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Mathematics for Physics

314 Derivation of Jacobian

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

Vector Function. Let

\[ \mathbf F = \begin{pmatrix} F_1 \\ F_2 \\ \vdots \\ F_m \end{pmatrix} \]

depend on

\[ \mathbf x = \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}. \]

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.

Small Input Change. Change the input by

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

The corresponding output change is

\[ \Delta\mathbf F = \mathbf F(\mathbf x+\Delta\mathbf x) - \mathbf F(\mathbf x). \]

where

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

Component Change. Each output component changes according to

\[ \Delta F_i = F_i(\mathbf x+\Delta\mathbf x) - F_i(\mathbf x). \]

For a differentiable function, the first-order Taylor expansion gives

\[ \begin{aligned} \Delta F_i &= F_i(\mathbf{x}+\Delta\mathbf{x})-F_i(\mathbf{x}) \\ &= F_i(x_1+\Delta x_1,\ldots,x_n+\Delta x_n) - F_i(x_1,\ldots,x_n) \\ &\approx \sum_{j=1}^{n} \frac{\partial F_i}{\partial x_j}\Delta x_j \\ &= \frac{\partial F_i}{\partial x_1}\Delta x_1 + \frac{\partial F_i}{\partial x_2}\Delta x_2 + \cdots + \frac{\partial F_i}{\partial x_n}\Delta x_n. \end{aligned} \]

where

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

Writing this for every output gives

\[ \Delta F_1 \approx \frac{\partial F_1}{\partial x_1}\Delta x_1 + \cdots + \frac{\partial F_1}{\partial x_n}\Delta x_n, \]

\[ \Delta F_2 \approx \frac{\partial F_2}{\partial x_1}\Delta x_1 + \cdots + \frac{\partial F_2}{\partial x_n}\Delta x_n, \]

\[ \vdots \]

\[ \Delta F_m \approx \frac{\partial F_m}{\partial x_1}\Delta x_1 + \cdots + \frac{\partial F_m}{\partial x_n}\Delta x_n. \]

Matrix Form. These equations combine into

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

The matrix multiplying \(\Delta\mathbf x\) is the Jacobian matrix:

\[ J_{\mathbf F} = \begin{pmatrix} \dfrac{\partial F_1}{\partial x_1} & \dfrac{\partial F_1}{\partial x_2} & \cdots & \dfrac{\partial F_1}{\partial x_n} \\ \dfrac{\partial F_2}{\partial x_1} & \dfrac{\partial F_2}{\partial x_2} & \cdots & \dfrac{\partial F_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac{\partial F_m}{\partial x_1} & \dfrac{\partial F_m}{\partial x_2} & \cdots & \dfrac{\partial F_m}{\partial x_n} \end{pmatrix}. \]

where

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

Therefore,

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

The Jacobian is the linear transformation that gives the first-order change in \(\mathbf F\) from a small change in \(\mathbf x\).

Two Dimensions. Let

\[ \mathbf F(x,y) = \begin{pmatrix} F_x(x,y) \\ F_y(x,y) \end{pmatrix}. \]

A small displacement is

\[ \Delta\mathbf x = \begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix}. \]

The component changes are

\[ \Delta F_x \approx \frac{\partial F_x}{\partial x}\Delta x + \frac{\partial F_x}{\partial y}\Delta y \]

and

\[ \Delta F_y \approx \frac{\partial F_y}{\partial x}\Delta x + \frac{\partial F_y}{\partial y}\Delta y. \]

Thus,

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

Hence,

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

Jacobian Determinant. When the input and output dimensions are equal, the Jacobian matrix has a determinant.

For two dimensions,

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

Expanding gives

\[ \det J_{\mathbf F} = \frac{\partial F_x}{\partial x} \frac{\partial F_y}{\partial y} - \frac{\partial F_x}{\partial y} \frac{\partial F_y}{\partial x}. \]

where

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

Area Scaling. Consider the coordinate displacements

\[ d\mathbf x_1 = \begin{pmatrix} dx \\ 0 \end{pmatrix}, \qquad d\mathbf x_2 = \begin{pmatrix} 0 \\ dy \end{pmatrix}. \]

Their original area is

\[ dA = dx\,dy. \]

The Jacobian maps these displacements to

\[ d\mathbf F_1 = J_{\mathbf F}d\mathbf x_1 \]

and

\[ d\mathbf F_2 = J_{\mathbf F}d\mathbf x_2. \]

Therefore,

\[ d\mathbf F_1 = \begin{pmatrix} \dfrac{\partial F_x}{\partial x}dx \\ \dfrac{\partial F_y}{\partial x}dx \end{pmatrix}, \]

\[ d\mathbf F_2 = \begin{pmatrix} \dfrac{\partial F_x}{\partial y}dy \\ \dfrac{\partial F_y}{\partial y}dy \end{pmatrix}. \]

These vectors form a parallelogram. Its signed area is

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

Factoring out \(dx\,dy\) gives

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

Therefore,

\[ dA' = \det(J_{\mathbf F})\,dA. \]

Thus, the Jacobian matrix gives the first-order transformation, while its determinant gives the corresponding area or volume scaling.

Hence,

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

314.1 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.