Functions as arguments (Operators)

Notice that if you have a function of a single variable, Mathematica knows what you mean by:

Partial derivatives are also pretty straightforward:

Sometimes, it is useful to define a function that operates on other functions.  So, for example, I could define a function to find the derivative matrix for functions from →:

(Notice that I needed to transpose the matrix to get the rows/columns to work out like we want.  I used MatrixForm just because it looks cool.)  However, you have to be really careful with these.  For example, if I just make one minor little change when calling it:

The problem is that, when we defined our function, we assumed that our variables would be called x and y, so any other choice of variables gets interpreted as a constant.  One way to fix this would be to explicitly specify your independent variables in the definition:

This now works, though at the cost of a bit of redundancy in calling the function.  (There are fancier tricks we could use to generalize this, but this is a good starting point.)

If you need to access one of the components of a vector, you can use the notation:

Problem

Define your own functions to compute the gradient (call it myGrad) and divergence (call it myDiv) of an arbitrary function of 3 variables.  (You may not use any of the built-in functions that handle these.)  Demonstrate that each of these actually work for a few different functions.

Created by Mathematica  (February 28, 2007)