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Jacobian Matrix or Jacobi Matrix 📂Vector Analysis

Jacobian Matrix or Jacobi Matrix

Definition

Let a multivariable vector function f:D→Rm\mathbf{f} : D \to \mathbb{R}^{m} defined by D⊂RnD \subset \mathbb{R}^{n} be defined for each scalar function f1,⋯ ,fm:D→Rf_{1} , \cdots , f_{m} : D \to \mathbb{R} as follows:

f(x1,⋯ ,xn):=[f1(x1,⋯ ,xn)⋮fm(x1,⋯ ,xn)] \mathbf{f} ( x_{1} , \cdots , x_{n} ) : = \begin{bmatrix} f_{1} ( x_{1} , \cdots , x_{n} ) \\ \vdots \\ f_{m} ( x_{1} , \cdots , x_{n} ) \end{bmatrix}

It is called the Jacobian matrix of f\mathbf{f}.

Description

The following notation is also often used:

J=∂(f1,…fm)∂(x1,…,xn) J = \dfrac{\partial (f_{1}, \dots f_{m})}{\partial (x_{1}, \dots, x_{n})}

The Jacobian matrix of f\mathbf{f} is also represented by defining an operator DD that makes Df:=JD \mathbf{f} := J. The name Jacobian matrix comes from the 19th-century German mathematician Carl Gustav Jacob Jacobi, so it’s correct to write and read it as Jacobian matrix, but in fact, JJ is very often read as ‘Jacobian’.

It is also called the total derivative, referring to the derivative of a multivariable vector function. Therefore, if a Jacobian matrix exists for a multivariable function, it is said to be differentiable, and conversely, a differentiable function f:R→Rf : \mathbb{R} \to \mathbb{R} can be thought of as having a Jacobian matrix of size 1×11 \times 1. In simple terms, the Jacobian matrix is the matrix of the derivative of a vector function.

Typically, it is first encountered in calculus together with polar coordinates, where

∫B∫Af(x,y)dxdy \int_{B} \int_{A} f(x,y) dx dy

if you change the Cartesian coordinates used to x=rcos⁡θx= r \cos \theta, y=rsin⁡θy= r \sin \theta as well known,

∫B∫Af(rcos⁡θ,rsin⁡θ)rdrdθ \int_{B} \int_{A} f( r \cos \theta , r \sin \theta ) r dr d \theta

an additional rr is attached as follows. This is because

[∂x∂r∂x∂θ∂y∂r∂y∂θ]=[cos⁡θsin⁡θ−rsin⁡θrcos⁡θ] \begin{bmatrix} {{\partial x } \over {\partial r }} & {{\partial x } \over {\partial \theta }} \\ {{\partial y } \over {\partial r }} & {{\partial y } \over {\partial \theta }} \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ -r \sin \theta & r \cos \theta \end{bmatrix}

the determinant of the matrix is derived as rcos⁡2θ+rsin⁡2θ=rr \cos^2 \theta + r \sin^2 \theta = r. In the same sense, the Jacobian matrix is essentially the same concept already encountered in high school when doing variable substitution for integration. For example,

∫01(27x3+9x2+3x)dx \int_{0}^{1} ( 27x^3 + 9 x^2 + 3 x ) dx

consider doing the substitution like 3x=y3x = y when calculating. If this is viewed as yy being the function y(x)=3xy(x) = 3x of xx, then its Jacobian matrix is

[∂3x∂x]=[3] \begin{bmatrix} {{\partial 3x } \over {\partial x }} \end{bmatrix} = \begin{bmatrix} 3 \end{bmatrix}

This is the same as differentiating both sides of 3x=y3x = y with respect to their variables to obtain 3dx=dy3dx = dy.

See Also