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Sahithyan's S2
Sahithyan's S2 — Methods of Mathematics

Jacobian Matrix

A generalization of the derivative for vector-valued functions.

Jacobian matrix for f:Rn→Rmf: \mathbb{R}^n \to \mathbb{R}^m is defined as:

Jpf(x)=(∂f1∂x1∂f1∂x2⋯∂f1∂xn∂f2∂x1∂f2∂x2⋯∂f2∂xn⋮⋮⋱⋮∂fm∂x1∂fm∂x2⋯∂fm∂xn)\mathbf{J}_p f(\mathbf{x}) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \cdots & \frac{\partial f_m}{\partial x_n} \end{pmatrix}

Jacobian of a transformation

Jacobian of the transformation T:x=x(u,v),  y=(u,v)T: x=x(u,v),\;y=(u,v) is:

∂(x,y)∂(u,v)=det⁡[∂x∂u∂x∂v∂y∂u∂y∂v]=∂x∂u∂y∂v−∂x∂v∂y∂u\frac{\partial(x, y)}{\partial(u, v)} = \det \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \frac{\partial x}{\partial u} \frac{\partial y}{\partial v} - \frac{\partial x}{\partial v} \frac{\partial y}{\partial u}

Approximation

First order approximation of f(x)f(x) around x0x_0 can be obtained by the Jacobian matrix.

A(x‾)=f(x0‾)+JpTf(x0‾)(x‾−x0‾)A(\boldsymbol{\underline{x}}) = f (\boldsymbol{\underline{x_0}} ) + \mathbf{J}_p^T f(\boldsymbol{\underline{x_0}} )(\underline{\boldsymbol{x}} − \underline{\boldsymbol{x_0}} )