We define vector-valued functions, whose image is a curve in the plane or space. We also show how to differentiate and integrate them

1. Sections 10.1–2
Vector-Valued Functions and Curves in Space
Derivatives and Integrals of Vector-Valued
Functions
Math 21a
February 20, 2008
Announcements
Problem Sessions:
Monday, 8:30, SC 103b (Sophie)
Thursday, 7:30, SC 103b (Jeremy)
Office hours Wednesday 2/20 2–4pm SC 323.

2. Outline
Vector-valued functions
Derivatives of vector-valued functions
Integrals of vector-valued functions

3. Recall
−→ −→
If P and Q are two points in the plane, u = OP, and v = OQ,
then the line through P and Q can be parametrized as
r(t) = tv + (1 − t)u

4. Recall
−→ −→
If P and Q are two points in the plane, u = OP, and v = OQ,
then the line through P and Q can be parametrized as
r(t) = tv + (1 − t)u
This is a function whose domain is R and whose range is a subset
of R3 (the line).

5. Definition
A vector-valued function or vector function is a function r(t)
whose domain is a set of real numbers and whose range is a set of
vectors.
We can split r(t) into its components
r(t) = f (t)i + g (t)j + h(t)k
Then f , g , and h are called the component functions of r.
The range of r is a curve in R2 or R3 .

6. Example
Given the plane curve described by the vector equation
r(t) = sin(t)i + 2 cos(t)j
(a) Sketch the plane curve.

7. r(t) = r(t) = sin(t)i + 2 cos(t)j
y
t r(t)
0 2j
π/2 i r(π/4)
π −2j x
3π/2 −i
2π 2j

8. Curves and functions
Example
Two particles travel along the space curves
r1 (t) = 3t, 7t − 12, t 2 r2 (t) = 4t − 3, t 2 , 5t − 6
Do the particles collide?

9. Curves and functions
Example
Two particles travel along the space curves
r1 (t) = 3t, 7t − 12, t 2 r2 (t) = 4t − 3, t 2 , 5t − 6
Do the particles collide?
Answer
Yes. r1 (3) = r2 (3).

10. Outline
Vector-valued functions
Derivatives of vector-valued functions
Integrals of vector-valued functions

11. Derivatives of vector-valued functions
Definition
Let r be a vector function.
The limit of r at a point a is defined componentwise:
lim r(t) = lim f (t), lim g (t), lim h(t)
t→a t→a t→a t→a
The derivative of r is defined in much the same way as it is
for real-valued functions:
dr r(t + h) − r(t)
= r (t) = lim
dt h→0 h

12. Example
Given r(t) = t, cos 2t, sin 2t , find r (t).

13. Example
Given r(t) = t, cos 2t, sin 2t , find r (t).
Answer
1, −2 sin 2t, 2 cos(2t)

14. Fact
If r(t) = f (t), g (t), h(t) , then
r (t) = f (t), g (t), h (t)

15. Fact
If r(t) = f (t), g (t), h(t) , then
r (t) = f (t), g (t), h (t)
Proof.
Follow your nose:
r(t + h) − r(t)
r (t) = lim
h→0 h
1
= lim [ f (t + η), g (t + η), h(t + η) − f (t), g (t), h(t) ]
η→0 η
1
= lim [ f (t + η) − f (t), g (t + η) − g (t), h(t + η) − h(t) ]
η→0 η
f (t + η) − f (t) g (t + η) − g (t) h(t + η) − h(t)
= lim , lim , lim
η→0 η η→0 η η→0 η
= f (t), g (t), h (t)

16. Example
Given the plane curve described by the vector equation
r(t) = sin(t)i + 2 cos(t)j
(a) Sketch the plane curve.
(b) Find r (t)

17. r(t) = r(t) = sin(t)i + 2 cos(t)j
r (t) = cos(t)i − 2 sin(t)j
y
t r(t)
0 2j
π/2 i r(π/4)
π −2j x
3π/2 −i
2π 2j

18. Example
Given the plane curve described by the vector equation
r(t) = sin(t)i + 2 cos(t)j
(a) Sketch the plane curve.
(b) Find r (t)
(c) Sketch the position vector r(π/4) and the tangent vector
r (π/4).

19. r(t) = r(t) = sin(t)i + 2 cos(t)j
r (t) = cos(t)i − 2 sin(t)j
y
t r(t)
0 2j
π/2 i r(π/4) r (π/4)
π −2j x
3π/2 −i
2π 2j

20. Rules for differentiation
Theorem
Let u and v be differentiable vector functions, c a scalar, and f a
real-valued function. Then:
d
1. [u(t) + v(t)] = u (t) + v (t)
dt
d
2. [cu(t)] = cu (t)
dt
d
3. [f (t)u(t)] = f (t)u(t) + f (t)u (t)
dt
d
4. [u(t) · v(t)] = u (t) · v(t) + u(t) · v (t)
dt
d
5. [u(t) × v(t)] = u (t) × v(t) + u(t) × v (t)
dt
d
6. [u(f (t))] = f (t)u (f (t))
dt

21. Leibniz rule for cross products
Let u = f1 (t), g1 (t), h1 (t) and v = f2 (t), g2 (t), h2 (t) . The first
component of u(t) × v(t) is
(u(t) × v(t)) · i = g1 h2 − g2 h1
Differentiating gives
(u(t) × v(t)) · i = g1 h2 + g1 h2 − g2 h1 − g2 h1
= g1 h2 − g2 h1 + g1 h2 − g2 h1
= (u (t) × v(t)) · i + (u(t) × v (t)) · i
= u (t) × v(t) + u(t) × v (t) · i

22. Meet the Mathematician: Isaac Newton
English, 1643–1727
Professor at Cambridge
(England)
Philosophiae Naturalis
Principia Mathematica
published 1687

23. Meet the Mathematician: Gottfried Leibniz
German, 1646–1716
Eminent philosopher as
well as mathematician
Contemporarily disgraced
by the calculus priority
dispute

24. Smooth curves
Example
Which of the following curves are smooth? That is, which curves
satisfy the property that r (t) = 0 for all t?
(a) r(t) = t 3 , t 4 , t 5
(b) r(t) = t 3 + t, t 4 , t 5
(c) r(t) = cos3 t, sin3 t

25. The first curve r(t) = t 3 , t 4 , t 5 has r (t) = 3t 2 , 4t 3 , 5t 4 , and is
not smooth at t = 0.
z
x y
Projecting r(t) onto the yz-plane gives y = z 4/5 , which is not
differentiable at 0.

26. If r(t) = t 3 + t, t 4 , t 5 , then r (t) = 3t 2 + 1, 4t 3 , 5t 4 , which is
never 0.
So this curve is smooth.

27. If r(t) = cos3 t, sin3 t , then
r (t) = −3 cos2 (t) sin(t), 3 sin2 (t) cos(t) . This is 0 when
cos t = 0 or sin t = 0, i.e., when t = π/2, π, 3π/2, 2π.
y
x

28. Outline
Vector-valued functions
Derivatives of vector-valued functions
Integrals of vector-valued functions

29. Integrals of vector-valued functions
Definition
Let r be a vector function defined on [a, b]. For each whole number
n, divide the interval [a, b] into n pieces of equal width ∆t.
Choose a point ti∗ on each subinterval and form the Riemann sum
n
Sn = r(ti∗ ) ∆t
i=1
Then define
b n
r(t) dt = lim Sn = lim r(ti∗ ) ∆t
a n→∞ n→∞
i=1
n n n
= lim f (ti∗ ) ∆ti + g (ti∗ ) ∆tj + h(ti∗ ) ∆
n→∞
i=1 i=1 i=1
b b b
= f (t) dt i + g (t) dt j + h(t) dt
a a a

30. Example
Given r(t) = t, cos 2t, sin 2t , find
π
r(t) dt
0

31. Example
Given r(t) = t, cos 2t, sin 2t , find
π
r(t) dt
0
Answer
π2
, 0, 0
2

32. FTC for vector functions
Theorem (Second Fundamental Theorem of Calculus)
If r(t) = R (t), then
b
r(t) dt = R(t)
a

33. FTC for vector functions
Theorem (Second Fundamental Theorem of Calculus)
If r(t) = R (t), then
b
r(t) dt = R(t)
a
Proof.
Let R(t) = F (t), G (t), H(t) . To say that R (t) = r(t) means
that F = f , G = g , and H = h. That and the componentwise
b
definition of r(t) dt are all you need.
a