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Math 251-copyright Joe Kahlig, 15A
Section 11.6: Vector Functions and Space Curves
Let r be a vector function whose domain is a real number t and result is a three-dimensional vector.
Let
r(t) = hf (t), g(t), h(t)i = f (t)i + g(t)j + h(t)k
where f (t), g(t), and h(t) are real valued functions and are called the component functions of r.
The limit of a vector function r is defined by taking the limits of its component functions:
lim r(t) = lim f (t), lim g(t), lim h(t)
t→a
t→a
t→a
t→a
A vector function r is continuous if and only if its component functions f (t), g(t), and h(t) are continuous.
Example: Given r(t) =
*
√
et − 1
t t + 5, t2 + 2,
t
+
a) Find the domain of r(t).
b) Find all t where r(t) is continuous.
c) Compute lim r(t).
t→0
Definition: Suppose that f (t), g(t), and h(t) are real valued functions on an interval I, then the set
C defined as :
C = {(x, y, z)|x = f (t), y = g(t), z = h(t)}
where t is a parameter and t varies in some interval, I, is called a space curve. The space curve C
can be traversed by the vector function r(t) = hf (t), g(t), h(t)i.
Math 251-copyright Joe Kahlig, 15A
Example: Describe the curve defined by the vector function. Indicate the direction of motion.
(a) r(t) = t, t2 , 0
(b) r(t) = t, t2 , c , where c is a constant.
(c) r(t) = t, t2 , t .
(d) r(t) = h2 + t, 2 + 3t, 4 − 2ti, 0 ≤ t ≤ 1
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Math 251-copyright Joe Kahlig, 15A
D
Example: Show that the curve r(t) = sin(t), 2 cos(t),
What does the space curve for r(t) look like?
√
E
3 sin(t) lies on both a plane and a sphere.
Theorem Let r(t) = hf (t), g(t), h(t)i = f (t)i + g(t)j + h(t)k, where f , g, and h are differentiable
functions, then r′ (t) = hf ′ (t), g′ (t), h′ (t)i = f ′ (t)i + g′ (t)j + h′ (t)k
Definition: The unit tangent vector at t is defined to be T(t) =
r′ (t)
|r′ (t)|
Theorem Suppose u and v are differentiable vector functions, c is a scalar, and f is a real-valued
function. Then.
d
[u(t) + v(t)] = u′ (t) + v′ (t)
dt
d
[cu(t)] = cu′ (t)
dt
d
[f (t)u(t)] = f ′ (t)u(t) + f (t)u′ (t)
dt
d
[u(t) · v(t)] = u′ (t) · v(t) + u(t) · v′ (t)
dt
d
[u(t) × v(t)] = u′ (t) × v(t) + u(t) × v′ (t)
dt
d
[u(f (t))] = f ′ (t)u′ (f (t))
dt
chain rule
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Math 251-copyright Joe Kahlig, 15A
Example: Given r(t) = 3t, e2t−4 , sin(tπ) .
(a) Find a tangent vector to the curve at t = 0.
(b) Find T(0).
(c) Find a tangent line to the curve at the point (6, 1, 0)
(d) compute
Z
r(t) dt