Section10.6Directional Derivatives and the Gradient
Motivating Questions
The partial derivatives of a function \(f\) tell us the rate of change of \(f\) in the direction of the coordinate axes. How can we measure the rate of change of \(f\) in other directions?
The partial derivatives of a function tell us the instantaneous rate at which the function changes as we hold all but one independent variable constant and allow the remaining independent variable to change. It is natural to wonder how we can measure the rate at which a function changes in directions other than parallel to a coordinate axes. In what follows, we investigate this question, and see how the rate of change in any given direction is connected to the rates of change given by the standard partial derivatives.
and suppose that \(f\) measures the temperature, in degrees Celsius, at a given point in the plane, where \(x\) and \(y\) are measured in feet. Assume that the positive \(x\)-axis points due east, while the positive \(y\)-axis points due north. A contour plot of \(f\) is shown in Figure 10.6.1
Suppose that a person is walking due east, and thus parallel to the \(x\)-axis. At what instantaneous rate is the temperature changing with respect to \(x\) at the moment the walker passes the point \((2,1)\text{?}\) What are the units on this rate of change?
Next, determine the instantaneous rate of change of temperature with respect to distance at the point \((2,1)\) if the person is instead walking due north. Again, include units on your result.
Now, rather than walking due east or due north, let’s suppose that the person is walking with velocity given by the vector \(\vv = \langle 3, 4 \rangle\text{,}\) where time is measured in seconds. Note that the person’s speed is thus \(| \vv | = 5\) feet per second. Find parametric equations for the person’s path; that is, parameterize the line through \((2,1)\) using the direction vector \(\vv = \langle 3, 4 \rangle\text{.}\) Let \(x(t)\) denote the \(x\)-coordinate of the line, and \(y(t)\) its \(y\)-coordinate. Make sure your parameterization places the walker at the point \((2,1)\) when \(t=0\text{.}\)
With the parameterization in (c), we can now view the temperature \(f\) as not only a function of \(x\) and \(y\text{,}\) but also of time, \(t\text{.}\) Hence, use the chain rule to determine the value of \(\frac{df}{dt}\bigm|_{t=0}\text{.}\) What are the units on your answer? What is the practical meaning of this result?
Given a function \(z = f(x,y)\text{,}\) the partial derivative \(f_x(x_0,y_0)\) measures the instantaneous rate of change of \(f\) as only the \(x\) variable changes; likewise, \(f_y(x_0,y_0)\) measures the rate of change of \(f\) at \((x_0,y_0)\) as only \(y\) changes. Note particularly that \(f_x(x_0,y_0)\) is measured in “units of \(f\) per unit of change in \(x\text{,}\)” and that the units on \(f_y(x_0,y_0)\) are similar.
In Preview Activity 10.6.1, we saw how we could measure the rate of change of \(f\) in a situation where both \(x\) and \(y\) were changing; in that activity, however, we found that this rate of change was measured in “units of \(f\) per unit of time.” In a given unit of time, we may move more than one unit of distance. In fact, in Preview Activity 10.6.1, in each unit increase in time we move a distance of \(| \vv | = 5\) feet. To generalize the notion of partial derivatives to any direction of our choice, we instead want to have a rate of change whose units are “units of \(f\) per unit of distance in the given direction.”
In this light, in order to formally define the derivative in a particular direction of motion, we want to represent the change in \(f\) for a given unit change in the direction of motion. We can represent this unit change in direction with a unit vector, say \(\vu = \langle u_1, u_2 \rangle\text{.}\) If we move a distance \(h\) in the direction of \(\vu\) from a fixed point \((x_0,y_0)\text{,}\) we then arrive at the new point \((x_0+u_1h, y_0+u_2h)\text{.}\) It now follows that the slope of the secant line to the curve on the surface through \((x_0,y_0)\) in the direction of \(\vu\) through the points \((x_0,y_0)\) and \((x_0+u_1h, y_0+u_2h)\) is
To get the instantaneous rate of change of \(f\) in the direction \(\vu = \langle u_1, u_2 \rangle\text{,}\) we must take the limit of the quantity in Equation (10.6.1) as \(h \to 0\text{.}\) Doing so results in the formal definition of the directional derivative.
Let \(f = f(x,y)\) be given. The derivative of \(f\) at the point \((x,y)\) in the direction of the unit vector \(\vu = \langle u_1, u_2 \rangle\) is denoted \(D_{\vu}f(x,y)\) and is given by
The quantity \(D_{\vu} f(x,y)\) is called a directional derivative. When we evaluate the directional derivative \(D_{\vu} f(x,y)\) at a point \((x_0, y_0)\text{,}\) the result \(D_{\vu} f(x_0,y_0)\) tells us the instantaneous rate at which \(f\) changes at \((x_0, y_0)\) per unit increase in the direction of the vector \(\vu\text{.}\) In addition, the quantity \(D_{\vu} f(x_0,y_0)\) tells us the slope of the line tangent to the surface in the direction of \(\vu\) at the point \((x_0,y_0,f(x_0,y_0))\text{.}\)
Subsection10.6.2Computing the Directional Derivative
In a similar way to how we developed shortcut rules for standard derivatives in single variable calculus, and for partial derivatives in multivariable calculus, we can also find a way to evaluate directional derivatives without resorting to the limit definition found in Equation (10.6.2). We do so using a very similar approach to our work in Preview Activity 10.6.1.
Suppose we consider the situation where we are interested in the instantaneous rate of change of \(f\) at a point \((x_0,y_0)\) in the direction \(\vu = \langle u_1, u_2 \rangle\text{,}\) where \(\vu\) is a unit vector. The variables \(x\) and \(y\) are therefore changing according to the parameterization
\begin{equation*}
x = x_0 + u_1t \ \ \ \ \ \text{ and } \ \ \ \ \ y = y_0 + u_2t.
\end{equation*}
Observe that \(\frac{dx}{dt} = u_1\) and \(\frac{dy}{dt} = u_2\) for all values of \(t\text{.}\) Since \(\vu\) is a unit vector, it follows that a point moving along this line moves one unit of distance per one unit of time; that is, each single unit of time corresponds to movement of a single unit of distance in that direction. This observation allows us to use the Chain Rule to calculate the directional derivative, which measures the instantaneous rate of change of \(f\) with respect to change in the direction \(\vu\text{.}\)
Note well: To use Equation (10.6.3), we must have a unit vector \(\vu = \langle u_1, u_2 \rangle\) in the direction of motion. In the event that we have a direction prescribed by a non-unit vector, we must first scale the vector to have length 1.
Use Equation (10.6.3) to determine \(D_{\vi} f(x,y)\) and \(D_{\vj} f(x,y)\text{.}\) What familiar function is \(D_{\vi} f\text{?}\) What familiar function is \(D_{\vj} f\text{?}\) (Recall that \(\vi\) is the unit vector in the positive \(x\)-direction and \(\vj\) is the unit vector in the positive \(y\)-direction.)
Use Equation (10.6.3) to find the derivative of \(f\) in the direction of the vector \(\vv = \langle 2, 3 \rangle\) at the point \((1,-1)\text{.}\) Remember that a unit direction vector is needed.
Via the Chain Rule, we have seen that for a given function \(f = f(x,y)\text{,}\) its instantaneous rate of change in the direction of a unit vector \(\vu = \langle u_1, u_2 \rangle\) is given by
we see that we may recast Equation (10.6.4) in a way that has geometric meaning. In particular, we see that \(D_{\vu}f(x_0,y_0)\) is the dot product of the vector \(\left\langle f_x(x_0,y_0), f_y(x_0,y_0) \right\rangle\) and the vector \(\vu\text{.}\)
For each of the following points \((x_0,y_0)\text{,}\) evaluate the gradient \(\nabla f(x_0,y_0)\) and sketch the gradient vector with its tail at \((x_0,y_0)\text{.}\) Some of the vectors are too long to fit onto the plot, but we’d like to draw them to scale; to do so, scale each vector by a factor of 1/4.
As a vector, \(\nabla f(x_0,y_0)\) defines a direction and a length. As we will soon see, both of these convey important information about the behavior of \(f\) near \((x_0,y_0)\text{.}\)
Remember that the dot product also conveys information about the angle between the two vectors. If \(\theta\) is the angle between \(\nabla
f(x_0,y_0)\) and \(\vu\) (where \(\vu\) is a unit vector), then we also have that
In particular, when \(\theta\) is a right angle, as shown on the left of Figure 10.6.4, then \(D_{\vu}f(x_0,y_0) = 0\text{,}\) because \(\cos(\theta) = 0\text{.}\) Since the value of the directional derivative is 0, this means that \(f\) is unchanging in this direction, and hence \(\vu\) must be tangent to the contour of \(f\) that passes through \((x_0,y_0)\text{.}\) In other words, \(\nabla f(x_0,y_0)\) is orthogonal to the contour through \((x_0,y_0)\text{.}\) This shows that the gradient vector at a given point is always perpendicular to the contour passing through the point, confirming that what we saw in part (c) of Activity 10.6.3 holds in general.
and therefore \(D_{\vu}f(x_0,y_0) > 0\text{,}\) as shown in the middle image in Figure 10.6.4. This means that \(f\) is increasing in any direction where \(\theta\) is acute. In a similar way, when \(\theta\) is an obtuse angle, then \(\cos(\theta) \lt 0\) so \(D_{\vu}f(x_0,y_0) \lt 0\text{,}\) as seen on the right in Figure 10.6.4. This means that \(f\) is decreasing in any direction for which \(\theta\) is obtuse.
Finally, as we can see in the following activity, we may also use the gradient to determine the directions in which the function is increasing and decreasing most rapidly.
In this activity we investigate how the gradient is related to the directions of greatest increase and decrease of a function. Let \(f\) be a differentiable function and \(\vu\) a unit vector.
Let \(\theta\) be the angle between \(\nabla f(x_0,y_0)\) and \(\vu\text{.}\) Use the relationship between the dot product and the angle between two vectors to explain why
At the point \((x_0,y_0)\text{,}\) the only quantity in Equation (10.6.6) that can change is \(\theta\) (which determines the direction \(\vu\) of travel). Explain why \(\theta = 0\) makes the quantity
When \(\theta = 0\text{,}\) in what direction does the unit vector \(\vu\) point relative to \(\nabla f(x_0,y_0)\text{?}\) Why? What does this tell us about the direction of greatest increase of \(f\) at the point \((x_0,y_0)\text{?}\)
State the unit vectors \(\vu\) and \(\vv\) (in terms of \(\nabla f(x_0,y_0)\)) that provide the directions of greatest increase and decrease for the function \(f\) at the point \((x_0,y_0)\text{.}\) What important assumption must we make regarding \(\nabla f(x_0,y_0)\) in order for these vectors to exist?
Having established in Activity 10.6.4 that the direction in which a function increases most rapidly at a point \((x_0,y_0)\) is the unit vector \(\vu\) in the direction of the gradient, (that is, \(\vu = \frac{1}{|\nabla f(x_0,y_0)|} \nabla f(x_0,y_0)\text{,}\) provided that \(\nabla f(x_0,y_0) \ne \vzero\)), it is also natural to ask, “in the direction of greatest increase for \(f\) at \((x_0,y_0)\text{,}\) what is the value of the rate of increase?” In this situation, we are asking for the value of \(D_{\vu} f(x_0,y_0)\) where \(\vu = \frac{1}{|\nabla f(x_0,y_0)|} \nabla f(x_0,y_0)\text{.}\)
Next, we recall two important facts about the dot product: (i) \(\vw\cdot (c \vv) = c (\vw \cdot \vv)\) for any scalar \(c\text{,}\) and (ii) \(\vw \cdot \vw = |\vw|^2\text{.}\) Applying these properties to the most recent equation involving the directional derivative, we find that
Finally, since \(\nabla f(x_0,y_0)\) is a nonzero vector, its length \(|\nabla f(x_0,y_0)|\) is a nonzero scalar, and thus we can simplify the preceding equation to establish that
Let \(f\) be a differentiable function and \((x_0,y_0)\) a point for which \(\nabla f(x_0,y_0) \ne \vzero\text{.}\) Then \(\nabla f(x_0,y_0)\) points in the direction of greatest increase of \(f\) at \((x_0,y_0)\text{,}\) and the instantaneous rate of change of \(f\) in that direction is the length of the gradient vector. That is, if \(\vu = \frac{1}{|\nabla f(x_0,y_0)|} \nabla f(x_0,y_0)\text{,}\) then \(\vu\) is a unit vector in the direction of greatest increase of \(f\) at \((x_0,y_0)\text{,}\) and \(D_{\vu} f(x_0,y_0) = |\nabla f(x_0,y_0)|\text{.}\)
Sketch the unit vector \(\vz = \left\langle-\frac1{\sqrt{2}},
-\frac1{\sqrt{2}}\right\rangle\) on Figure 10.6.5 with its tail at \((1,2)\text{.}\) Now find the directional derivative \(D_{\vz}f(1,2)\text{.}\)
Consider the vector \(\vv = \langle 2,-1\rangle\) and sketch \(\vv\) on Figure 10.6.5 with its tail at \((1,2)\text{.}\) Find a unit vector \(\vw\) pointing in the same direction of \(\vv\text{.}\) Without computing \(D_{\vw}f(1,2)\text{,}\) what do you know about the sign of this directional derivative? Now verify your observation by computing \(D_{\vw}f(1,2)\text{.}\)
Suppose you are standing at the point \((3,3)\text{.}\) In which direction should you move to cause \(f\) to increase as rapidly as possible? At what rate does \(f\) increase in this direction?
The gradient finds many natural applications. For example, situations often arise — for instance, constructing a road through the mountains or planning the flow of water across a landscape — where we are interested in knowing the direction in which a function is increasing or decreasing most rapidly.
For example, consider a two-dimensional version of how a heat-seeking missile might work.(This application is borrowed from United States Air Force Academy Department of Mathematical Sciences.) Suppose that the temperature surrounding a fighter jet can be modeled by the function \(T\) defined by
where \((x,y)\) is a point in the plane of the fighter jet and \(T(x,y)\) is measured in degrees Celsius. Some contours and gradients \(\nabla T\) are shown on the left in Figure 10.6.6.
A heat-seeking missile will always travel in the direction in which the temperature increases most rapidly; that is, it will always travel in the direction of the gradient \(\nabla T\text{.}\) If a missile is fired from the point \((2,4)\text{,}\) then its path will be that shown on the right in Figure 10.6.6.
In the final activity of this section, we consider several questions related to this context of a heat-seeking missile, and foreshadow some upcoming work in Section 10.7.
The temperature \(T(x,y)\) has its maximum value at the fighter jet’s location. State the fighter jet’s location and explain how Figure 10.6.6 tells you this.
Suppose that a different function \(f\) has a local maximum value at \((x_0,y_0)\text{.}\) Sketch the behavior of some possible contours near this point. What is \(\nabla f(x_0,y_0)\text{?}\) (Hint: What is the direction of greatest increase in \(f\) at the local maximum?)
Suppose that a function \(g\) has a local minimum value at \((x_0,y_0)\text{.}\) Sketch the behavior of some possible contours near this point. What is \(\nabla g(x_0,y_0)\text{?}\)
for those values of \(x\) and \(y\) for which the limit exists. In addition, \(D_{\vu}f(x,y)\) measures the slope of the graph of \(f\) when we move in the direction \(\vu\text{.}\) Alternatively, \(D_{\vu} f(x_0,y_0)\) measures the instantaneous rate of change of \(f\) in the direction \(\vu\) at \((x_0,y_0)\text{.}\)
At any point where the gradient is nonzero, gradient is orthogonal to the contour through that point and points in the direction in which \(f\) increases most rapidly; moreover, the slope of \(f\) in this direction equals the length of the gradient \(|\nabla f(x_0,y_0)|\text{.}\) Similarly, the opposite of the gradient points in the direction of greatest decrease, and that rate of decrease is the opposite of the length of the gradient.
Find the rate of change of the function \(f\) at the point (5, -5,3) in the direction \(\mathbf u = \langle -2/\sqrt{38}, -5/\sqrt{38}, -3/\sqrt{38} \rangle\text{.}\)
If \(f \left( x, y \right) = -4 x^{2} + 2 y^{2}\text{,}\) find the value of the directional derivative at the point \(\left( -3, -4 \right)\) in the direction given by the angle \(\theta = \frac{2 \pi}{1}\text{.}\)
Find the directional derivative of \(\displaystyle f(x,y,z) = 2xy+z^{2}\) at the point \((-2,1,2)\) in the direction of the maximum rate of change of \(f\text{.}\)
The temperature at a point (x,y,z) is given by \(\displaystyle
T(x,y,z) = 200e^{-x^2 -y^2/4 - z^2/9}\text{,}\) where \(T\) is measured in degrees Celsius and x,y, and z in meters. There are lots of places to make silly errors in this problem; just try to keep track of what needs to be a unit vector.
The concentration of salt in a fluid at \((x,y,z)\) is given by \(F(x,y,z) = 3x^{2}+4y^{4}+2x^{2}z^{2}\) mg/cm\({}^3\text{.}\) You are at the point \((1,1,-1)\text{.}\)
At a certain point on a heated metal plate, the greatest rate of temperature increase, 5 degrees Celsius per meter, is toward the northeast. If an object at this point moves directly north, at what rate is the temperature increasing?
Let \(E(x,y) = \frac{100}{1+(x-5)^2 + 4(y-2.5)^2}\) represent the elevation on a land mass at location \((x,y)\text{.}\) Suppose that \(E\text{,}\)\(x\text{,}\) and \(y\) are all measured in meters.
Let \(\vu\) be a unit vector in the direction of \(\langle -4,3 \rangle\text{.}\) Determine \(D_{\vu} E(3,4)\text{.}\) What is the practical meaning of \(D_{\vu} E(3,4)\) and what are its units?
Find the direction of greatest increase in \(f\) at the point \((1,2)\text{.}\) Explain. A graph of the surface defined by \(f\) is shown at left in Figure 10.6.7. Illustrate this direction on the surface.
The properties of the gradient that we have observed for functions of two variables also hold for functions of more variables. In this problem, we consider a situation where there are three independent variables. Suppose that the temperature in a region of space is described by
and that you are standing at the point \((1,2,-1)\text{.}\)
Find the instantaneous rate of change of the temperature in the direction of \(\vv=\langle 0, 1, 2\rangle\) at the point \((1,2,-1)\text{.}\) Remember that you should first find a unit vector in the direction of \(\vv\text{.}\)
