Jacobian

| Main Article
| Discussion
| Related Articles  [?]
| Bibliography  [?]
| External Links  [?]
| Citable Version  [?]

|
|
|
|
|
|
|
|

This editable Main Article is under development and subject to a disclaimer.

[edit intro]

In mathematics, the Jacobi matrix is the matrix of first-order partial derivatives of the (vector-valued) function:

𝐟:ℝn->ℝm

(often f maps only from and to appropriate subsets of these spaces). The Jacobi matrix is m × n and consists of m rows of the first-order partial derivatives of f with respect to x1, ...,xn, respectively. This matrix is also known as the functional matrix of Jacobi. The determinant of the Jacobi matrix for n = m is known as the Jacobian. The Jacobi matrix and its determinant have several uses in mathematics:

* For m = 1, the Jacobi matrix appears in the second (linear) term of the Taylor series of f. Here the Jacobi matrix is 1 × n (the gradient of f, a row vector).

* The Jacobian appears as the weight (measure) in multi-dimensional integrals over generalized coordinates, i.e, over non-Cartesian coordinates.

* The inverse function theorem states that if m = n and f is continuously differentiable, then f is invertible in the neighborhood of a point x0 if and only if the Jacobian at x0 is non-zero.

The Jacobi matrix and its determinant are named after the German mathematician Carl Gustav Jacob Jacobi (1804 - 1851).

## Contents

* 1 Definition
* 1.1 Example
* 2 Coordinate transformation
* 3 Multiple integration
* 4 Geometric interpretation of the Jacobian
* 5 Reference

## Definition[edit]

Let f be a map of an open subset T of ℝn into ℝm with continuous first partial derivatives,

𝐟:T->ℝm.

That is if

𝐭=(t1,t2,…,tn)∈T⊂ℝn,

then

x1=f1(t1,t2,…,tn)x2=f2(t1,t2,…,tn)⋯⋯xm=fm(t1,t2,…,tn),

with

𝐱=(x1,x2,…,xm)∈ℝm.

The m × n functional matrix of Jacobi consists of partial derivatives

(∂f1∂t1∂f1∂t2…∂f1∂tn∂f2∂t1∂f2∂t2……⋱∂fm∂t1……∂fm∂tn).

The determinant (which is only defined for square matrices) of this matrix is usually written as (take m = n),

𝐉𝐟(𝐭)or∂(f1,f2,…,fn)∂(t1,t2,…,tn).

### Example[edit]

Let T be the subset {r, θ, φ | r > 0, 0 < θ<π, 0 <φ <2π} in ℝ3 and let f be defined by

x1≡x=f1(r,θ,ϕ)=rsin⁡θcos⁡ϕx2≡y=f2(r,θ,ϕ)=rsin⁡θsin⁡ϕx3≡z=f3(r,θ,ϕ)=rcos⁡θ

The Jacobi matrix is

(sin⁡θcos⁡ϕrcos⁡θcos⁡ϕ−rsin⁡θsin⁡ϕsin⁡θsin⁡ϕrcos⁡θsin⁡ϕrsin⁡θcos⁡ϕcos⁡θ−rsin⁡θ0)

Its determinant can be obtained most conveniently by a Laplace expansion along the third row

cos⁡θ|rcos⁡θcos⁡ϕ−rsin⁡θsin⁡ϕrcos⁡θsin⁡ϕrsin⁡θcos⁡ϕ|+rsin⁡θ|sin⁡θcos⁡ϕ−rsin⁡θsin⁡ϕsin⁡θsin⁡ϕrsin⁡θcos⁡ϕ|=r2(cos⁡θ)2sin⁡θ+r2(sin⁡θ)3=r2sin⁡θ

The quantities {r, θ, φ} are known as spherical polar coordinates and its Jacobian is r2sinθ.

## Coordinate transformation[edit]

Let T⊂ℝn. The map 𝐟:T->ℝn, is a coordinate transformation if (i) f has continuous first derivatives on T (ii) f is one-to-one on T and (iii) the Jacobian of f is not equal to zero on T.

## Multiple integration[edit]

It can be proved [1] that

∫𝐟(𝐭)ϕ(𝐱)d𝐱=∫Tϕ(𝐟(𝐭))𝐉𝐟(𝐭)d𝐭.

As an example we consider the spherical polar coordinates mentioned above. Here x = f(t) ≡ f(r, θ, φ) covers all of ℝ3, while T is the region {r > 0, 0 < θ<π, 0 <φ <2π}. Hence the theorem states that

∭ℝ3ϕ(𝐱)d𝐱=∫0∞∫0π∫02πϕ(𝐱(r,θ,ϕ))r2sin⁡θdrdθdϕ.

## Geometric interpretation of the Jacobian[edit]

The Jacobian has a geometric interpretation which we expound for the example of n = 3.

The following is a vector of infinitesimal length in the direction of increase in t1,

d𝐠1≡limΔt1->0𝐟(t1+Δt1,t2,t3)−𝐟(t1,t2,t3)Δt1Δt1=∂𝐟∂t1dt1

Similarly, we define

d𝐠2≡∂𝐟∂t2dt2,d𝐠3≡∂𝐟∂t3dt3

The scalar triple product of these three vectors gives the volume of an infinitesimally small parallelepiped,

dV=d𝐠1⋅(d𝐠2×d𝐠3)=∂𝐟∂t1⋅(∂𝐟∂t2×∂𝐟∂t3)dt1dt2dt3

The components of the first vector are given by

∂𝐟∂t1≡(∂x∂t1,∂y∂t1,∂z∂t1)≡(∂f1∂t1,∂f2∂t1,∂f3∂t1)

and similar expressions hold for the components of the other two derivatives. It has been shown in the article on the scalar triple product that

∂𝐟∂t1⋅(∂𝐟∂t2×∂𝐟∂t3)=|∂f1∂t1∂f2∂t1∂f3∂t1∂f1∂t2∂f2∂t2∂f3∂t2∂f1∂t3∂f2∂t3∂f3∂t3|≡∂(f1,f2,f3)∂(t1,t2,t3)≡𝐉𝐟(𝐭).

Note that a determinant is invariant under transposition (interchange of rows and columns), so that the transposed determinant being given is of no concern. Finally.

dV=∂(f1,f2,f3)∂(t1,t2,t3)dt1dt2dt3≡𝐉𝐟(𝐭)d𝐭.

## Reference[edit]

1. ↑ T. M. Apostol, Mathematical Analysis, Addison-Wesley, 2nd ed. (1974), sec. 15.10