Type a command or search the notes. Tab cycles completions. Up and down recall history when blank. Escape dismisses, then leaves the prompt.
MATH
[1251]

Definitions from Change of variables and polar coordinates

All definitions from Change of variables and polar coordinates.

Back to Change of variables and polar coordinates

Jacobian matrix

The Jacobian matrix of gg is

Jg(u,v)=(∂x∂u∂x∂v∂y∂u∂y∂v).J_g(u,v)= \begin{pmatrix} \dfrac{\partial x}{\partial u}& \dfrac{\partial x}{\partial v}\\[8pt] \dfrac{\partial y}{\partial u}& \dfrac{\partial y}{\partial v} \end{pmatrix}.
  1. [1]:Jacobian determinant

Jacobian determinant

The Jacobian determinant is

det⁡Jg(u,v)=∂(x,y)∂(u,v).\det J_g(u,v) =\frac{\partial(x,y)}{\partial(u,v)}.

Its absolute value is the local area-scaling factor:

dA=dx dy=∣det⁡Jg(u,v)∣ du dv.dA=dx\,dy =\left|\det J_g(u,v)\right|\,du\,dv.

The determinant is negative when the map reverses orientation. Area is non-negative, so the change-of-variables formula uses its absolute value.

To see why it measures area, take small increments Δu,Δv\Delta u,\Delta v:

g(u+Δu,v+Δv)−g(u,v)≈Jg(u,v)(ΔuΔv).g(u+\Delta u,v+\Delta v)-g(u,v) \approx J_g(u,v) \begin{pmatrix}\Delta u\\ \Delta v\end{pmatrix}.

The linear map Jg(u,v)J_g(u,v) sends a small coordinate rectangle of area Δu Δv\Delta u\,\Delta v to a parallelogram of area approximately ∣det⁡Jg(u,v)∣Δu Δv|\det J_g(u,v)|\Delta u\,\Delta v. Taking the limit gives the area element dA=∣det⁡Jg(u,v)∣ du dvdA=|\det J_g(u,v)|\,du\,dv.

  1. [1]:Jacobian matrix
  2. [2]:Euler's Basel sum
  3. [3]:Change of variables

On this page

Jacobian matrixJacobian determinant