What linear map describes a nonlinear change nearby?

Source scope: §4.3, PDF pages 27, 28. Numerical examples, experiments, added derivations, and practice are teaching supplements unless identified as source results.

Zoom in on a curved map

Let $g(x,y)=(x^2,y+xy)^T$. At $(1,1)$ the output is $(1,2)^T$. Increase $x$ by $0.1$ and hold $y$ fixed: the new output is $(1.21,2.1)^T$. The first-order prediction will be $(1.2,2.1)^T$; the missing $0.01$ is the quadratic term in $(1+0.1)^2$.

For $g:\mathbb R^n\to\mathbb R^m$, the Jacobian is an $m\times n$ matrix:

A row fixes one output and varies the input coordinate. A column describes how all outputs respond to one input coordinate. This matches the matrix-of-a-map convention: columns are images of input directions.

Use it as a local map

For the example,

Teaching supplement. When $g$ is differentiable, $g(a+h)=g(a)+J_g(a)h+o(\|h\|)$. With $a=(1,1)$ and $h=(s,t)$, the true increment is $(2s+s^2,s+2t+st)$, while the linear prediction is $(2s,s+2t)$. The error $(s^2,st)$ becomes small relative to $\|h\|$ as $h\to0$. Having partial derivatives alone is not enough to guarantee differentiability.

For a scalar output ($m=1$), $J_f$ is a $1\times n$ row and the source notes

The transpose expresses a convention with consequences: a Jacobian maps input increments forward; the column gradient participates in dot products and later propagates derivatives backward through a composite function.

Transfer

For $h(x,y,z)=(x+z,yz)$, the Jacobian is $\begin{bmatrix}1&0&1\\0&z&y\end{bmatrix}$, a $2\times3$ matrix. It takes a three-coordinate input increment to a two-coordinate output increment.

Explore the relationship