The derivative of a function tells you a great deal about how the function behaves. For a real-valued function of a single variable, the derivative - which is the slope of the tangent line - shows how fast the function is increasing or decreasing. For a function of several variables, the derivative is a linear transformation defined by the Jacobian matrix. You can apply the tools of linear algebra to the Jacobian to get a picture of the behavior of the function near a point.

Before describing the Jacobian, take a look at a simple example of a function of one variable.

Suppose you want to approximate f(x) at π/3. The best linear approximation to f(x) at this point is given by the function L(x), whose graph is the tangent line to f(x). L(x) is defined by 

(Note: The green wavy line under some variables indicates that a variable has been previously defined. That feature can be turned off in the Preferences dialog.)

For points x close to π/3, L(x) is approximately equal to f(x). For example,

The graph below shows the functions f(x) and L(x).

The Jacobian Matrix

In higher dimensions, things get more complicated. The derivative of a function of several variables is not a number - it is a linear transformation, defined by the matrix of partial derivatives. This matrix is called the Jacobian. Just as in the one-dimensional case, you can use the Jacobian matrix to compute the best linear approximation to the function near a point.

Here's how the Jacobian is defined. Suppose F(x) is a function from Rⁿ to R^m defined by the following m coordinate functions: 

The Jacobian of F is the m-by-n matrix of partial derivatives of the coordinate functions.

Here's an example involving a function from R² to R². First, set the global variable ORIGIN equal to 1, so that the numbering of the array entries for the function starts at 1.

For a vector

the function is defined as follows:

To insert the subscripts of x in the expressions above, press [ and then type the subscript in the placeholder that appears.

You can compute the Jacobian of F using the function Jacob.

For any value of x, the Jacobian is a 2-by-2 matrix. For example, if

is the zero vector - that is, the origin in R² - the Jacobian at a is

The derivative of F at a is the linear transformation from R² to R² defined as multiplication by J.

Just as for a function of one variable, the best linear approximation to F at a is the function

In this example, since F(a) happens to equal 0 and a is the zero vector, L(x) simplifies to

So, for any vector x close to the origin, F(x) is approximately equal to J·x. For example,

Eigenvalues and Eigenvectors of the Jacobian

You can use some basic tools of linear algebra to analyze the behavior of F near the origin. First, compute the eigenvalues of J, which are 2 and 1/2. 

The function eigenvec returns an eigenvector corresponding to the eigenvalue 2.

By definition, this means that J·ev2 = 2·ev2.

Now, multiply ev2 by a small scalar.

The resulting vector v is an eigenvector that is close to the origin.

But since v is close to the origin, F(v) is approximately equal to J·v, which equals 2·v.

This tells you that for any small eigenvector v, corresponding to eigenvalue 2, F(v) is approximately 2·v.

Similarly, for any small eigenvector w corresponding to eigenvalue 1/2, F(w) is approximately (1/2)·w.

The following graph shows the vectors v and w, and their images F(v) and F(w).

Since any vector can be written as a linear combination of the eigenvectors v and w, you now have a good picture of the behavior of F close to the origin.

Here's another example.

In this case, since the eigenvalues are complex numbers of length 1, the Jacobian corresponds to a rotation in the plane. The angle of rotation, in the counter-clockwise direction, is

So, for any vector v close to the origin, F(v) is approximately equal to v rotated by π/3.

As these examples show, the Jacobian is a powerful tool for studying the behavior of vector functions.