Section 13.6 Directional Derivatives
Partial derivatives give us an understanding of how a surface changes when we move in the and directions. We made the comparison to standing in a rolling meadow and heading due east: the amount of rise/fall in doing so is comparable to Likewise, the rise/fall in moving due north is comparable to The steeper the slope, the greater in magnitude
But what if we didn’t move due north or east? What if we needed to move northeast and wanted to measure the amount of rise/fall? Partial derivatives alone cannot measure this. This section investigates directional derivatives, which do measure this rate of change.
Subsection 13.6.1 Functions of Two Variables
We begin with a definition.
The partial derivatives and are defined with similar limits, but only or varies with not both. Here both and vary with a weighted determined by a particular unit vector This may look a bit intimidating but in reality it is not too difficult to deal with; it often just requires extra algebra. However, the following theorem reduces this algebraic load.
Theorem 13.6.2. Directional Derivatives.
Example 13.6.3. Computing directional derivatives.
- toward the point
- in the direction of
and - toward the origin.
Solution.
The surface is plotted in Figure 13.6.4, where the point is indicated in the -plane as well as the point which lies on the surface of We find that and and
- Let
be the unit vector that points from the point to the point as shown in the figure. The vector the unit vector in this direction is Thus the directional derivative of at in the direction of isThus the instantaneous rate of change in moving from the point on the surface in the direction of (which points toward the point ) is about Moving in this direction moves one steeply downward. - We seek the directional derivative in the direction of
The unit vector in this direction is Thus the directional derivative of at in the direction of isStarting on the surface of at and moving in the direction of (or ) results in no instantaneous change in -value. This is analogous to standing on the side of a hill and choosing a direction to walk that does not change the elevation. One neither walks up nor down, rather just “along the side” of the hill. Finding these directions of “no elevation change” is important. - At
the direction towards the origin is given by the vector the unit vector in this direction is The directional derivative of at in the direction of the origin isMoving towards the origin means “walking uphill” quite steeply, with an initial slope of about
As we study directional derivatives, it will help to make an important connection between the unit vector that describes the direction and the partial derivatives and We start with a definition and follow this with a Key Idea.
Definition 13.6.5. Gradient.
To simplify notation, we often express the gradient as The gradient allows us to compute directional derivatives in terms of a dot product.
Key Idea 13.6.6. The Gradient and Directional Derivatives.
The properties of the dot product previously studied allow us to investigate the properties of the directional derivative. Given that the directional derivative gives the instantaneous rate of change of when moving in the direction of three questions naturally arise:
- In what direction(s) is the change in
the greatest (i.e., the “steepest uphill”)? - In what direction(s) is the change in
the least (i.e., the “steepest downhill”)? - In what direction(s) is there no change in
Using the key property of the dot product, we have
where is the angle between the gradient and (Since is a unit vector, ) This equation allows us to answer the three questions stated previously.
- Equation (13.6.1) is maximized when
i.e., when the gradient and have the same direction. We conclude the gradient points in the direction of greatest change. - Equation (13.6.1) is minimized when
i.e., when the gradient and have opposite directions. We conclude the gradient points in the opposite direction of the least change. - Equation (13.6.1) is 0 when
i.e., when the gradient and are orthogonal to each other. We conclude the gradient is orthogonal to directions of no change.
This result is rather amazing. Once again imagine standing in a rolling meadow and face the direction that leads you steepest uphill. Then the direction that leads steepest downhill is directly behind you, and side-stepping either left or right (i.e., moving perpendicularly to the direction you face) does not change your elevation at all.
Recall that a level curve is defined as a curve in the -plane along which the -values of a function do not change. Let a surface be given, and let’s represent one such level curve as a vector-valued function, As the output of does not change along this curve, for all for some constant
This last equality states the gradient is orthogonal to the derivative of meaning the gradient is orthogonal to the graph of Our conclusion: at any point on a surface, the gradient at that point is orthogonal to the level curve that passes through that point.
We restate these ideas in a theorem, then use them in an example.
Theorem 13.6.7. The Gradient and Directional Derivatives.
- The maximum value of
is the direction of maximal increase is - The minimum value of
is the direction of minimal increase is - At
is orthogonal to the level curve passing through
Example 13.6.8. Finding directions of maximal and minimal increase.
Let and let Find the directions of maximal/minimal increase, and find a direction where the instantaneous rate of change is 0.
Solution.
We begin by finding the gradient. and thus
Thus the direction of maximal increase is In this direction, the instantaneous rate of change is
Figure 13.6.9 shows the surface plotted from two different perspectives. In each, the gradient is drawn at with a dashed line (because of the nature of this surface, the gradient points “into” the surface). Let be the unit vector in the direction of at Each graph of the figure also contains the vector This vector has a “run” of 1 (because in the -plane it moves 1 unit) and a “rise” of hence we can think of it as a vector with slope of in the direction of helping us visualize how “steep” the surface is in its steepest direction.
The direction of minimal increase is in this direction the instantaneous rate of change is
Any direction orthogonal to is a direction of no change. We have two choices: the direction of and the direction of The unit vector in the direction of is shown in each graph of the figure as well. The level curve at is drawn: recall that along this curve the -values do not change. Since is a direction of no -change, this vector is tangent to the level curve at
Example 13.6.10. Understanding when .
Solution.
We find At we have According to Theorem 13.6.7, this is the direction of maximal increase. However, is directionless; it has no displacement. And regardless of the unit vector chosen,
Figure 13.6.11 helps us understand what this means. We can see that lies at the top of a paraboloid. In all directions, the instantaneous rate of change is 0.
So what is the direction of maximal increase? It is fine to give an answer of as this indicates that all directional derivatives are 0.
The fact that the gradient of a surface always points in the direction of steepest increase/decrease is very useful, as illustrated in the following example.
Example 13.6.12. The flow of water downhill.
Consider the surface given by the graph of Water is poured on the surface at What path does it take as it flows downhill?
Solution.
Let be the vector-valued function describing the path of the water in the -plane; we seek and We know that water will always flow downhill in the steepest direction; therefore, at any point on its path, it will be moving in the direction of (We ignore the physical effects of momentum on the water.) Thus will be parallel to and there is some constant such that
We find and write as and as Then
This implies
As equals both expressions, we have
To find an explicit relationship between and we can integrate both sides with respect to Recall from our study of differentials that Thus:
Now raise both sides as a power of
As the water started at the point we can solve for
Thus the water follows the curve in the -plane. The surface and the path of the water is graphed in Figure 13.6.13.(a). In Figure 13.6.13.(b), the level curves of the surface are plotted in the -plane, along with the curve Notice how the path intersects the level curves at right angles. As the path follows the gradient downhill, this reinforces the fact that the gradient is orthogonal to level curves.
Subsection 13.6.2 Functions of Three Variables
The concepts of directional derivatives and the gradient are easily extended to three (and more) variables. We combine the concepts behind Definitions 13.6.1 and 13.6.5 and Theorem 13.6.2 into one set of definitions.
Definition 13.6.14. Directional Derivatives and Gradient with Three Variables.
The same properties of the gradient given in Theorem 13.6.7, when is a function of two variables, hold for a function of three variables.
Theorem 13.6.15. The Gradient and Directional Derivatives with Three Variables.
- The maximum value of
is obtained when the angle between and is 0, i.e., the direction of maximal increase is - The minimum value of
is obtained when the angle between and is i.e., the direction of minimal increase is when and are orthogonal.
We interpret the third statement of the theorem as “the gradient is orthogonal to level surfaces,” the three-variable analogue to level curves.
Example 13.6.16. Finding directional derivatives with functions of three variables.
If a point source is radiating energy, the intensity at a given point in space is inversely proportional to the square of the distance between and That is, when for some constant
Let let be a unit vector, and let Measure distances in inches. Find the directional derivative of at in the direction of and find the direction of greatest intensity increase at
Solution.
We need the gradient meaning we need and Each partial derivative requires a simple application of the Quotient Rule, giving
The directional derivative tells us that moving in the direction of from results in a decrease in intensity of about units per inch. (The intensity is decreasing as moves one farther from the origin than )
The gradient gives the direction of greatest intensity increase. Notice that
That is, the gradient at is pointing in the direction of that is, towards the origin. That should make intuitive sense: the greatest increase in intensity is found by moving towards to source of the energy.
The directional derivative allows us to find the instantaneous rate of change in any direction at a point. We can use these instantaneous rates of change to define lines and planes that are tangent to a surface at a point, which is the topic of the next section.
Exercises 13.6.3 Exercises
Terms and Concepts
1.
What is the difference between a directional derivative and a partial derivative?
2.
3.
4.
The gradient is to level curves.
5.
The gradient points in the direction of increase.
6.
It is generally more informative to view the directional derivative not as the result of a limit, but rather as the result of a product.
Problems
Exercise Group.
Exercise Group.
In the following exercises, a function and a point are given. Find the directional derivative of in the indicated directions. Note: these are the same functions as in Exercises 13.6.3.7–13.6.3.12.
13.
(a)
In the direction of
(b)
In the direction toward the point
14.
15.
(a)
In the direction of
(b)
In the direction toward the point
16.
17.
(a)
In the direction of
(b)
In the direction toward the point
18.
Exercise Group.
In the following exercises, a function and a point are given. Investigate the directions of maximal increase and decrease, as indicated.
Note: these are the same functions and points as in Exercises 13–18.