Math 217: Vector Functions (Ch. 13)
Lecture 4 (Sep. 12)
Vector Functions and Space Curves (reading: 13.1)
Vector-valued function:
r(t) = hf (t), g(t), h(t)i = f (t)î + g(t)ĵ + h(t)k̂.
Limits: Definition: the limit of r(t) as t ! a is
lim r(t) = hlim f (t), lim g(t), lim h(t)i
t!a
t!a
t!a
t!a
(if it exists).
Continuity: Definition: the vector function r(t) is continuous at a if
lim r(t) = r(a),
t!a
and is continuous on an interval I if it is continuous at each a 2 I.
Definition: A space curve is the image of a continuous vector function defined on
some interval I:
C = {r(t) | t 2 I} ⇢ R3 .
If r(t) = hf (t), g(t), h(t)i, then parametric equations of C are
x = f (t),
y = g(t),
1
z = h(t).
Example: find a vector function which parameterizes the straight line through r0 in
the direction v.
Example: suppose the position of a particle at time t is given by r(t) = hcos(t) sin(t), sin2 (t), cos(t)i.
Sketch the particle’s trajectory.
Example: parameterize the curve of intersection of the parabolic cylinder z = y 2 and
the plane x + y = 1.
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Derivatives and Integrals of Vector Functions (reading: 13.2)
Definition: the derivative of the vector function r(t) at a is
r(a + h)
h!0
h
r0 (a) = lim
r(a)
,
if it exists (in which case we say r is di↵erentiable at a).
Clearly, if r = hf, g, hi, then r is di↵erentiable at a , f , g, and h, are all di↵erentiable
at a, and, if so,
r0 (a) = hf 0 (a), g 0 (a), h0 (a)i.
Remark: We define higher derivatives r00 , r000 , etc., in the same way.
Geometrically: if r(t) describes the space curve C, then r0 (a) is a vector tangent to
the curve C at r(a) (provided r0 (a) 6= 0).
Definition: Suppose a vector function r(t) parameterizes a space curve C. If r(t) is
di↵erentiable at t = a, and r0 (a) 6= 0, then
1. the tangent line to C at r(a) is given parametrically by r(a) + tr0 (a)
2. the unit tangent to C at r(a) is the vector T(a) = r0 (a)/|r0 (a)|.
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Example: Find the unit tangent to the curve given by r(t) = Rhcos(t), sin(t), 0i.
Definition: We say the curve described by r(t), a  t  b is smooth if r0 (t) is
continuous on (a, b), and r0 (t) 6= 0 for all a < t < b.
Example: A point on a rolling disk:
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Di↵erentiation rules for vector functions
1. [u(t) + v(t)]0 = u0 (t) + v0 (t)
2. [cu(t)]0 = cu0 (t)
3. [f (t)u(t)]0 = f (t)u0 (t) + f 0 (t)u(t)
4. [u(t) · v(t)]0 = u(t) · v0 (t) + u0 (t) · v(t)
5. [u(t) ⇥ v(t)]0 = u(t) ⇥ v0 (t) + u0 (t) ⇥ v(t)
6. [u(f (t))]0 = f 0 (t)u0 (f (t))
Proof of (5), for example:
Example: Suppose the image of a vector function r(t) lies on the sphere of radius R
centred at the origin. Show that the tangent vector to the curve is perpendicular to
r(t).
Integrals: to integrate a vector function, we just integrate each component function.
Definition: If r(t) = hf (t), g(t), h(t)i is continuous for a  t  b, we define the
definite integral of r over [a, b] as
⌧Z b
Z b
Z b
Z b
r(t)dt :=
f (t)dt,
g(t)dt,
h(t)dt .
a
a
a
0
a
By the fundamental theorem of calculus, if R (t) = r(t), then
Z b
r(t)dt = R(b) R(a).
a
Example:
Z
(et î + 2tĵ + ln(t)k̂)dt =
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