Three things about the gradient vector

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 $F:\R ^n\to \R$: $$\grad F = \vector {\pp [F]{x_1},\pp [F]{x_2},\dots ,\pp [F]{x_n}}$$ This is a vector-valued function of $n$ 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 $n$ variables is a vector that lives in $\R ^n$. 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 $(x,y)$-values to get the greatest initial change in the $z$-value.

Third: The gradient vector is orthogonal to level sets. In particular, given $F:\R ^2\to \R$, the gradient vector $\grad F\in \R ^2$ is always orthogonal to the level curves $c = F(x,y)$. Moreover, given $F:\R ^3\to \R$, $\grad F \in \R ^3$ is always orthogonal to level surfaces.

Computing the gradient vector

Given a function of several variables, say $F:\R ^2\to \R$, the gradient, when evaluated at a point in the domain of $F$, is a vector in $\R ^2$. We can see this in the interactive below.

The gradient at each point is a vector pointing in the $(x,y)$-plane. You compute the gradient vector, by writing the vector: $$\grad F = \vector {\pp [F]{x_1},\pp [F]{x_2},\dots ,\pp [F]{x_n}}$$ You’ve done this sort of direct computation many times before. So now, try your hand at these puzzlers:

The initial greatest increase

Given a function $F:\R ^n\to \R$ and point in $\R ^n$, the gradient vector tells you which initial direction to leave the point in order to get the greatest increase in $F$. 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: $$D_\uvec {u}(F) = \grad {F} \dotp \uvec {u}$$ To make this change as large as possible, $\uvec {u}$ must be the same direction as $\grad F$. 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 $F:\R ^2\to \R$ that is described by a table of values.

Let $F:\R ^2\to \R$ be a differentiable function described by the following table of values:

Estimate $\grad F(3,5)$.

We estimate $\grad F(3,5)$ by estimating the partial derivatives. To estimate $F^{(1,0)}(3,5)$, we examine the change in $F(x,5)$ between $x=4$ and $x=3$: $$\frac {F(4,5)-F\left (\answer [given]{3},5\right )}{\answer [given]{4}-3}= \answer [given]{-1}$$ We should also examine the change in $F(x,5)$ between $x=3$ and $x=2$: $$\frac {F(3,5)-F\left (\answer [given]{2},5\right )}{\answer [given]{3}-2} =\answer [given]{-5}$$ Now if we average these values together, we see: $$\eval {\pp {x} F(x,y)}_{(x,y)=(3,5)} \approx \answer [given]{-3}$$ On the other hand, using a similar procedure, we find that: $$\eval {\pp {y} F(x,y)}_{(x,y)=(3,5)} \approx \answer [given]{4}$$ Thus the gradient is $$\grad F(3,5) = \vector {\answer [given]{-3},\answer [given]{4}}$$ Note if you leave the point $(3,5)$ in the direction of $\grad F(3,5) = \vector {\answer [given]{-3},\answer [given]{4}}$, you head toward $F(2,6)= \answer [given]{16}$, the greatest initial increase from $(3,5)$.

Here is a plot of an elliptic paraboloid $G(x,y) = x^2 + y^2$ along with a vector attached to a point on the surface:

True or false: The vector above could be the gradient vector for $G$ at the given point.

True False

The answer is “False.” Here the graph of the function is three dimensional. The gradient vector is in one less dimension than the function’s graph. Hence the gradient of $G$ is in fact always a two dimensional vector.

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.

Suppose you have a differentiable function $F:\R ^2\to \R$ with the following set of level curves. You should interpolate reasonable values of the function $F$ between the level curves which are shown:

Consider the points $A$, $B$, and $C$ on the surface $z=F(x,y)$. Where $|\grad F|$ largest? The magnitude of the gradient vector of $F$ is largest at point $\answer [format=string]{B}$. Where is $|\grad F|$ smallest? The magnitude of the gradient vector of $F$ is smallest at point $\answer [format=string]{C}$.

Now, stand back. We’re going to do some serious calculus. Just read, relax and enjoy.

Consider the surface given by $F(x,y)= 20-x^2-2y^2$:

Water is poured on the surface at $(1,1/4)$. What path does it take as it flows downhill?

Let $\vec {w}(t) = \vector {x(t), y(t)}$ be the vector-valued function describing the path of the water in the $(x,y)$-plane. We seek $x(t)$ and $y(t)$. We know that water will always flow downhill in the initial steepest direction. Therefore, at any point on its path, it will be moving in the direction of $$-\grad F(x,y)$$ We’ll ignore the physical effects of momentum on the water. Thus $\vec {w}(t)$ will be parallel to $\grad F$. Ah! This means there is some scalar function $c(t)$ such that $$c(t)\grad F(x(t),y(t)) = \vec {w}'(t) = \vector {x'(t), y'(t)}.$$ Computing the gradient, $$\grad F(x(t),y(t)) = \vector {-2x(t), -4y(t)}$$ Then $$\begin{align*} c(t)\cdot \grad F(x(t),y(t)) &= \vector{ x'(t), y'(t)}\\ c(t)\cdot \vector{-2x(t),-4y(t)} &= \vector{ x'(t), y'(t)}\\ \vector{-2c(t)x(t),-4c(t)y(t)} &= \vector{ x'(t), y'(t)}\\ \end{align*}$$

This implies $$-2c(t)x(t) = x'(t) \quad \text {and} \quad -4c(t)y(t) =y'(t)$$ so $$c(t) = -\frac {x'(t)}{2x(t)} \quad \text {and} \quad c(t) =-\frac {y'(t)}{4y(t)}.$$ Now recall that the differentials $\d x = x'(t) \d t$, and $\d y=y'(t)\d t$, so we may write

$$\begin{align*} \int \frac{1}{2x}x'(t)\d t &=\int \frac{1}{4y} y'(t)\d t \\ \int \frac{1}{2x}\d x &=\int\frac{1}{4y}\d y \\ \frac{1}{2}\ln|x| +C &= \frac{1}{4}\ln|y|\\ 2\ln|x| + C &= \ln|y|\\ \ln|x^2| + C &= \ln|y| \end{align*}$$

Raising $e$ to the left-hand and right hand sides, we see

$$\begin{align*} e^{\ln|x^2| + C} &= e^{\ln|y|}\\ x^2\cdot e^C &= |y|, \end{align*}$$

setting $K = e^C$, we write $$K\cdot x^2 = y.$$ We are so close to being done, $y=K\cdot x^2$, this is the path described in the $(x,y)$-plane. Since the water started at the point $(1,1/4)$, we can solve for $K$: $$K\cdot 1^2 = \frac 14 \quad \Rightarrow \quad K = \frac 14.$$ Thus the water follows the curve $$y=x^2/4$$ in the $(x,y)$-plane.

Orthogonality and the gradient

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 $\vec {c}(t)=\vector {x(t),y(t)}$ runs along a level surface for the surface $F(x,y)$. If we ask ourselves: “What is the change in $F$ as $t$ varies?” We must conclude that $$\dd {t} F(\vec {c}(t)) = 0$$ since the value of $F$ doesn’t change on the curve drawn by $\vec {c}$ (remember, $\vec {c}$ draws a level curve). On the other hand, by the chain rule: $$\dd {t} F(\vec {c}(t)) = \grad F(\vec {c}(t)) \dotp \vec {c}'(t)$$ The vector $\vec {c}'$ is tangent to the curve drawn by $\vec {c}$, and putting the two equations above together we see $$0 = \grad F(\vec {c}(t)) \dotp \vec {c}'(t)$$ so $\grad F(\vec {c}(t))$ must be orthogonal to $\vec {c}'$, and hence orthogonal to the curve drawn by $\vec {c}$.

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 $F:\R ^2\to \R$ 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.

Find an implicit equation for the tangent plane to the elliptic paraboloid $z = x^2 + y^2$

at $\vec {p} = \vector {2,3,13}$.

Consider $$F(x,y,z) =x^2 +y^2 -z$$ and imagine the elliptic paraboloid as the level surface $$F(x,y,z) = \answer [given]{0}$$ Remember, the gradient is perpendicular to level surfaces. We’ll use this fact to find a normal vector to the surface, and with this vector we’ll find the tangent plane. The gradient is: $$\begin{align*} \grad F(x,y,z) &= \vector{\pp[F]{x}, \pp[F]{y},\pp[F]{z}}\\ &= \vector{\answer[given]{2x},\answer[given]{2y}, \answer[given]{-1}}. \end{align*}$$

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 $$\vec {n}\dotp (\vec {x}-\vec {p}) = 0$$ where $\vec {p} = \vector {2,3,13}$ and

$$\begin{align*} \vec{n} &= \grad F(\vec{p})\\ &=\vector{\answer[given]{4}, \answer[given]{6},\answer[given]{-1}} \end{align*}$$

Thus the equation of the plane tangent to the ellipsoid at $\vec {p}$ is: $$\answer [given]{4}(x-2) + 6(y-3) - \left (z-\answer [given]{13}\right ) = \answer [given]{0}$$

Now let’s see a more in-depth problem.

Summary

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.