Proof of the Laplacian as a sum of second partial derivatives

Builds on Laplacian Divergence Gradient Differentiable function of several variables Partial derivative Higher partial derivatives

Proof

We unfold the definitions.

Let $f \colon U \to \R$ be twice differentiable on the open set $U \subseteq \R^n$ . Then $f$ is differentiable at each point of $U$ .
By the coordinate form of the gradient, $\nabla f = (g_1, \dots, g_n)$ on $U$ , where $g_i = \partial f / \partial x_i$ is a partial derivative.
The map $x \mapsto Df(x)$ is differentiable on $U$ , since $f$ is twice differentiable. Its entries in the standard basis are $g_1, \dots, g_n$ , so each $g_i$ is differentiable on $U$ .
Hence each partial derivative $\partial g_i / \partial x_i$ exists on $U$ . By the definition of higher partial derivatives, $\partial g_i / \partial x_i = \partial^2 f / \partial x_i^2$ .
By the definition of the divergence, $\operatorname{div}(\nabla f) = \sum_{i=1}^n \partial g_i / \partial x_i$ .
By the definition of the Laplacian and steps 4 and 5, $\Delta f = \operatorname{div}(\nabla f) = \frac{\partial^2 f}{\partial x_1^2} + \dots + \frac{\partial^2 f}{\partial x_n^2}$ .

Idea

The gradient collects the first partial derivatives of $f$ . The divergence then differentiates the $i$ -th of them in the direction $x_i$ and adds the results. So only the unmixed second derivatives appear.

Reading path

To understand this page from the ground up, read these 307 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 that open sets are the sets that are neighbourhoods of all their points, 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 of the triangle inequality for norms, 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 derivative as a little-o approximation, Proof that a linear map between normed spaces is continuous if and only if it is bounded, 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 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 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 of the limit of a product of sequences, 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 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 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 triangle inequality for real numbers, 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 of the existence of the infimum by negation, Proof of the equivalence of norms in finite dimension by compactness of the unit sphere, Proof that every spanning set contains a basis, by Zorn's lemma, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, 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 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 Archimedean property in an Archimedean ordered field, Proof that a continuous real function on a compact set attains its bounds, 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 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 mean value theorem from Rolle's theorem, Proof of continuity via open sets in metric spaces, Proof of the constant multiple rule.

Basis concept
Complex numbers object
Construction of the reals from Cauchy sequences concept
Coordinate space object
Coordinate vector concept
Definition by recursion concept
Element definition
Euclidean metric example
Euclidean space object
Extended real number line object
Function concept
Integers object
Limit point of a set definition
Linear isomorphism concept
Linear map concept
Local maximum definition
Mathematical induction concept
Natural numbers object
Norm concept
Ordered field definition
Peano axioms concept
Proof of Zorn's lemma proof
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Vector space definition
p-metric on R^n example
Real and imaginary parts definition
Set definition
Fréchet derivative definition
Having a directional derivative definition
Total derivative concept
Function of several variables concept
Vector-valued function concept
Ring of integers statement
Limit of a function definition
Local minimum definition
Successor function definition
Archimedean ordered field 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 combination definition
Linear map definition
Linear subspace definition
Norm definition
p-norm on R^n example
Empty set object
Family of sets definition
Ordered pair definition
Partial derivative concept
Intersection definition
Limit of a function of several variables concept
Union definition
Continuity at a point definition
Derivative definition
Addition of natural numbers definition
Biconditional definition
Truth table concept
Modus ponens concept
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
Existential quantifier definition
Universal quantifier definition
Bounded sequence definition
Limit of a sequence definition
Monotone sequence definition
Linear independence definition
Linear isomorphism definition
Linear span definition
Multilinear map definition
Completeness axiom statement
Normed vector space definition
Cartesian product definition
Jacobian matrix concept
Partial derivative definition
Continuity in R^n definition
Differentiable function of several variables definition
Continuous function definition
Differentiable function definition
Union of a family of sets definition
Subset definition
Proof of the commutativity of addition by induction proof
Archimedean property statement
Mathematical proof concept
Order on the natural numbers definition
Propositional logic concept
Proof of the derivative as a little-o approximation proof
The rationals form an ordered field statement
Uniqueness quantifier definition
Convergent sequence definition
Sequential continuity definition
Spanning set definition
Higher partial derivatives definition
Jacobian matrix definition
Binary relation definition
Differentiable on an open set definition
Universal set definition
Topological space definition
Gradient definition
Proof of the derivative of a map into R^n by components proof
Derivative as a little-o approximation statement
Order on the reals definition
Commutativity of addition statement
Tautology definition
Proof of strong induction from ordinary induction proof
Finite-dimensional vector space definition
Basis definition
Iterated derivative definition
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Sequential compactness definition
Subspace topology definition
Complement definition
Function definition
Interval definition
Logical equivalence definition
Strong induction statement
Coordinate vector definition
Preorder definition
Choice function definition
Codomain definition
Function composition definition
Identity function definition
Injective function definition
Image of a set definition
Preimage of a set definition
Proof of the epsilon-delta form of continuity proof
Proof of the epsilon-delta form of the limit of a function proof
Metric space definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Left inverse definition
Right inverse definition
Axiom of choice statement
Surjective function definition
Epsilon-delta definition of continuity statement
Epsilon-delta definition of a limit statement
Closed ball definition
Convergence of a sequence in a metric space definition
Lipschitz map definition
Open ball definition
Triangle inequality for a metric statement
Double negation law statement
Well-ordering principle statement
Diameter of a set definition
Maximal element definition
Total order definition
Upper bound definition
Bijective function definition
Metric topology definition
Bounded set definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Chain definition
Proof that a function is bijective iff it has a two-sided inverse proof
Open set definition
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
Bijective iff it has a two-sided inverse statement
Hausdorff space definition
Closed set definition
Open cover definition
Continuous map of topological spaces definition
Proof that open sets are the sets that are neighbourhoods of all their points proof
Proof that the derivative gives the directional derivatives proof
Zorn's lemma statement
Equivalent metrics definition
Proof of the description of the open sets of a metric space proof
The reals form an ordered field statement
Compact space definition
Proof that a set is open if and only if its complement is closed proof
Derivative gives the directional derivatives statement
Open sets characterised by neighbourhoods statement
Proof that the coordinate map is a linear isomorphism proof
Equivalent norms definition
Open sets of a metric space statement
Absolute value definition
Proof that every spanning set contains a basis, by Zorn's lemma proof
Proof of the approximation property of the supremum proof
Proof of the existence of the infimum by negation proof
Proof of the existence of square roots of nonnegative reals proof
Complement of an open set is closed statement
Compact set definition
Continuous linear map definition
Continuity at a point of a topological space definition
Proof that closed sets contain their limits proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof that open balls are open proof
Isomorphism with the coordinate space statement
Proof of the epsilon-N definition of the limit of a sequence proof
Every spanning set contains a basis 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
Closed sets contain their limits statement
Triangle inequality statement
Absolute value is multiplicative 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
Compactness via open covers statement
Modulus of a complex number definition
Continuity via closed sets statement
Proof of the limit of a sum of sequences proof
Proof of the uniqueness of limits proof
Proof that differentiable functions are continuous proof
Proof of epsilon-delta continuity at a point in metric spaces proof
Proof of continuity via open sets in metric spaces proof
Proof that limits preserve non-strict inequalities proof
Proof of the Lebesgue number lemma by a finite cover of balls proof
Proof of the squeeze theorem for sequences 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 continuous image of a compact set is compact proof
Differentiable functions are continuous statement
Proof that compact subsets of a metric space are bounded proof
Epsilon-delta continuity in metric spaces statement
Lebesgue number lemma statement
Metric spaces are Hausdorff statement
Proof that compactness and sequential compactness agree in metric spaces proof
Proof of the triangle inequality for norms proof
Limit of a sum of sequences statement
Proof of the sequence characterisation of uniform continuity proof
Uniqueness of limits statement
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
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Compactness and sequential compactness agree in metric spaces statement
Compact subsets of a metric space are bounded statement
Continuity via open sets statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
There is a natural number n with 1/n below any positive epsilon statement
Triangle inequality for norms statement
Proof that a decreasing sequence bounded below converges proof
Proof that a linear map between normed spaces is continuous if and only if it is bounded 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
Limit of a product of sequences statement
Decreasing sequence bounded below converges statement
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Bounded linear maps are continuous statement
Proof of the linearity of differentiation proof
Bolzano–Weierstrass theorem statement
Proof of the derivative of a map into a product proof
Linearity of differentiation statement
Proof of the Heine–Borel theorem by sequential compactness proof
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Proof of the constant multiple rule proof
Boundedness theorem statement
Heine–Borel theorem statement
Constant multiple rule statement
Derivative of a map into R^n by components statement
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
Continuous real function on a compact set attains its bounds 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
Fermat's theorem on stationary points statement
Proof of Rolle's theorem proof
Equivalence of norms in finite dimension statement
Rolle's theorem statement
Proof that linear maps on finite-dimensional normed spaces are continuous proof
Linear maps on finite-dimensional spaces are continuous statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem statement
Proof that continuous partial derivatives imply differentiability proof
Continuous partial derivatives imply differentiability statement
Proof that C^k functions are characterised by continuous derivatives proof
C^k function characterised by continuous derivatives statement
Vector field definition
Divergence definition
Laplacian definition
Proof of the Laplacian as a sum of second partial derivatives proof

Where this leads

73 pages build on this one.

Tag	`5HB2ECHQFD`
URI	`m2://5HB2ECHQFD`
Kind	proof
Writing	W0 stub
MSC	35J05
Proves	Laplacian as a sum of second partial derivatives
Uses	Laplacian, Divergence, Gradient, Differentiable function of several variables, Partial derivative, Higher partial derivatives
Changed	4 November 2026

Proof of the Laplacian as a sum of second partial derivatives

Proof

Idea

Reading path

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