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Mathematics for Physics

323 Jacobian

A matrix of first partial derivatives of a vector-valued map that is used to describe how small input changes produce output changes.

Jacobian matrix. For \(n\) inputs and \(m\) outputs, the Jacobian is the \(m\times n\) matrix of partial derivatives.

\[ J_{\mathbf{F}} = \begin{pmatrix} \dfrac{\partial F_{1}}{\partial x_{1}} & \cdots & \dfrac{\partial F_{1}}{\partial x_{n}} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial F_{m}}{\partial x_{1}} & \cdots & \dfrac{\partial F_{m}}{\partial x_{n}} \end{pmatrix} \]

where

  • \(\mathbf{F}\) is the vector-valued map.
  • \(F_{1},\ldots,F_{m}\) are the outputs.
  • \(x_{1},\ldots,x_{n}\) are the inputs.
  • \(J_{\mathbf{F}}\) is the Jacobian matrix.

Linear approximation. A small input change produces an approximate output change by multiplication with the Jacobian.

\[ \Delta\mathbf{F} \approx J_{\mathbf{F}}\,\Delta\mathbf{x} \]

where

  • \(\Delta\mathbf{x}\) is the input change.
  • \(\Delta\mathbf{F}\) is the output change.

Two-dimensional form. For two inputs and two outputs,

\[ J_{\mathbf{F}} = \begin{pmatrix} \dfrac{\partial F_{x}}{\partial x} & \dfrac{\partial F_{x}}{\partial y} \\ \dfrac{\partial F_{y}}{\partial x} & \dfrac{\partial F_{y}}{\partial y} \end{pmatrix} \]

Jacobian determinant. For equal numbers of inputs and outputs, the Jacobian has a determinant. In two dimensions,

\[ \det J_{\mathbf{F}} = \dfrac{\partial F_{x}}{\partial x}\dfrac{\partial F_{y}}{\partial y} - \dfrac{\partial F_{x}}{\partial y}\dfrac{\partial F_{y}}{\partial x} \]

Area scaling. The absolute value of the determinant gives the factor by which infinitesimal areas are scaled. Its sign records orientation.

\[ dA' = \left|\det J_{\mathbf{F}}\right|\,dA \]

where

  • \(dA\) is the original infinitesimal area.
  • \(dA'\) is the transformed infinitesimal area.

Note: Also called the Jacobian matrix. The determinant of a square Jacobian matrix is called the Jacobian determinant.

323.1 References

  1. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 5th ed. Matrix Editions, 2015. §1.7 — Jacobian matrix as the matrix of partial derivatives of a map \(f:U\subset\mathbb{R}^{n}\to\mathbb{R}^{m}\).
  2. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 5th ed. Matrix Editions, 2015. §4.9–4.10 — Jacobian determinant when \(m=n\); local volume scaling by \(\lvert\det[D\phi(x)]\rvert\).
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists. 7th ed. Academic Press, 2013. §4.4 — Jacobian as the determinant of the partial-derivative matrix; \(d\tau=J\,du_{1}\,du_{2}\cdots\).
  4. Boas, M. L. Mathematical Methods in the Physical Sciences. 3rd ed. Wiley, 2005. Chapter 5, §4 — two-dimensional Jacobian determinant; \(dA=\lvert J\rvert\,ds\,dt\) and the three-dimensional volume factor.
  5. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering. 3rd ed. Cambridge University Press, 2006. Chapter 6 — Jacobian determinants in two and three dimensions; change of variables in multiple integrals.
  6. Kreyszig, E. Advanced Engineering Mathematics. 10th ed. Wiley, 2011. §10.3 — \(J=\partial(u,v)/\partial(x,y)\) and \(dx\,dy=\lvert J\rvert\,du\,dv\).
  7. Stewart, J., Clegg, D., & Watson, S. Calculus: Early Transcendentals. Section 15.9 — Jacobian determinant and change of variables in multiple integrals.