville's formula

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