Jacobians

  • Introduction

To get an idea about Jacobians, one requires knowledge of matrices, determinants, functions and partial differentiation.

I) Matrices and Determinants :

A matrix is a rectangular arrangement of numbers in {m} rows and {n} columns. When {m =n}, the matrix is known as a square matrix. Determinants are defined for square matrices. The simplest square matrix is

{A = \begin {bmatrix} a & b \\ c & d \end {bmatrix}, |A| = ad -bc}

One can expand a {3 \times 3} determinant using cofactors.

Note that Jacobians are determinants whose elements are partial derivatives.

II) Functions of Several Variables :

This is often the case in real world applications. For example, the temperature at a given point may be a function of space {(x,y,z)} and time {t}. So {T} depends on {4} variables.

III) Partial Differentiation : 

Read more here.

  • Intuitive Idea

Let {f} be a function of {x}. Then,by definition of differentiation,

{f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}}

By removing the limit, we can approximate the left hand side to right hand side.

{f(x + \Delta x) \approx f(x) + f'(x) \Delta x}

Now, instead of {x}, if we have a vector function {X}, then

{f(\underbrace{X}_{n \times 1} + \underbrace{\Delta X}_{n\times 1}) \approx}

{\underbrace{f(X)}_{m \times 1} + \underbrace{f'(X)}_{?} \underbrace{\Delta X}_{n \times 1}}

The {?} thing is the Jacobian. This means, while dealing with derivatives of functions of several variables, to get an approximation, one needs to differentiate each function w.r.t. each independent variable. This is what Jacobians are for.

  • Definition

Jacobians are termed as functional determinants. Let {u,v} be functions of independent variables {x,y}. Then the determinant

{\begin {vmatrix} \frac {\partial u}{\partial x} & \frac {\partial u}{\partial y} \\ \frac {\partial v}{\partial x} & \frac {\partial v}{\partial y}\end {vmatrix} = \begin {vmatrix} u_x & u_v \\ v_x & v_y \end {vmatrix}}

is the Jacobian, which is sometimes denoted by

{ J = \frac {\partial (u,v)}{\partial (x,y)}}

Similarly, for functions {u,v,w} of {x,y,z},

{\begin {vmatrix}u_x & u_y & u_z \\ v_x & v_y & v_z \\ w_x & w_y & w_z \end {vmatrix}}

While changing the coordinate system from one to another, Jacobians are useful. Thus,

{dxdydz = r^2 sin^2 (\theta) dr d \theta d \phi}

And {r^2 sin^2 (\theta)} is

{J = \frac {\partial (x,y,z)}{\partial (r, \theta, \phi)}}

  • Chain Rule

Let

{x = f_1 (u,v), \ y = f_2 (u,v) \ and \ u = g_1 (s,t), \ v = g_2 (s,t)}

Then,

{\frac {\partial (x,y)}{\partial (u,v)} \times \frac {\partial (u,v)}{\partial (s,t)} = \frac {\partial (x,y)}{\partial (s,t)}}

A corollary to this is,

{\frac {\partial (x,y)}{\partial (u,v)} \times \frac {\partial (u,v)}{\partial (x,y)}=1}

  • Jacobians of Implicit Functions

Let {u_1,u_2,u_3} be implicit functions of {x_1,x_2,x_3} such that following equations are satisfied :

{f_1 (u_1, u_2, u_3, x_1, x_2, x_3) = 0 , \ f_2 (u_1, u_2, u_3, x_1, x_2, x_3) = 0 , \ f_3 (u_1, u_2, u_3, x_1, x_2, x_3) = 0 }

Then, for {j =1,2,3},

{\frac {\partial f_j}{\partial x_j} + \sum \limits_{i = 1}^{3} \frac {\partial f_j}{\partial u_i} \frac {\partial u_i}{\partial x_j}}

Useful to solve problems :

{\frac {\partial (u_1, u_2)}{\partial (x_1, x_2)} = (-1)^2 \times \frac {\frac {\partial (f_1, f_2)}{\partial (x_1, x_2)}} {\frac {\partial (f_1, f_2)}{\partial (u_1, u_2)}}}

and

{\frac {\partial (u_1, u_2, u_3)}{\partial (x_1, x_2, x_2)} = (-1)^3 \times \frac {\frac {\partial (f_1, f_2, f_3)}{\partial (x_1, x_2, x_3)}} {\frac {\partial (f_1, f_2, f_3)}{\partial (u_1, u_2, u_3)}}}

  • Partial Derivatives of Implicit Functions

Let {u_1,u_2} be implicit functions of {x_1,x_2} such that

{f_1 (u_1, u_2, x_1, x_2) = 0 , \ f_2 (u_1, u_2, x_1, x_2) = 0}

Then,

{\frac {\partial u_1}{\partial x_1} = (-1) \times \frac {\frac {\partial (f_1,f_2)}{\partial (x_1, u_2)}}{\frac {\partial (f_1,f_2)}{\partial (u_1, u_2)}}}

Similarly, one can get {\frac {\partial u_1}{\partial x_2}} and other derivatives.

  • Functional Dependence

For functional dependence of {f_1 (x_1, x_2)} and {f_2(x_1,x_2)},

{\frac {\partial (f_1, f_2)}{\partial (x_1, x_2)} = 0}

For dependence of 3 functions, one has to equate the {3 \times 3} Jacobian to {0}.

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