In a first course in calculus, many students encounter a image similar to the following:
Such an illustration highlights a key property of the single variable derivative: it’s the best linear approximation of a function at a point. For functions of more than one variable, the derivative exhibits this same characteristic, yet there is no obvious corresponding picture. What would an analogous visualization look like for a multivariable function?
For the past few weeks, I’ve been working towards a visualization of multivariable functions and their derivatives. Check out the end result here, or read on to hear about my process. I assume some knowledge of calculus and mathematical notation.
Visualizing vector fields in the plane
To make matters simple, I narrowed my focus to functions . The derivative of such a function is also a transform . Thus to build a visual representation of the derivative, I first needed a general purpose visualization of vector fields in the plane.
Consider a particular function given by
What does look like? A common representation of such functions selects a uniformly-spaced set of points in , and draws an arrow at each point representing the magnitude and direction of the vector field:
This is sometimes called a vector plot. For static mediums like a textbook or chalk board, this is a intuitive visualization of . However, I wanted to make use of additional visual techniques made possible with computer graphics. I came up with the following:
- Draw a line in .
- Consider the line as a discrete collection of points (a polyline).
- Apply a function to those points.
- Draw a new line through the transformed points.
Performing those steps on a grid of lines in with the function from above produces the following visual:
Neat! The transformed grid gives some sense of how deforms and stretches the Euclidean plane. As another example, the linear map given by
yields the following picture when applied to a grid around the origin:
As one might expect, the linear map sends linear subspaces to linear subspaces (straight lines to straight lines). The visualization has a rather vivid aesthetic—there is no curvature in the transformed result. This will be helpful for understanding the multivariable derivative, which is always a linear map.
To give the visualization more object constancy, I animate between the starting and ending states of each line as well. As a fun aside, the data of such an animation are computed by what mathematicians call a straight line homotopy; as a coder, this is a tween function of SVG
path elements. If you would like to peek under the hood of the visualization, be sure to check out the main d3.js-based drawing method here.
The multivariable derivative
Inspired by the common single variable visualization of the derivative, I wanted to plot functions alongside a derivative-based approximation function. What does this approximation look like in the multivariable case?
If it exists, the derivative of a multivariable function at a point is a linear function such that given by
approximates well near . More precisely, we require that
If such a linear transform exists, it is unique and is given by the Jacobian matrix of partial derivatives
Suppose we choose and compute for the particular given before. Plotting both and together (with in green) yields the following visual:
Immediately we can see the essential properties of the derivative: near the chosen point , the function closely approximates . Moreover, this approximation is linear; the grid transformed by consists only of straight lines, indicating that it is a linear function. Be sure to check out the full animated version of this visualization to see different functions at work!
Extending the visualization to complex functions
Conveniently, this visualization can also be adapted to visualizing functions . Just like functions of real numbers, complex functions can be differentiated and have an approximation function
where and is the derivative of .
Typically a point in the complex plane is written as for some . We could alternatively notate this point as , which looks just like a point in ! These points operate under a different arithmetic, but can be fed to the same visualization algorithm. In this case, we get a depiction of the points on the real-imaginary plane. Here, for example, is a visualization of the complex exponential function and its derivative:
Next up I’m planning a 3D-printable version of this same visualization. The idea is to perform similar deformation of a lattice in . More on this coming soon!