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Jacobian Matrix#

Definition: Jacobian Matrix

Let \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be a real vector function.

If \(f\) is totally differentiable at \(\boldsymbol{p} \in \mathcal{D}\), then its Jacobian matrix at \(\boldsymbol{p}\) is the matrix representation of its total derivative there with respect to the standard bases of \(\mathbb{R}^m\) and \(\mathbb{R}^n\).

Notation

\[Df(\boldsymbol{p}) \qquad J_f(\boldsymbol{p}) \qquad \boldsymbol{J}_f(\boldsymbol{p})\]

Definition: Regularity

Let \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be a real vector function and let \(\boldsymbol{p}\) be an interior point of \(\boldsymbol{p}\).

We say that \(f\) is regular if it is totally differentiable at \(\boldsymbol{p}\) and its Jacobian matrix there is invertible.

Theorem: Jacobian via Partial Derivatives

Let \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be a real vector function.

If \(f\) is totally differentiable at \(\boldsymbol{p} \in \mathcal{D}\), then its Jacobian matrix \(J_f(\boldsymbol{p})\) is given by the partial derivatives of \(f\)'s component functions \(f_1, \dotsc, f_n\) as follows:

\[J_f(\boldsymbol{p}) = \begin{bmatrix}\partial_1 f_1(\boldsymbol{p}) & \cdots & \partial_m f_1(\boldsymbol{p}) \\ \vdots & \ddots & \vdots \\ \partial_1 f_n(\boldsymbol{p}) & \cdots & \partial_m f_n(\boldsymbol{p})\end{bmatrix}\]
Example

Consider the real vector function \(f: \mathbb{R}^2 \to \mathbb{R}^3\) defined as follows:

\[f\left(x, y\right) = \begin{bmatrix} x^2 y \\ \sin y \\ x^2 + y^2\end{bmatrix}\]

It is totally differentiable on \(\mathbb{R}^2\) with the following Jacobian matrix:

\[J_f\left(x, y\right) = \begin{bmatrix}2xy & x^2 \\ 0 & \cos y \\ 2x & 2y\end{bmatrix}\]
Example: \(f(\boldsymbol{x}) = \boldsymbol{A} \boldsymbol{x}\)

Consider the real vector function \(f: \mathbb{R}^m \to \mathbb{R}^n\) defined as

\[f(\boldsymbol{x}) = \boldsymbol{A}\boldsymbol{x}\]

for some fixed real matrix \(\boldsymbol{A} \in \mathbb{R}^{n\times m}\). It is totally differentiable on \(\mathbb{R}^m\) with the following Jacobian matrix:

\[J_f(\boldsymbol{x}) = \boldsymbol{A}\]
Example

Let \(f: \mathbb{R}^n \to \mathbb{R}\) be a real scalar field which is twice totally differentiable.

The Jacobian matrix of \(f\)'s gradient is \(f\)'s Hessian matrix:

\[J_{\nabla f}(\boldsymbol{x}) = H_f (\boldsymbol{x})\]
Example: Polar Coordinate Transformation

Consider the coordinate transformation \(f: [0,+\infty)\times [0, 2\uppi) \to \mathbb{R}^2\) from polar coordinates:

\[f\left(\rho, \varphi\right) = \begin{bmatrix} \rho \cos \varphi \\ \rho \sin \varphi \end{bmatrix}\]

It is totally differentiable on \((0, +\infty) \times (0, 2 \uppi)\) with the following Jacobian matrix:

\[J_f\left(\rho, \varphi\right) = \begin{bmatrix}\cos \varphi & - \rho \sin \varphi \\ \sin \varphi & \rho \cos \varphi\end{bmatrix}\]
Example: Cylindrical Coordinate Transformation

Consider the coordinate transformation \(f: [0,+\infty)\times [0, 2\uppi) \times \mathbb{R} \to \mathbb{R}^3\) of cylindrical coordinates:

\[f\left(\rho, \varphi, z \right) = \begin{bmatrix} \rho \cos \varphi \\ \rho \sin \varphi \\ z\end{bmatrix}\]

It is totally differentiable on \((0, +\infty) \times (0, 2 \uppi) \times \mathbb{R}\) with the following Jacobian matrix:

\[J_f\left(\rho, \varphi, z\right) = \begin{bmatrix}\cos \varphi & - \rho \sin \varphi & 0 \\ \sin \varphi & \rho \cos \varphi & 0 \\ 0 & 0 & 1\end{bmatrix}\]
Example: Spherical Coordinate Transformation

Consider the coordinate transformation \(f: [0,+\infty)\times [0, \uppi] \times [0, 2\uppi) \to \mathbb{R}^3\) of spherical coordinates:

\[f\left(r, \theta, \varphi\right) = \begin{bmatrix} r \sin \theta \cos \varphi \\ r \sin \theta \sin \varphi \\ r \cos \theta\end{bmatrix}\]

It is totally differentiable on \((0,+\infty)\times (0, \uppi) \times (0, 2\uppi)\) with the following Jacobian matrix:

\[J_f\left(r, \theta, \varphi\right) = \begin{bmatrix}\sin \theta \cos \varphi & r \cos \theta \cos \varphi & -r \sin \theta \sin \varphi \\ \sin \theta \sin \varphi & r \cos \theta \sin \varphi & r \sin \theta \cos \varphi \\ \cos \theta & - r \sin \theta & 0\end{bmatrix}\]
Proof

TODO

Theorem: Jacobian via Gradients

Let \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be a real vector function.

If \(f\) is totally differentiable at \(\boldsymbol{p} \in \mathcal{D}\), then the rows of its Jacobian matrix \(J_f(\boldsymbol{p})\) are the gradients of \(f\)'s component functions \(f_1, \dotsc, f_n\):

\[J_f(\boldsymbol{p}) = \begin{bmatrix} - & (\nabla f_1 (\boldsymbol{p}))^\mathsf{T} & - \\ - & \vdots & - \\ - & (\nabla f_n(\boldsymbol{p}))^\mathsf{T} & - \end{bmatrix}\]
Proof

TODO

Theorem: Jacobian of Linear Combination

Let \(f:\mathcal{D}_f \subseteq \mathbb{R}^m \to \mathbb{R}^n\) and \(g: \mathcal{D}_g \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be real vector functions.

If \(f\) and \(g\) are totally differentiable at \(\boldsymbol{p} \in \mathcal{D}_f \cap \mathcal{D}_g\), then the Jacobian matrix of \(\lambda f + \mu g\) is given by the Jacobian matrices of \(f\) and \(g\) as

\[J_{\lambda f + \mu g}(\boldsymbol{p}) = \lambda J_{f}(\boldsymbol{p}) + \mu J_g (\boldsymbol{p})\]

for all \(\lambda, \mu \in \mathbb{R}\).

Proof

TODO

Theorem: Chain Rule for Jacobian Matrices

Let \(g: \mathcal{D}_g \subseteq \mathbb{R}^m \to \mathbb{R}^n\) and \(f: \mathcal{D}_f \subseteq \mathbb{R}^n \to \mathbb{R}^p\) be real vector functions.

If \(g\) is totally differentiable at \(\boldsymbol{p} \in \mathcal{D}_g\) and \(f\) is totally differentiable at \(g(\boldsymbol{p}) \in \mathcal{D}_f\), then the Jacobian matrix of the composition \(f \circ g\) is given by the matrix product of the Jacobian matrices of \(f\) and \(g\) as follows:

\[J_{f \circ g}(\boldsymbol{p}) = J_{f}(g(\boldsymbol{p}))J_g(\boldsymbol{p})\]
Proof

TODO

Mean Value Inequality

Let \(f: \mathcal{D} \subseteq \mathbb{R}^m \to \mathbb{R}^n\) be a real vector function and let \(\boldsymbol{a}, \boldsymbol{b} \in \mathcal{D}\) such that \(L = \{\boldsymbol{a} + t(\boldsymbol{b} - \boldsymbol{a}) \mid t \in [0,1]\} \subseteq \mathcal{D}\).

If \(f\) is continuous on \(L\) and totally differentiable on \(\operatorname{int} L\), then

\[||f(\boldsymbol{b}) - f(\boldsymbol{a})|| \le \sup_{\boldsymbol{p} \in \operatorname{int} L} \{||J_f(\boldsymbol{p})||\} \, ||\boldsymbol{b} - \boldsymbol{a}||,\]

where the matrix norm can be induced by any vector norms on \(\mathbb{R}^m\) and \(\mathbb{R}^n\).

Proof

TODO