Proof of the second-order Taylor expansion

Builds on Taylor's theorem in several variables Multi-index notation Gradient Hessian matrix Schwarz's theorem for C^2 functions C^k function characterised by continuous derivatives Continuous partial derivatives imply differentiability

Proof

We take $k = 2$ in Taylor's theorem and rewrite the terms with matrices.

Let $U$ , $f$ and $a$ be as in the statement. Write $\partial_i$ for the partial derivative in the direction $x_i$ . Since $f$ is of class C^2, it is differentiable at $a$ , so $\nabla f(a)$ is defined.
By Taylor's theorem in several variables with $k = 2$ , we have $f(a + h) = \sum_{\abs{\alpha} \le 2} \frac{D^\alpha f(a)}{\alpha!} h^\alpha + R_2(h)$ with $R_2(h) / \norm{h}^2 \to 0$ as $h \to 0$ .
It suffices to show that the sum in step 2 equals $f(a) + \nabla f(a) \cdot h + \tfrac{1}{2} h^{\mathsf{T}} H_f(a) h$ . Then $R(h) = R_2(h)$ has the required limit.
The only multi-index with $\abs{\alpha} = 0$ is $\alpha = 0$ . Its term is $f(a)$ .
The multi-indices with $\abs{\alpha} = 1$ are the $e_i$ , with $e_i! = 1$ and $h^{e_i} = h_i$ . Their terms add up to $\sum_i \partial_i f(a) h_i$ . This is $\nabla f(a) \cdot h$ , since the gradient is the vector of partial derivatives.
The multi-indices with $\abs{\alpha} = 2$ are $2 e_i$ for each $i$ , and $e_i + e_j$ for each pair $i < j$ .
For $\alpha = 2 e_i$ we have $\alpha! = 2$ , so the term is $\tfrac{1}{2} \partial_i \partial_i f(a) \, h_i^2$ . For $\alpha = e_i + e_j$ with $i < j$ we have $\alpha! = 1$ , so the term is $\partial_i \partial_j f(a) \, h_i h_j$ .
Let $H = H_f(a)$ , the Hessian matrix, with entries $H_{ij} = \partial_i \partial_j f(a)$ . By Schwarz's theorem, $H_{ij} = H_{ji}$ .
Then $\tfrac{1}{2} h^{\mathsf{T}} H h = \tfrac{1}{2} \sum_{i, j} H_{ij} h_i h_j = \tfrac{1}{2} \sum_i H_{ii} h_i^2 + \tfrac{1}{2} \sum_{i < j} (H_{ij} + H_{ji}) h_i h_j$ .
By step 8 the last sum is $\sum_{i < j} H_{ij} h_i h_j$ . So $\tfrac{1}{2} h^{\mathsf{T}} H h$ equals the sum of the terms in step 7.
Steps 4, 5, 7 and 10 give the identity of step 3, which proves the expansion.

Idea

The second-order expansion is the case $k = 2$ of Taylor's theorem. The only work is bookkeeping: the terms of order two in multi-index form match $\tfrac{1}{2} h^{\mathsf{T}} H h$ , because the Hessian is symmetric and each mixed term appears twice in the matrix product.

Reading path

To understand this page from the ground up, read these 401 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, Proof of integration by parts, 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 power rule by induction, Proof of the triangle inequality for norms, 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 of the chain rule with an auxiliary difference quotient, Proof of the derivative as a little-o approximation, Proof of the equality of complex numbers, Proof that continuous functions on a closed bounded interval are uniformly continuous, 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 of the linearity of the derivative, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the derivative of a continuous linear map, 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 of the mean value theorem for integrals, 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 Riemann's criterion for integrability, Proof of the sequential characterization of limits, Proof of Taylor's theorem with the integral remainder by the fundamental theorem of calculus, 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 of the first fundamental theorem of calculus, Proof of Schwarz's theorem for C^2 functions by the mean value theorem, 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 the linearity of the integral, Proof of Zorn's lemma, Proof that metric spaces are Hausdorff, Proof that the derivative gives the directional derivatives, Proof of the second fundamental theorem of calculus, Proof of the intermediate value theorem by the supremum, Proof that differentiable functions are continuous, Proof of Taylor's theorem in several variables by restriction to a line, 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 composition of continuous functions is continuous, Proof of the additivity of the integral over intervals, 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 product rule, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the symmetry of second derivatives by a second difference, Proof that open balls are open, Proof of the monotonicity of the integral, 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 fundamental theorem of calculus, Proof of the mean value theorem from Rolle's theorem, Proof that sums and products of continuous functions are continuous, Proof of continuity via open sets in metric spaces, Proof that continuous functions are Riemann integrable, Proof of the mean value inequality, Proof of the constant multiple rule.

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Proof of the epsilon-N definition of the limit of a sequence proof
Every spanning set contains a basis statement
Approximation property of the supremum statement
Existence of the infimum statement
Existence of square roots of nonnegative reals statement
Closed sets contain their limits statement
Proof of continuity via closed sets proof
Proof that compactness of a subset can be tested with open sets of the ambient space proof
Triangle inequality statement
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Epsilon-N definition of the limit of a sequence statement
Proof of the intermediate value theorem by the supremum proof
Proof that the reals are Archimedean from the completeness axiom proof
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Continuity via closed sets statement
Proof of the limit of a sum of sequences proof
Compactness via open covers statement
Proof that differentiable functions are continuous proof
Proof of the uniqueness of limits proof
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Proof that metric spaces are Hausdorff proof
Intermediate value theorem statement
Proof of continuity via open sets in metric spaces 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 linearity of the integral 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 a complex number times its conjugate is the square of its modulus proof
Proof that compact subsets of a Hausdorff space are closed proof
Proof that a continuous image of a compact set is compact proof
Proof that compact subsets of a metric space are bounded proof
Metric spaces are Hausdorff statement
Differentiable functions are continuous statement
Epsilon-delta continuity in metric spaces statement
Proof that compactness and sequential compactness agree in metric spaces proof
Lebesgue number lemma statement
Limit of a sum of sequences statement
Proof of the sequence characterisation of uniform continuity proof
Proof of the triangle inequality for norms proof
Uniqueness of limits statement
Linearity of the integral statement
Limits preserve non-strict inequalities statement
Proof of the monotonicity of the integral proof
Squeeze theorem statement
Proof that there is a natural number n with 1/n below any positive epsilon proof
Convergent sequences are bounded statement
Subsequences of a convergent sequence converge to the same limit statement
Proof that Darboux and Riemann integrals agree proof
Monotone convergence theorem for sequences statement
A complex number times its conjugate statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Compact subsets of a metric space are bounded statement
Compactness and sequential compactness agree in metric spaces statement
Continuity via open sets statement
Proof of the limit of a product of sequences proof
Epsilon-delta uniform continuity statement
Monotonicity of the integral statement
Darboux and Riemann integrals agree 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 of the Cauchy–Schwarz inequality proof
Proof that continuity is equivalent to sequential continuity proof
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
Proof of Riemann's criterion for integrability proof
Lipschitz maps are uniformly continuous statement
Continuity is equivalent to sequential continuity statement
Sequential characterization of limits statement
Riemann's criterion for integrability statement
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Cauchy–Schwarz inequality statement
Bounded linear maps are continuous statement
Proof of the chain rule with an auxiliary difference quotient proof
Proof of the product rule proof
Proof of the linearity of differentiation proof
Proof that sums and products of continuous functions are continuous proof
Bolzano–Weierstrass theorem statement
Proof of the additivity of the integral over intervals proof
Proof of the derivative of a map into a product proof
Product of continuous functions is continuous statement
Chain rule statement
Product rule statement
Proof of the linearity of the derivative proof
Linearity of differentiation statement
Proof of the Heine–Borel theorem by sequential compactness proof
Proof that continuous functions on a closed bounded interval are uniformly continuous proof
Operator norm definition
Additivity of the integral over intervals statement
Proof of the boundedness theorem by Bolzano–Weierstrass proof
Derivative of a map into a product statement
Linearity of the derivative statement
Proof of the constant multiple rule proof
Heine–Borel theorem statement
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
Constant multiple rule statement
Derivative of a map into R^n by components statement
Power rule 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
Proof that continuous functions are Riemann integrable proof
Fermat's theorem on stationary points statement
Continuous functions are Riemann integrable 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
Proof of the mean value theorem for integrals proof
Linear maps on finite-dimensional spaces are continuous statement
Proof of the mean value theorem from Rolle's theorem proof
Mean value theorem for integrals statement
Mean value theorem statement
First fundamental theorem of calculus statement
Proof of the chain rule in several variables proof
Proof that continuous partial derivatives imply differentiability proof
Chain rule in several variables statement
Proof of the second fundamental theorem of calculus proof
Continuous partial derivatives imply differentiability statement
Proof of the chain rule for a composition with a curve proof
Second fundamental theorem of calculus statement
Proof of the fundamental theorem of calculus proof
Chain rule for a composition with a curve statement
Proof of integration by parts proof
Proof that C^k functions are characterised by continuous derivatives proof
C^k function characterised by continuous derivatives statement
Integration by parts statement
Fundamental theorem of calculus statement
Proof of the mean value inequality proof
Proof of Schwarz's theorem for C^2 functions by the mean value theorem proof
Proof of Taylor's theorem with the integral remainder by the fundamental theorem of calculus proof
Mean value inequality statement
Taylor's theorem with the integral remainder statement
Proof of the symmetry of second derivatives by a second difference proof
Symmetry of second derivatives statement
Schwarz's theorem for C^2 functions statement
Proof of Taylor's theorem in several variables by restriction to a line proof
Taylor's theorem in several variables statement
Proof of the second-order Taylor expansion proof

Where this leads

3 pages build on this one.

Tag	`8GDF8T2RR0`
URI	`m2://8GDF8T2RR0`
Kind	proof
Writing	W0 stub
MSC	26B12
Proves	Second-order Taylor expansion
Uses	Taylor's theorem in several variables, Multi-index notation, Gradient, Hessian matrix, Schwarz's theorem for C^2 functions, C^k function characterised by continuous derivatives, Continuous partial derivatives imply differentiability
Changed	4 November 2026

Proof of the second-order Taylor expansion

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