Third-Order Derivative Tensor¶

Definition¶

For a scalar function $f\colon A\subseteq\mathbb{R}^n\to\mathbb{R}$ with third-order partial derivatives continuous on an open neighbourhood of $\boldsymbol{a}$ (that is, $f$ is $C^3$ there), the third-order derivative tensor is the array

It represents the trilinear map

Interpretation¶

The third derivative tensor stores all third-order partial derivatives at a point, in the same way that the Hessian stores all second-order ones. Third-order partials carry three indices, so they fill a cube rather than a matrix.

For $n=2$ the cube has $2^3=8$ entries $f_{ijk}$, short for $f_{x_ix_jx_k}(\boldsymbol{a})$: $i$ picks the row, $j$ the column and $k$ the layer, with $k=1$ in front and $k=2$ at the back. For a $C^3$ function, entries whose indices are rearrangements of each other are equal (higher-order symmetry of partials), for example $f_{112}=f_{121}=f_{211}$$f_{211}$, so only four different values remain: $f_{111}$, $f_{112}$, $f_{122}$ and $f_{222}$.

The array is used by feeding it vectors. Trilinear means linear in each of the three slots separately, just as $\boldsymbol{u}^{\top}\boldsymbol{H}_f(\boldsymbol{a})\boldsymbol{v}$ is linear in $\boldsymbol{u}$ and in $\boldsymbol{v}$. With the same displacement $\boldsymbol{h}$ in all three slots, $\tfrac{1}{3!}\boldsymbol{D}^3f(\boldsymbol{a})[\boldsymbol{h},\boldsymbol{h},\boldsymbol{h}]$ is the cubic term of the Taylor polynomial (tensor contraction for Taylor terms).

For $f(x,y)=x^3+xy^2$ from the Examples, $f_{111}=6$$6$, the three arrangements of $f_{122}$ equal $2$ and all other entries are $0$. The sum is therefore $6h_1^3+3\cdot2\,h_1h_2^2$$2h_1{h}_2^2$, and the cubic term is $h_1^3+h_1h_2^2=f(\boldsymbol{h})$$f(\mathbf{h})$: for a polynomial whose terms all have degree $3$, the cubic Taylor term is the polynomial itself. A common slip is to count each mixed value once. In the sum, $f_{122}$, $f_{212}$ and $f_{221}$ are separate entries with the same value, which is where the factor $3$ comes from.

Key Results¶

• For $n$ variables, the raw array has $n^3$ entries.
• If the third partial derivatives are continuous, many entries are equal by symmetry.
• Taylor's third-order term is obtained by feeding the displacement vector into the tensor three times.

Examples¶

• For $f(x,y)=x^3+xy^2$$x{y}^2$, entries include $f_{xxx}=6$$6$, $f_{xyy}=2$$2$, and $f_{yyy}=0$$0$.
• For $f(x,y,z)=xyz$$xyz$, all mixed third partials with one $x$, one $y$, and one $z$ equal $1$.

Prerequisites¶

• Higher-Order Partial Derivative
• Vector

Related Concepts¶

• Higher-Order Symmetry of Partials
• Tensor Contraction for Taylor Terms
• Hessian Matrix

Sources¶

• Math IV: Multivariate Calculus · Week 2, Lecture 1

Aliases: third-order tensor of derivatives Keywords: third derivative tensor, array of third partial derivatives Symbols: real numbers, subset, sum

Last updated: 27 October 2026 11:52