Understanding Jacobians
The Jacobian Matrix
Consider a function that maps reals to reals,
Let’s try expanding this concept to vector spaces. For a function
What do we do when we have a function mapping vector spaces to vector spaces?
Consider the function
Let’s try to solve this by decomposing the function
If we collect all these approximations into a vector by representing the term
I’ve written the functions out in matrix form for clarity.
Here,
The Jacobian Matrix thus, is an analog of the gradient vector for functions that
map vector spaces to vector spaces. Everything that we can do using gradients
can be done in a more general form using the Jacobian Matrix. Consider the
condition for differentiability of a multivariate scalar function
here,
Relating Jacobian Matrices and Transformation Matrices
If you notice, the Jacobian Matrix need not be square; for the special case
Therefore, this acts like a linear transformation between the infinitesimal
elements in the space of
Scaling factor and the Jacobian Determinant
Recall that for any linear transformation, the determinant of the transformation gives us the scaling factor, that is the ratio of the change in ‘volume’ occupied by the vector. This is also known as a dilation transformation (because just the size is involved, without worrying about orientation).
I’ll provide a proof for the statement highlighted above in
In algebra terms, we get
The volume of a parallelepiped is given by the determinants of the 3 edge vectors,
and hence the volume of the transformed cube is
This is the main principle that allows us to use the Jacobian in multiple integrals while changing variables: it scales up or down the size of the area or volume element we are using proportionately to the change of variables.
References:
- What is the Jacobian Matrix, a good MSE thread on the Jacobian Matrix
- Jacobian Matrix and Determinant Wikipedia page, of course
- A Quora question on Jacobian matrices with another very nice answer
Personal comments:
I thoroughly enjoyed writing so much ‘hard’ mathematics on this blog after a long time (last proper math post was on 11th September of Last year, and other math notes in the interim were published on the MathNotes site). A lot of calc textbooks don’t go into detail on Jacobians, instead just using them like a gift of god that fell out of the sky. The bare minimum they would provide would be a diagram of domain transformation, and the cliched example of converting to polar integrals (the disc is transformed into a rectangle), but that would still not make intuitive sense: why the determinant? And why this weird matrix? were the questions that popped up in my head, and I hope I’ve done justice to those questions in this article.