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Mathematics for Physics

315 Derivation of Jacobian

\[ \begin{align*} J_{\mathbf F} &= \dfrac{\partial(F_1,\ldots,F_m)}{\partial(x_1,\ldots,x_n)} = \begin{pmatrix} \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n} \end{pmatrix} \end{align*} \]

\[ \begin{align*} \mathbf F &= \begin{pmatrix} F_1 \\ F_2 \\ \vdots \\ F_m \end{pmatrix} \\ \mathbf x &= \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} \end{align*} \]

Start from a vector-valued map and its input

where

  • \(\mathbf F\) is a vector-valued function.
  • \(F_i\) is an output component.
  • \(\mathbf x\) is the input vector.
  • \(x_j\) is an input coordinate.
  • \(m\) is the number of outputs.
  • \(n\) is the number of inputs.

\[ \begin{align*} \Delta\mathbf x &= \begin{pmatrix} \Delta x_1 \\ \Delta x_2 \\ \vdots \\ \Delta x_n \end{pmatrix} \end{align*} \]

Change the input by a small increment

\[ \begin{align*} \Delta\mathbf F &= \mathbf F(\mathbf x+\Delta\mathbf x)-\mathbf F(\mathbf x) \end{align*} \]

Form the corresponding output change

where

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

\[ \begin{align*} \Delta F_i &= F_i(\mathbf x+\Delta\mathbf x)-F_i(\mathbf x) \end{align*} \]

Write the change in one output component

\[ \begin{align*} \Delta F_i &\approx \sum_{j=1}^{n} \dfrac{\partial F_i}{\partial x_j}\Delta x_j \end{align*} \]

Keep the first-order Taylor increment

where

  • \(\Delta F_i\) is one output change.
  • \(\partial F_i/\partial x_j\) measures coordinate sensitivity.
  • Higher-order terms vanish faster than \(\Delta\mathbf x\).

\[ \begin{align*} \Delta F_1 &\approx \dfrac{\partial F_1}{\partial x_1}\Delta x_1 \\ &\quad +\cdots \\ &\quad +\dfrac{\partial F_1}{\partial x_n}\Delta x_n \\ \Delta F_m &\approx \dfrac{\partial F_m}{\partial x_1}\Delta x_1 \\ &\quad +\cdots \\ &\quad +\dfrac{\partial F_m}{\partial x_n}\Delta x_n \end{align*} \]

Repeat the increment for every output component

\[ \begin{align*} \begin{pmatrix} \Delta F_1 \\ \Delta F_2 \\ \vdots \\ \Delta F_m \end{pmatrix} &\approx \begin{pmatrix} \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n} \end{pmatrix} \begin{pmatrix} \Delta x_1 \\ \Delta x_2 \\ \vdots \\ \Delta x_n \end{pmatrix} \end{align*} \]

Collect the component increments into a matrix product

\[ \begin{align*} J_{\mathbf F} &= \begin{pmatrix} \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n} \end{pmatrix} \end{align*} \]

Name that matrix the Jacobian of F

where

  • \(J_{\mathbf F}\) is the Jacobian matrix.
  • Rows correspond to output components.
  • Columns correspond to input coordinates.
  • Each entry is a partial derivative.

\[ \begin{align*} \Delta\mathbf F &\approx J_{\mathbf F}\Delta\mathbf x \end{align*} \]

Read the first-order change as a linear map of the input increment

\[ \begin{align*} \mathbf F(x,y) &= \begin{pmatrix} F_x(x,y) \\ F_y(x,y) \end{pmatrix} \\ \Delta\mathbf x &= \begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix} \end{align*} \]

Specialize to two input and two output coordinates

\[ \begin{align*} \Delta F_x &\approx \dfrac{\partial F_x}{\partial x}\Delta x+\dfrac{\partial F_x}{\partial y}\Delta y \\ \Delta F_y &\approx \dfrac{\partial F_y}{\partial x}\Delta x+\dfrac{\partial F_y}{\partial y}\Delta y \end{align*} \]

Write the two first-order component changes

\[ \begin{align*} \begin{pmatrix} \Delta F_x \\ \Delta F_y \end{pmatrix} &\approx \begin{pmatrix} \dfrac{\partial F_x}{\partial x} & \dfrac{\partial F_x}{\partial y} \\ \dfrac{\partial F_y}{\partial x} & \dfrac{\partial F_y}{\partial y} \end{pmatrix} \begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix} \end{align*} \]

Collect those increments into a two-by-two product

\[ \begin{align*} J_{\mathbf F} &= \begin{pmatrix} \dfrac{\partial F_x}{\partial x} & \dfrac{\partial F_x}{\partial y} \\ \dfrac{\partial F_y}{\partial x} & \dfrac{\partial F_y}{\partial y} \end{pmatrix} \end{align*} \]

Read off the two-dimensional Jacobian

\[ \begin{align*} \det J_{\mathbf F} &= \begin{vmatrix} \dfrac{\partial F_x}{\partial x} & \dfrac{\partial F_x}{\partial y} \\ \dfrac{\partial F_y}{\partial x} & \dfrac{\partial F_y}{\partial y} \end{vmatrix} \end{align*} \]

Form the Jacobian determinant when the dimensions match

\[ \begin{align*} \det J_{\mathbf F} &= \dfrac{\partial F_x}{\partial x}\dfrac{\partial F_y}{\partial y} - \dfrac{\partial F_x}{\partial y}\dfrac{\partial F_y}{\partial x} \end{align*} \]

Expand the two-by-two determinant

where

  • \(\det J_{\mathbf F}\) is the Jacobian determinant.
  • Its magnitude gives local area scaling.
  • Its sign gives orientation change.

\[ \begin{align*} d\mathbf x_1 &= \begin{pmatrix} dx \\ 0 \end{pmatrix} \\ d\mathbf x_2 &= \begin{pmatrix} 0 \\ dy \end{pmatrix} \end{align*} \]

Take the two coordinate edge displacements

\[ \begin{align*} dA &= dx\,dy \end{align*} \]

Form the original area they span

\[ \begin{align*} d\mathbf F_1 &= J_{\mathbf F}\,d\mathbf x_1 \\ d\mathbf F_2 &= J_{\mathbf F}\,d\mathbf x_2 \end{align*} \]

Map those edges by the Jacobian

\[ \begin{align*} d\mathbf F_1 &= \begin{pmatrix} \dfrac{\partial F_x}{\partial x}dx \\ \dfrac{\partial F_y}{\partial x}dx \end{pmatrix} \\ d\mathbf F_2 &= \begin{pmatrix} \dfrac{\partial F_x}{\partial y}dy \\ \dfrac{\partial F_y}{\partial y}dy \end{pmatrix} \end{align*} \]

Write the images of the two edges

\[ \begin{align*} dA' &= \begin{vmatrix} \dfrac{\partial F_x}{\partial x}dx & \dfrac{\partial F_x}{\partial y}dy \\ \dfrac{\partial F_y}{\partial x}dx & \dfrac{\partial F_y}{\partial y}dy \end{vmatrix} \end{align*} \]

Compute the signed area of the image parallelogram

\[ \begin{align*} dA' &= \det(J_{\mathbf F})\,dA \end{align*} \]

Factor out the original area

Hence,

\[ \begin{align*} J_{\mathbf F} &= \dfrac{\partial(F_1,\ldots,F_m)}{\partial(x_1,\ldots,x_n)} = \begin{pmatrix} \dfrac{\partial F_1}{\partial x_1} & \cdots & \dfrac{\partial F_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \dfrac{\partial F_m}{\partial x_1} & \cdots & \dfrac{\partial F_m}{\partial x_n} \end{pmatrix} \end{align*} \]

315.1 References

  1. Hubbard, J. H., & Hubbard, B. B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Prentice Hall, 1998, §1.7, pp. 107–112.

  2. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering, 3rd ed. Cambridge University Press, 2006, §5.7, pp. 160–161.

  3. Riley, K. F., Hobson, M. P., & Bence, S. J. Mathematical Methods for Physics and Engineering, 3rd ed. Cambridge University Press, 2006, §6.4, pp. 199–206.