Lecture Note - Calculus of Vector-Valued Functions
Michael Wang
1. Basic Calculus of Vector-Valued Functions
Definition 1. (Limit of a Vector-Valued Function). A vector-valued function r(t) approaches
the limit u (a vector) as t approaches t0 if limt→t0 kr(t) − uk = 0. In this case, we write
lim r(t) = u
t→t0
Theorem 1. (Vector-Valued Limits are Computed Componentwise). A vector-valued function r(t) = hx(t), y(t), z(t)i approaches a limit as t → t0 iff each component approaches a limit, and
in this case,
D
E
lim r(t) = lim x(t), lim y(t), lim z(t)
t→t0
t→t0
t→t0
t→t0
Proof. Let u = ha, b, ci and consider the square of the length
kr(t) − uk2 = (x(t) − a)2 + (y(t) − b)2 + (z(t) − c)2
The term on the left approaches zero iff each term on the right approaches zero. It follows that
kr(t) − uk approaches zero iff |x(t) − a|, |y(t) − b|, and |z(t) − c| tend to zero. Therefore, r(t)
approaches a limit u as t → t0 iff x(t), y(t), and z(t) converge to the components a, b, and c.
Continuity of vector-valued functions is defined in the same way as in the scalar case.
Definition 2. A vector-valued function r(t) = hx(t), y(t), z(t)i is continuous at t0 if
lim r(t) = r(t0 )
t→t0
Definition 3. The derivative of r(t) is the limit of the difference quotient:
r(t + h) − r(t)
d
r(t) = lim
(1)
h→0
dt
h
Theorem 2. (Vector-Valued Derivatives are Computed Componentwise). A vector-valued
function r(t) = hx(t), y(t), z(t)i is differentiable iff each component is differentiable. In this case,
r0 (t) =
r0 (t) =
d
r(t) = hx0 (t), y 0 (t), z 0 (t)i
dt
Differentiation Rules
Assume that r(t), r1 (t), and r2 (t) are differentiable. Then
• Sum Rule: (r1 (t) + r2 (t))0 = r01 (t) + r02 (t).
• Constant Multiple Rule: For any constant c, (cr(t))0 = cr0 (t)>
• Product Rule: For any differentiable scalar-valued function f (t),
d
(f (t)r(t)) = f (t)r0 (t) + f 0 (t)r(t)
dt
• Chain Rule: For any differentiable scalar-valued function g(t),
d
r(g(t)) = g 0 (t)r0 (g(t))
dt
1
Theorem 3. (Product Rule for Dot and Cross Products). Assume that r1 (t) and r2 (t) are
differentiable. Then
Dot Products:
Cross Products:
d
r1 (t) · r2 (t) = r1 (t) · r02 (t) + r01 (t) · r2 (t)
dt
d
r1 (t) × r2 (t) = r1 (t) × r02 (t) + r01 (t) × r2 (t)
dt
(2)
(3)
2. The Derivative as a Tangent Vector
We refer to r0 (t0 ) as the tangent vector or the velocity vector at r(t0 ). The tangent vector r0 (t0 )
(if it is nonzero) is a direction vector for the tangent line to the curve. THerefore, the tangent line
has vector parametrization:
L(t) = r(t0 ) + tr0 (t0 )
Tangent line at r(t0 ) :
(4)
3. Vector-Valued Integration
The integral of a vector-valued function can be defined in terms of Riemann sums. We will define it
more simply via componentwise integration. In other words,
Z
b
r(t)dt =
DZ
a
b
Z
x(t)dt,
a
b
Z
y(t)dt,
a
b
z(t)dt
E
a
Vector-valued integrals obey the same linearity rules as scalar-valued integrals.
An antiderivative of r(t) is a vector-valued function R(t) such that R0 (t) = r(t).
Theorem 4. If R1 (t) and R2 (t) are differentiable and R01 (t) = R02 (t), then
R1 (t) = R2 (t) + c
for some constant vector c.
The general antiderivative of r(t) is written
Z
r(t)dt = R(t) + c
where c = hc1 , c2 , c3 i is an arbitrary constant vector.
Fundamental Theorem of Calculus for Vector-Valued Functions
If r(t) is continuous on [a, b], and R(t) is an antiderivative of r(t), then
Z b
r(t)dt = R(b) − R(a)
a
2