Theory of Jacobians in 2D

Foundational Definitions

We have seen that the Jacobian matrix for a transformation

F:  x(u,v), y(u,v)

 from R² to R² is defined by

This matrix is seen as a transformation  from R² to R² that is the best linear approximation of the transformation F.  When working with Jacobians, we usually write a point as a column vector.  For example, instead of writing the point (2,3) we write it as

In general we define

It is easier to understand this by looking at an example.

Example

Let F be the transformation defined by

x(u,v)  =  u² - 3uv,   y(u,v)  =  u³ +5v²

Find J_F evaluated at the point (-1,2).

Solution

First find the Jacobian by calculating the partial derivatives.

Now plug in (-1,2) to get:

Finally, multiply by the column vector:

Addition and Scalar Multiplication of Jacobians and Matrices

Let F and G both be differentiable transformations from R² to R² and let a be a real number.  Then if we think of a point (x,y) as a vector, we can define the sum of the two transformations as follows.

and

This is just doing the obvious thing as we see in the following example.

Example Let

F:  x(u,v) = u - 2v,    y(u,v) = u⁴ - u²v

G:  x(u,v) = u²,    y(u,v) = 2uv - v

Then

F + G:  x(u,v) = u - 2v + u² ,    y(u,v) = u⁴ - u²v + 2uv - v

and

5F:  x(u,v) = 5u - 10v,    y(u,v) = 5u⁴ - 5u²v

We define addition and scalar multiplication of 2x2 matrices and follows.  Let a be a real number and let

Then

Example

Several properties of Jacobians follow from properties of matrices.  The proofs are left as exercises.

Properties of Jacobians

Let F and G be differentiable transformations from R² to R² and let a be a real number.  Then

1. J_F+G  = J_F + J_G
   
2. J_aF  =  aJ_F

Multiplication of Matrices and Jacobians

Unlike scalar multiplication and addition of matrices, multiplication of two matrices is not as intuitive.  Although the definition looks contrived, it is just what is needed to describe the Jacobian of the composition of transformations.  To multiply two matrices, we think of the first matrix as a collection of row vectors and the second matrix as a collection of column vectors.  Then the ij^th entry of the product matrix is the dot product of the i^th row vector of the first matrix with the j^th column vector of the second matrix.  This gives us the following definition. 

Definition of Matrix Multiplication

Let A and B be 2x2 matrices.  Then

Example  Let

Find AB

Solution

Now that we know how to multiply two matrices, we can state the main result about the composition of two transformations.

Theorem (Composition of Transformations)

Let F and G be two differentiable transformations  from R² to R² with Jacobian matrices J_F = A  and J_G = B  and let FoG be the transformation  from R² to R² that represents the composition of these two transformations.  Then

J_FoG  =  AB

We leave the proof for you to do as an exercise.

We can think of the composition of two transformations as follows.  Consider G as the transformation that takes (u,v) to (s,t) and F as the transformation that takes (s,t) to (x,y).  Then the composition transformation takes (u,v) to (x,y).  Be careful about the order of operations.  Since these are functions, writing FoG means do G first and then F.

Exercises

For Exercises 1-4, find J_F(P) for the given transformation F at the given point P.

1.  F:  x(u,v) = u² - 3v,     y(u,v) = 4uv,          P = (3,-2)

2.  F:  x(u,v) = ve^u,     y(u,v) = 2u + v²,          P = (0,5)

3.  F:  x(u,v) = ln(u + v),     y(u,v) = u^v,          P = (3,-2)

4.  F:  x(u,v) = sin(u - v),     y(u,v) = ucos(v),          P = (0,p/6)

For Exercises 5, 6, and 7, calculate the Jacobian of H in two ways.  First perform the arithmetic on the transformation and calculate the Jacobian of the result.  Then calculate the Jacobian of each and perform the arithmetic on the resulting matrices.

5.  H  =  F + G,     F: x(u,v) = 2u - 3v,     y(u,v) = uv²           G:  x(u,v) = u² - 3v²,     y(u,v) = 5u - v

6.  H  =  F - G,     F: x(u,v) = cos(uv),     y(u,v) = 3u - 4v           G:  x(u,v) = 6u - 2v,     y(u,v) = 5u²

7.  H  =  2F - 3G,     F: x(u,v) = 3ue^v,     y(u,v) = ln v           G:  x(u,v) = v - 2ue^v,     y(u,v) = e^u

For Exercises 8 and 9, calculate the Jacobian of H = F o G in two ways.  First find the composition of the two transformations and then calculate the Jacobian of the result.  Then find the Jacobian of each and multiply the two matrices.

8.  F: x(u,v) = u - 3v²,     y(u,v) = u³           G:  u(s,t) = s²e^t,     v(s,t) = 4s - t

9.  F: x(u,v) = 2uv,     y(u,v) = cos(u + 2v)           G:  u(s,t) = s/t,     v(s,t) = t²

10.  Let  F: x(u,v) = 3u - v,  y(u,v) = u²v³      G:  u(s,t) = 3t, v(s,t) = e^st      H:  s(m,n) = n², t(m,n) = m/n

          Demonstrate that the Jacobian of the composition of the three transformations is the product of the
          three Jacobians, that is show that J_FoGoH = J_F J_G J_H 

11.  Prove that J_F+G  = J_F + J_G  for any differentiable transformations F and G from R² to R².

12.  Prove that J_aF  = aJ_F  for any differentiable transformation F and real number a.

13.  Prove that J_aF+bG  = aJ_F + bJ_G  for any differentiable transformations F and G from R² to R² and real numbers a and b.

14.  Prove that J_FoG  =  J_F J_G for any differentiable transformations F and G from R² to R².