**1.** Suppose we have a scalar function of several variables

(1)

Let be some unit vector. How does change in the direction?

We can consider this quantity as a function

(2)

What does this look like?

**2.** Lets restrict our attention to the smallest non-boring case: . Then we write . We have

(3)

Expanding this to first order in lets us write

(4)

But what does this look like? It’s simply

(5)

**Example 1.** * What is the derivative of in the direction of at ?*

*
**
*Solution*: We find the directional derivative is *

(6)

We compute

(7)

and

(8)

Thus we have

(9)

We plug in ,

(10)

Then we evaluate to get

(11)

This gives us the directional derivative of .

**3.** We denote

(12)

and call it the **“Gradient”**. Note we will write interchangeably with the vector arrow , and they mean the same thing. The vector arrow doesn’t add anything semantically, it’s just different syntax.

The directional derivative is then

(13)

Note that the gradient acting on a scalar function produces a vector-valued function of several variables, but we can also take the dot product of the gradient with such a monstrosity.

**4. Question:** What is a vector-valued function of several variables?

For us, in practice, we think of this as a *Vector Field*: a “function” which assigns to each point a vector. Each vector-component is a function, usually smooth (i.e., infinitely differentiable).

(Again, just as we warned the reader with vectors, this too is a lie. A vector field is a bit more than just a function , it’s a more complicated beast which is studied further in differential geometry.)

**5. Meaning of Gradient.** Consider a family of level curves . The gradient points towards the direction of increasing . How can we see this? Well, consider the function

(14)

We see its gradient is

(15)

Lets draw a few level-curves and see what the vectors point to:

We see the vectors point towards .

**Exercise 1.** Consider the function defined by . What is its gradient?

**Exercise 2.** Let be defined by . Find its gradient.

**Exercise 3.** Let . Find its derivative in the direction at the point .

**Exercise 4.** Let . Find its gradient.

### Like this:

Like Loading...

*Related*

## About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.

Pingback: Lagrange Multipliers | My Math Blog