Chapter 7

Calculus of Vector-Valued Functions

A parametric curve is made of $𝑛$ component functions, which are familiar functions $ℝ \to ℝ$ from single-variable calculus. Thus, the calculus of curves is simply the calculus of single-variable functions performed $𝑛$ times!

In the last chapter, we learned how to construct and interpret these curves. We now ask the usual questions of calculus: What happens as the parameter approaches a particular value? How quickly is the curve changing? And how can we recover a vector-valued function from its rate of change?

7.1Differentiation of Vector-Valued Functions

Before defining the derivative, we need one brief observation about limits. Taking limits of a parametric curve is just taking the limit of each component function.

Definition 7.1 (Limits of Parametric Curves). If $⃗ 𝑟 (𝑡) = (𝑥 (𝑡), 𝑦 (𝑡), 𝑧 (𝑡))$ is a parametric curve, then limits are computed componentwise:

l i m 𝑡 \to 𝑎 ⃗ 𝑟 (𝑡) = ⟨ l i m 𝑡 \to 𝑎 𝑥 (𝑡), l i m 𝑡 \to 𝑎 𝑦 (𝑡), l i m 𝑡 \to 𝑎 𝑧 (𝑡) ⟩

Example 7.2 (Limits of Parametric Curves). Let $⃗ 𝑟 (𝑡)$ be the following parametric curve

⃗ 𝑟 (𝑡) = ⟨ 1 𝑡 + 1, s i n (𝑡) 𝑡, 3 𝑡 + 𝑡 2 𝑡 ⟩

Compute the limit $l i m 𝑡 \to 0 ⃗ 𝑟 (𝑡)$ .

Computing the limit componentwise we see we just need to evaluate three limits:

l i m 𝑡 \to 0 1 𝑡 + 1

l i m 𝑡 \to 0 s i n (𝑡) 𝑡

l i m 𝑡 \to 0 3 𝑡 + 𝑡 2 𝑡

The first of these is continuous at zero, so we can plug in directly. The second is the standard limit $l i m 𝑡 \to 0 s i n (𝑡) / 𝑡 = 1$ (or can be evaluated with L'Hôpital's rule), and the third requires canceling a $𝑡$ from the numerator and denominator before plugging in. Thus,

l i m 𝑡 \to 0 ⃗ 𝑟 (𝑡) = ⟨ 1, 1, 3 ⟩

Recall the single variable definition of the derivative:

𝑓' (𝑡) = l i m ℎ \to 0 𝑓 (𝑡 + ℎ) - 𝑓 (𝑡) ℎ

The same definition works for parametric curves:

Definition 7.3 (Differentiating Curves). The derivative of a parametric curve $⃗ 𝑟 (𝑡)$ at a point $𝑡$ is given by the following limit, provided it exists:

⃗ 𝑟' (𝑡) = l i m ℎ \to 0 ⃗ 𝑟 (𝑡 + ℎ) - ⃗ 𝑟 (𝑡) ℎ .

The numerator here is a vector connecting two points of the curve that get closer together as $ℎ \to 0$ . We then rescale this vector by dividing by $ℎ$ to keep its length from collapsing. In the limit, it converges to a vector $⃗ 𝑟' (𝑡)$ tangent to the curve whenever $⃗ 𝑟' (𝑡) \neq ⃗ 0$ .

Figure 7.1 The blue arrow is the numerator on its own, and it shrinks away to nothing as

ℎ \to 0

— a limit worth noting. Dividing by

ℎ

is what saves it: the teal arrow keeps its length as it turns, and settles onto the dashed tangent line. Drag the base point and the same thing happens everywhere along the curve.

However, we don't need to calculate the derivative using this limit statement every time! We can use the fact that limits distribute over the components of the function to prove that we can also take the derivative one component at a time.

Theorem 7.4 (Differentiating Curves Componentwise). If $⃗ 𝑟 (𝑡) = (𝑥 (𝑡), 𝑦 (𝑡), 𝑧 (𝑡))$ is a parametric curve, then

⃗ 𝑟' (𝑡) = ⟨ 𝑥' (𝑡), 𝑦' (𝑡), 𝑧' (𝑡) ⟩

Figure 7.2 Three ordinary graphs on the left, one arrow on the right, and the same three numbers in both. The steepness of

𝑥 (𝑡)

at this instant is the first component of

⃗ 𝑟' (𝑡)

, the steepness of

𝑦 (𝑡)

the second, and

𝑧 (𝑡)

the third. Nothing new has to be invented to differentiate a curve: it is single-variable calculus, done once per slot.

Once we know the derivative, the tangent line is an immediate calculation. For example, let

⃗ 𝑟 (𝑡) = ⟨ c o s 𝑡, s i n 𝑡, 𝑡 ⟩ .

At $𝑡 0 = 0$ , we have

⃗ 𝑟 (0) = ⟨ 1, 0, 0 ⟩, ⃗ 𝑟' (0) = ⟨ 0, 1, 1 ⟩ .

Therefore the tangent line is

⃗ ℓ (𝑠) = ⃗ 𝑟 (0) + 𝑠 ⃗ 𝑟' (0) = ⟨ 1, 0, 0 ⟩ + 𝑠 ⟨ 0, 1, 1 ⟩ .

Because of this, it is straightforward to show that differentiation of vector functions obeys the familiar laws of single-variable calculus: you can break it up over sums and pull out scalars. But now there are three types of products: do we multiply the vector function by a scalar function, or take its dot or cross product with another vector?

Theorem 7.5 (Differentiation Product Laws).

(𝑓 (𝑡) ⃗ 𝑟 (𝑡))' = 𝑓' (𝑡) ⃗ 𝑟 (𝑡) + 𝑓 (𝑡) ⃗ 𝑟' (𝑡)

(⃗ 𝑐 (𝑡) \cdot ⃗ 𝑟 (𝑡))' = ⃗ 𝑐' (𝑡) \cdot ⃗ 𝑟 (𝑡) + ⃗ 𝑐 (𝑡) \cdot ⃗ 𝑟' (𝑡)

(⃗ 𝑐 (𝑡) \times ⃗ 𝑟 (𝑡))' = ⃗ 𝑐' (𝑡) \times ⃗ 𝑟 (𝑡) + ⃗ 𝑐 (𝑡) \times ⃗ 𝑟' (𝑡)

The cross-product rule applies to vector-valued functions in $ℝ 3$ .

There is also a chain rule: although we cannot generally compose one curve inside another, we can plug a scalar function in as the parameter of a curve.

Theorem 7.6 (The Chain Rule).

(⃗ 𝑟 (𝑓 (𝑡)))' = ⃗ 𝑟' (𝑓 (𝑡)) 𝑓' (𝑡)

You probably notice a similarity to the single variable calculus versions in all of these: they're as close as possible, except now being about vector functions! But these simple looking rules actually provide us a new powerful set of tools, they tell us about the rate of change at the same time as dot and cross products - which we know can measure areas and angles!

As one quick example, we'll prove a very useful fact about curves defined by vectors of constant length.

Theorem 7.7 (Curves on the Sphere). If a curve $⃗ 𝑟 (𝑡)$ never changes in length, so $‖ ⃗ 𝑟 (𝑡) ‖ = 𝑘$ for all $𝑡$ , then $⃗ 𝑟' (𝑡)$ is orthogonal to $⃗ 𝑟 (𝑡)$ .

Proof. Since the magnitude of a vector can be calculated from its dot product, we see that

⃗ 𝑟 \cdot ⃗ 𝑟 = 𝑘 2

Taking the derivative of this with the product rule, we find

⃗ 𝑟' \cdot ⃗ 𝑟 + ⃗ 𝑟 \cdot ⃗ 𝑟' = 0

The dot product is commutative (order does not matter), so we can rearrange the left-hand side:

2 ⃗ 𝑟 \cdot ⃗ 𝑟' = 0

Dividing by two shows that the dot product of $⃗ 𝑟$ and $⃗ 𝑟'$ is zero, so these two vectors are orthogonal. ∎

Figure 7.3 While the curve stays on the sphere its length never changes, and

⃗ 𝑟'

is perpendicular to

⃗ 𝑟

at every point—the dot product sits at zero without being asked to. Turn up the slider and the length starts to breathe, the curve lifts off the sphere, and the product leaves zero at exactly the moments the length is changing. Differentiating

⃗ 𝑟 \cdot ⃗ 𝑟 = 𝑅 2

says the same thing in one line. Chapter 9 uses this again, with

⃗ 𝑇

in place of

⃗ 𝑟

7.2Motion and Integration

When the parameter $𝑡$ represents time, the derivative has a familiar physical meaning. If $⃗ 𝑟 (𝑡)$ is position, then

p o s i t i o n ⃗ 𝑟 (𝑡) ⟶ v e l o c i t y ⃗ 𝑣 (𝑡) = ⃗ 𝑟' (𝑡) ⟶ a c c e l e r a t i o n ⃗ 𝑎 (𝑡) = ⃗ 𝑟 ″ (𝑡) .

Velocity records both the direction and rate of motion. Its magnitude

s p e e d = ‖ ⃗ 𝑣 (𝑡) ‖

records only how fast the particle is moving.

Figure 7.4 Velocity lies along the curve, always: it is the derivative of position, so it can point nowhere else. Acceleration is under no such obligation. Watch where it leans on the bends and where it swings forward as the motion speeds up—the two halves of that behaviour are what Chapter 9 separates.

The story of vector valued integration is similarly straightforward. Recall the definition of integration via Riemann sum:

\int 𝑏 𝑎 𝑓 (𝑥) 𝑑 𝑥 = l i m 𝑁 \to \infty 𝑁 \sum 𝑖 = 1 𝑓 (𝑥 𝑖) Δ 𝑥

We can apply the same definition to vector-valued functions because it is composed only of addition and scalar multiplication.

Definition 7.8 (Integrating Vector-Valued Functions). If $⃗ 𝑟$ is a vector-valued function, its vector valued integral on the interval $[𝑎, 𝑏]$ is defined by the following Riemann sum:

\int 𝑏 𝑎 ⃗ 𝑟 (𝑡) 𝑑 𝑡 = l i m 𝑁 \to \infty 𝑁 \sum 𝑖 = 1 ⃗ 𝑟 (𝑡 𝑖) Δ 𝑡

But wait! Both scalar multiplication and addition are operations we can perform componentwise on a vector. So we can break this large Riemann sum into a standard Riemann sum in each component. Taking the limit tells us that we can integrate vector functions componentwise.

Theorem 7.9 (Integrating Curves Componentwise). If $⃗ 𝑟 (𝑡) = ⟨ 𝑥 (𝑡), 𝑦 (𝑡), 𝑧 (𝑡) ⟩$ , the vector valued integral of $⃗ 𝑟$ on $[𝑎, 𝑏]$ is given by

\int 𝑏 𝑎 ⃗ 𝑟 (𝑡) 𝑑 𝑡 = ⟨ \int 𝑏 𝑎 𝑥 (𝑡) 𝑑 𝑡, \int 𝑏 𝑎 𝑦 (𝑡) 𝑑 𝑡, \int 𝑏 𝑎 𝑧 (𝑡) 𝑑 𝑡 ⟩

The Riemann sum in the definition has a concrete meaning when the vector function is velocity: each term is approximately the displacement during one short interval of time.

Figure 7.5 Take the velocity at a few instants, multiply each by the time it acts for, and lay the results head to tail. Even four steps land close; add more and the chain folds onto the path while its endpoint closes on

⃗ 𝑟 (𝑏) - ⃗ 𝑟 (𝑎)

. That is all

\int 𝑏 𝑎 ⃗ 𝑣 𝑑 𝑡

is—a Riemann sum whose terms happen to be vectors, which is why it can be evaluated one component at a time.

We will only find limited use of this in our class, as there are other types of integrals along curves that will prove more important. Nonetheless this does show up in many applications of multivariable calculus to engineering and physics, where one may wish to recover position from velocity, or velocity from acceleration.

Example 7.10 (Displacement from Velocity). If $⃗ 𝑣 (𝑡)$ gives the velocity of a particle at time $𝑡$ , then $\int 𝑏 𝑎 ⃗ 𝑣 (𝑡) 𝑑 𝑡$ is the displacement vector for its net travel between $𝑡 = 𝑎$ and $𝑡 = 𝑏$ .

An integral gives a change, so recovering the function itself also requires an initial value. If the velocity $⃗ 𝑣 (𝑡 0)$ and position $⃗ 𝑟 (𝑡 0)$ are known, then

⃗ 𝑣 (𝑡) = ⃗ 𝑣 (𝑡 0) + \int 𝑡 𝑡 0 ⃗ 𝑎 (𝑢) 𝑑 𝑢

and

⃗ 𝑟 (𝑡) = ⃗ 𝑟 (𝑡 0) + \int 𝑡 𝑡 0 ⃗ 𝑣 (𝑢) 𝑑 𝑢 .

The same is true for acceleration: our phones contain sensors called three-axis accelerometers. These measure acceleration as a function of time:

⃗ 𝑎 (𝑡) = ⟨ 𝑎 𝑥 (𝑡), 𝑎 𝑦 (𝑡), 𝑎 𝑧 (𝑡) ⟩

But this is not what software running on the phone wants or needs: it cares about your position in space, for example when using augmented reality. Given the necessary initial conditions, it can integrate acceleration to get velocity and then integrate again to get position. For any engineers in the class—if you have accelerometers in the Hive, this could be a fun project to code. Write a small Python program to numerically integrate (i.e., compute a Riemann sum from) the output data of an accelerometer and track your hand's position as you move it around.

Figure 7.6 Your phone contains this figure. A three-axis accelerometer reports the two squiggles on the left and nothing else; the path on the right is not measured but computed, by integrating that signal once for velocity and again for position, starting from a known position and velocity. With a perfect sensor the reconstruction is exact. Add noise and it drifts—slowly at first, then without recovery, because the errors are integrated twice along with everything else.

Real accelerometer data also includes gravity, measurement noise, and errors in the phone's orientation. Because those errors are integrated along with the signal, a small velocity error can grow into a large position error. The idea is simple; making it reliable is the hard part.

The magnitude $‖ ⃗ 𝑟' (𝑡) ‖$ gave us speed. In the next chapter, we use speed to measure distance along a curve and then use that distance to build scalar line integrals.