The Jacobian

What is the Jacobian?

The Jacobian of a function f is a row vector whose individual elements are partial derivatives of each variable in f.

$f(x, y, z) = x^2y + 3z$

The Jacobian is$J = [\frac{\delta f}{\delta x}, \frac{\delta f}{\delta y}, \frac{\delta f}{\delta z}]$ or $J = [2xy, x^2, 3]$in the current case.

What it means

The Jacobian vector points in the direction of the steepest slope of the function f.

Graphics

If we plot the Jacobians of a function, it will be something like this. Here x is the horizontal, y the vertical, and z the colors. The bright yellow represents the steepest slope and the dark blue the deepest valley.

Representing the 2D image into 3D gives a better view to see the peaks and valleys.

If we change the graph into a contour plot by drawing lines along the regions of the same height and overlay the Jacobians, we see that the directions of all Jacobians are pointing away from valleys and pointing towards peaks. The larger the Jacobian values are, the steeper the slope.

Jacobian matrices

If more than one variable is applied, we can build a matrix of Jacobian vectors on top of each other.

$u(x, y) = x - 2y \\ v(x, y) = - 2x + 3y$

$\large J_u = \begin{pmatrix} \frac{\delta u}{\delta x} & \frac{\delta u}{\delta y} \end{pmatrix}$

$\large J_v = \begin{pmatrix} \frac{\delta v}{\delta x} & \frac{\delta v}{\delta y} \end{pmatrix}$

$\Large J = \begin{pmatrix} \frac{\delta u}{\delta x} & \frac{\delta u}{\delta y} \\ \\ \frac{\delta v}{\delta x} & \frac{\delta v}{\delta y} \end{pmatrix}$

which gives us

$J = \begin{pmatrix} 1 & -2 \\ -2 & 3 \end{pmatrix}$ and if applied to (2, 3), the result is

$J_{(2, 3)} = \begin{pmatrix} 1 & -2 \\ -2 & 3 \end{pmatrix} \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} -4 \\ 5 \end{pmatrix}$

Jacobian applied to non-linear functions

The above example shows how Jacobian matrices can be applied to linear functions to find the slopes, but this does not mean that non-linear functions behave nicely. For more complex and non-linear functions, Jacobians can be still applied to small scales.

Applying to the transformation between Cartesian to polar coordinate systems

• small regions of space will scale

However, often they may still be smooth, which means that if we zoom in close enough, we can consider each little region of space to be approximately linear and therefore, by adding up all the contributions from the Jacobian determinants at each point in space, we can still calculate the change in the size of a region after transformation. A classic example of this occurs when transforming between cartesian and polar coordinate systems. So, if we have a vector expressed in terms of a radius r, and the angle up from the x-axis is theta, but we'd like them expressed in terms of x and y instead. We can write the following expressions just by thinking about trigonometry. Now, we can build the Jacobian matrix and take its determinant. The fact that the result is simply the radius r, and not the function theta, tells us that as we move along r, away from the origin, small regions of space will scale as a function of r, which I hope will make a lot of sense to you when we look at our little animation here.