vector-valued function∗
pahio†
2013-03-22 3:05:44
Let n be a positive integer greater than 1. A function F from a subset T
of R to the Cartesian product Rn is called a vector-valued function of one real
variable. Such a function joins to any real number t of T a coordinate vector
F (t) = (f1 (t), . . . , fn (t)).
Hence one may say that the vector-valued function F is composed of n real
functions t 7→ fi (t), the values of which at t are the components of F (t).
Therefore the function F itself may be written in the component form
F = (f1 , . . . , fn ).
(1)
Example. The ellipse
.
{(a cos t, b sin t) .. t ∈ R}
is the value set of a vector-valued function R → R2 (t is the eccentric anomaly).
Limit, derivative and integral of the function (1) are defined componentwise
through the equations
• lim F (t) := lim f1 (t), . . . , lim fn (t)
t→t0
t→t0
t→t0
• F 0 (t) := (f10 (t), . . . , fn0 (t))
Z
•
b
Z
F (t) dt :=
a
b
Z
f1 (t) dt, . . . ,
a
b
!
fn (t) dt
a
The function F is said to be continuous, differentiable or integrable on an interval [a, b] if every component of F has such a property.
∗ hVectorvaluedFunctioni created: h2013-03-2i by: hpahioi version: h41915i Privacy
setting: h1i hDefinitioni h26A36i h26A42i h26A24i
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1
Example. If F is continuous on [a, b], the set
.
γ := {F (t) .. t ∈ [a, b]}
(2)
is a (continuous) curve in Rn . It follows from the above definition of the derivative F 0 (t) that F 0 (t) is the limit of the expression
1
[F (t+h) − F (t)]
h
(3)
as h → 0. Geometrically, the vector (3) is parallel to the line segment connecting (the end points of the position vectors of) the points F (t+h) and F (t). If F
is differentiable in t, the direction of this line segment then tends infinitely the
direction of the tangent line of γ in the point F (t). Accordingly, the direction
of the tangent line is determined by the derivative vector F 0 (t).
2