Jacobian Matrices and Determinants: Linearization, Change of Variables, and Invertibility

All flashcards

What does the Jacobian matrix represent at a differentiable point?

The linear map that gives the best first-order approximation to changes in the function's output.

How do Jacobians combine under composition?

For differentiable $f$ and $g$, $J_{g\circ f}(x)=J_g(f(x))J_f(x)$.

What does the absolute Jacobian determinant measure?

The local volume-scaling factor of a square transformation.

What factor does polar-to-Cartesian conversion contribute to area?

The Jacobian determinant is $r$, so the area element is $r\,dr\,d\varphi$ for $r\geq 0$.

What is the Hessian of a scalar-valued function?

The matrix of second partial derivatives, equivalently the Jacobian of its gradient.

Quiz questions

1. For a function $F:\mathbb{R}^n\to\mathbb{R}^m$, when is its Jacobian determinant defined?Answer: When the Jacobian is square, so $m=n$.
2. For differentiable $f:\mathbb{R}^n\to\mathbb{R}^m$ and $g:\mathbb{R}^m\to\mathbb{R}^k$, which formula gives the Jacobian of $g\circ f$?Answer: $J_{g\circ f}(x)=J_g(f(x))J_f(x)$
3. In a change-of-variables formula for volume, which Jacobian factor is used?Answer: $|\det J_F(u)|$
4. For $x=r\cos\varphi$ and $y=r\sin\varphi$, what factor appears in the polar area element?Answer: $r$
5. If $f:\mathbb{R}^n\to\mathbb{R}^n$ is continuously differentiable near $p$ and $\det J_f(p)\ne 0$, what does the inverse function theorem guarantee?Answer: A differentiable inverse for $f$ on some neighborhood of $p$.