Tensor Contraction for Taylor Terms¶

Definition¶

Let $\boldsymbol{D}^k f(\boldsymbol{a})$ be the $k$ th derivative of a scalar function at $\boldsymbol{a}$ , viewed as a $k$ -linear map. Its contraction with a displacement $\boldsymbol{h}\in\mathbb{R}^n$ is

\boldsymbol{D}^k f(\boldsymbol{a})[\boldsymbol{h},\ldots,\boldsymbol{h}].

In a Taylor polynomial, the corresponding $k$ th-order contribution is

\frac{1}{k!}\boldsymbol{D}^k f(\boldsymbol{a})[\boldsymbol{h},\ldots,\boldsymbol{h}].

Interpretation¶

A derivative tensor is useful once it is evaluated on a concrete displacement. Contraction turns the stored derivative information into the actual Taylor correction in that direction.

A 2 by 2 by 2 cube of third-order partial derivatives f ijk, with front layer k = 1 and back layer k = 2, with row and column directions labelled i and j and layers labelled k.

$\boldsymbol{D}^3f(\boldsymbol{a})=\big(f_{x_ix_jx_k}(\boldsymbol{a})\big)_{i,j,k}$ has $n^3$ entries (here $n=2$ $2$ : a $2\times2\times2$ cube, front $k=1$ $1$ , back $k=2$ $2$ ), symmetric under any permutation of $(i,j,k)$ for $C^3$ $f$ . Contraction with $\boldsymbol{h}$ , $\displaystyle \boldsymbol{D}^3f(\boldsymbol{a})[\boldsymbol{h},\boldsymbol{h},\boldsymbol{h}]=\sum_{i,j,k}f_{x_ix_jx_k}(\boldsymbol{a})\,h_ih_jh_k$ $i, j, k \sum f_{x_{i} x_{j} x_{k}} (a) h_{i} h_{j} h_{k}$ , gives the cubic Taylor term $\tfrac{1}{3!}\boldsymbol{D}^3f(\boldsymbol{a})[\boldsymbol{h},\boldsymbol{h},\boldsymbol{h}]$ .

The third-order derivative tensor is the case $k=3$ $3$ : inserting the displacement in all three slots and summing over its indices produces one scalar cubic contribution, with coefficient $1/3!$ in the Taylor formula.

Key Results¶

For $k=2$ $2$ , the contraction is the quadratic form $\boldsymbol{h}^{\top} \boldsymbol{H}_f(\boldsymbol{a})\boldsymbol{h}$ .
For $k=3$ $3$ , the contraction sums all third partial derivatives weighted by three components of $\boldsymbol{h}$ .
The factorial corrects for repeated counting in the Taylor expansion.

Examples¶

If $\boldsymbol{H}=\begin{pmatrix}2&0\\0&6\end{pmatrix}$ and $\boldsymbol{h}=(u,v)$ $(u, v)$ , then $\boldsymbol{D}^2f(\boldsymbol{a})[\boldsymbol{h},\boldsymbol{h}]=2u^2+6v^2$ $6 v^{2}$ .
For $f(x,y)=x^3+xy^2$ at the origin and $\boldsymbol{h}=(u,v)$ $(u, v)$ , the third-order contraction is $6u^3+6uv^2$ $6 u v^{2}$ .

Prerequisites¶

Sources¶

Math IV: Multivariate Calculus · Week 2, Lecture 1

Aliases: contracting derivative tensors Keywords: tensor contraction, higher-order Taylor term Symbols: element of, real numbers, sum

Last updated: 27 October 2026 11:52