The gradient is the fundamental notion of a derivative for a function of several variables.
We have now learned much about the gradient vector. However, there are three things you must know about the gradient vector:
First: You must know how to compute the gradient vector. Remember given a function : This is a vector-valued function of variables. This means when you compute the gradient, you should express it as a vector!
Second: The gradient vector points in the initial direction of greatest increase for a function. Remember, the gradient vector of a function of variables is a vector that lives in . The gradient vector tells you how to immediately change the values of the inputs of a function to find the initial greatest increase in the output of the function. We can see this in the interactive below.
The gradient at each point shows you which direction to change the -values to get the greatest initial change in the -value.
Third: The gradient vector is orthogonal to level sets. In particular, given , the gradient vector is always orthogonal to the level curves . Moreover, given , is always orthogonal to level surfaces.
Given a function of several variables, say , the gradient, when evaluated at a point in the domain of , is a vector in . We can see this in the interactive below.
The gradient at each point is a vector pointing in the -plane. You compute the gradient vector, by writing the vector: You’ve done this sort of direct computation many times before. So now, try your hand at these puzzlers:
Given a function and point in , the gradient vector tells you which initial direction to leave the point in order to get the greatest increase in . Why is this so? Well, to compute the change in the output of a function when changing the inputs in a specific direction, we should use the directional derivative. Recall: To make this change as large as possible, must be the same direction as . Hence, it is the gradient vector that points in the initial direction of greatest increase for the function.
We can directly witness that the gradient vector points in the initial direction of greatest increase by looking at a differentiable function that is described by a table of values.
So far we have mostly talked about the direction of the gradient vector. Now let’s talk about the magnitude of the gradient vector. The magnitude of the gradient vector tells you “how fast” the function is increasing.
Now, stand back. We’re going to do some serious calculus. Just read, relax and enjoy.
Water is poured on the surface at . What path does it take as it flows downhill?
This implies so Now recall that the differentials , and , so we may write
Raising to the left-hand and right hand sides, we see
setting , we write We are so close to being done, , this is the path described in the -plane. Since the water started at the point , we can solve for : Thus the water follows the curve in the -plane.
- First, that the negative of the gradient points in the initial direction of greatest decrease.
- Second, is just to observe how the problem combines many aspects of calculus.
Now that we know gradient vectors point in the initial direction of the greatest increase of the function, let’s think about the geometry of the gradient vector. Previously we used the chain rule to show that the gradient vector is always orthogonal to level sets. The argument went like this: Suppose that a vector-valued function runs along a level surface for the surface . If we ask ourselves: “What is the change in as varies?” We must conclude that since the value of doesn’t change on the curve drawn by (remember, draws a level curve). On the other hand, by the chain rule: The vector is tangent to the curve drawn by , and putting the two equations above together we see so must be orthogonal to , and hence orthogonal to the curve drawn by .
The explanation we just gave is a good one, but let’s give one more. In this book, we are always thinking about differentiable functions. Remember, a function is differentiable if one can “zoom-in” and eventually the function will look like a plane. So let’s imagine that we’ve “zoomed-in” on a differentiable function and it looks like a plane. The contour plot of a plane looks like a bunch of parallel lines:
The fact that the gradient is always orthogonal to level surfaces is very powerful. In fact it gives new (easier!) solutions to old problems. Let’s use this fact to find a plane tangent to a surface.
Since this vector is normal to the surface, we can use it to find an implicit formula for the tangent plane to the surface by computing where and
Thus the equation of the plane tangent to the ellipsoid at is:
Now let’s see a more in-depth problem.
But we know that the -component of the vector above must be . So, write with me:
So is parallel to . Hence one formula for the plane is
To conclude, we will repeat ourselves: There are three things you must know about the gradient vector:
First: You must know how to compute the gradient vector. Second: The gradient vector points in the initial direction of greatest increase for a function. Third: The gradient vector is orthogonal to level sets.