## Directional Derivatives Part 1: Gradients.

### May 17, 2010

Last time we talked about derivatives on surfaces. We noted that there were two main derivatives that we care about, and called the *partial derivative in the x-direction* and the *partial derivative in the y-direction*. To calculate the partial in the x-direction for some function of x and y, we simply make y a constant and derive with respect to x.

“But!…” you might begin, “But in calculus, we had a function’s graph and we had a graph of the derivative! Both were usually functions! Don’t we have anything like that for surfaces?”

Well, yes. We do. Sort of. Because, at every point, we have an x and y partial derivative, we can assign to each point of our surface an ordered pair of partial derivatives. Namely, at the point , we have the ordered pair of partial derivatives . Now, a relatively nice way to represent this kind of thing is as a **vector field**. A vector field is sort of like a regular graph, but for each point on the graph we assign a vector.

Here is an image (blatantly stolen from wikipedia) of a surface with its associated vector field below it.

Cool, but how do we do this in practice? It seems like a real drag to find out this vector thing at every point. It turns out to not be so bad, actually. In fact, there’s even a cool little symbol that we can use while we’re doing this.

We define the symbol by the following application to a function: . This is sort of obvious: we want the direction derivative vector field, so let’s take the partials at each point. Let’s do an example or two.

- Let . Then we have . If we want the associated vector at the point , we need only plug in this point into : we get .
- Let . Then we have . If we want the associated vector at the point , we need only plug in this point into : we get .

We call this taking the **gradient of f**, and this is a very important part of finding a derivative in any direction! The gradient of other functions is also defined in the same way: if we are given or some other function of some-number-of-variables, we can define the gradient by simply taking the partial derivatives in each of the main directions (for g, above, we’d take it in the x, y, z, a, b, and c direction) and make an ordered pair out of it.

Next post I’ll discuss directional derivatives some more. For now, here are some exercises!

**Exercises: **

Let’s let , , and .

- Find the gradient of f at the point .
- Find the gradient of g at the point .
- Find the gradient of h at the point .
- Suppose we’re given . What is the gradient of this function? In particular, what is the gradient of a function of one variable? Is this cool (y/n)?