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What is the Jacobian?

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Illustration of area change
For a differentiable map f:U⊂Rn→Rmf:U\subset\mathbb{R}^n\to\mathbb{R}^m, the Jacobian matrix collects all its first partial derivatives. When m=nm=n, the determinant of this matrix is called the Jacobian determinant. Its absolute value gives the local area or volume scale factor, while its sign records whether orientation is preserved or reversed. The Jacobian takes its name from the German mathematician Carl Jacobi (1804–1851).
JACOBIAN MATRIX AND DETERMINANT
Let U⊂RnU\subset\mathbb{R}^n be open, and let f=(f1,…,fm):U→Rmf=(f_1,\ldots,f_m):U\to\mathbb{R}^m be differentiable. At a point a∈U\mathbf{a}\in U, the Jacobian matrix of ff is the m×nm\times n matrix
Jf(a)=(∂f1∂x1(a)⋯∂f1∂xn(a)⋮⋱⋮∂fm∂x1(a)⋯∂fm∂xn(a)).\begin{equation*}J_f(\mathbf{a})=\begin{pmatrix}\frac{\partial f_1}{\partial x_1}(\mathbf{a}) & \cdots & \frac{\partial f_1}{\partial x_n}(\mathbf{a}) \\\vdots & \ddots & \vdots \\\frac{\partial f_m}{\partial x_1}(\mathbf{a}) & \cdots & \frac{\partial f_m}{\partial x_n}(\mathbf{a})\end{pmatrix}.\end{equation*}
When m=nm=n, its determinant is the Jacobian determinant, written in the same notation used below for changes of variables:
∂(f1,…,fn)∂(x1,…,xn)(a)=det⁡Jf(a).\begin{equation*}\frac{\partial(f_1,\ldots,f_n)}{\partial(x_1,\ldots,x_n)}(\mathbf{a})=\det J_f(\mathbf{a}).\end{equation*}
JACOBIAN MATRIX AND DETERMINANT IN TWO DIMENSIONS
Let u=u(x,y)u=u(x,y) and v=v(x,y)v=v(x,y) be differentiable functions. We define the Jacobian matrix
(∂u∂x∂u∂y∂v∂x∂v∂y).\begin{equation*}\begin{pmatrix}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}.\end{equation*}
Its determinant, called the Jacobian determinant, is
∂(u,v)∂(x,y)=det⁡(∂u∂x∂u∂y∂v∂x∂v∂y).\begin{equation*}\frac{\partial(u,v)}{\partial(x,y)}=\det\begin{pmatrix}\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}\end{pmatrix}.\end{equation*}
Illustration of how area changes under the polar-coordinate mapping
Consider the polar-coordinate transformation x=rcos⁡θx=r\cos\theta, y=rsin⁡θy=r\sin\theta. Its Jacobian determinant is
∂(x,y)∂(r,θ)=det⁡(∂x∂r∂x∂θ∂y∂r∂y∂θ)=det⁡(cos⁡θ−rsin⁡θsin⁡θrcos⁡θ)=r(cos⁡2θ+sin⁡2θ)=r.\begin{equation*}\frac{\partial(x,y)}{\partial(r,\theta)} = \det\begin{pmatrix}\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\\frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta}\end{pmatrix} = \det\begin{pmatrix}\cos\theta & -r\sin\theta \\\sin\theta & r\cos\theta\end{pmatrix} = r(\cos^2\theta + \sin^2\theta) = r.\end{equation*}
Then the area transforms as
dx dy=∣∂(x,y)∂(r,θ)∣dr dθ=r dr dθ.\begin{equation*}dx\,dy = \left|\frac{\partial(x,y)}{\partial(r,\theta)}\right|dr\,d\theta = r\,dr\,d\theta.\end{equation*}
For the unit disk D={(x,y):x2+y2≤1}D=\{(x,y):x^2+y^2\le 1\}, this gives its area directly:
∬D1 dx dy=∫02π∫01r dr dθ=∫02π12 dθ=π.\begin{equation*}\iint_D 1\,dx\,dy = \int_0^{2\pi}\int_0^1 r\,dr\,d\theta = \int_0^{2\pi}\frac{1}{2}\,d\theta = \pi.\end{equation*}

References

  • [HubbardHubbard2015]Hubbard, John H., and Barbara Burke Hubbard. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Fifth edition. Matrix Editions, 2015.
  • [StewartEtAl2021]Stewart, James, Daniel K. Clegg, and Saleem Watson. Calculus: Early Transcendentals, Metric Edition. Ninth edition. Cengage, 2021.