1. Suppose we have a scalar function of several variables
Let be some unit vector. How does change in the direction?
We can consider this quantity as a function
What does this look like?
2. Lets restrict our attention to the smallest non-boring case: . Then we write . We have
Expanding this to first order in lets us write
But what does this look like? It’s simply
Example 1. What is the derivative of in the direction of at ?
Solution: We find the directional derivative is
Thus we have
We plug in ,
Then we evaluate to get
This gives us the directional derivative of .
3. We denote
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
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
We see its gradient is
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.