,
and
are the components of the vector r(t) , then f, g, and h are
real valued functions called the component functions of
r and we can write:Contents [VF.1] [VF.2] [VF.3] [VF.4]
In general, a function is a rule that assigns to each element in the domain an
element in the range. A vector-valued function, or
vector function, is simply a function whose domain is a set
of real numbers and whose range is a set of vectors. This means for every
number t in the domain of r there is a unique vector in 3-D
space denoted by r(t). If
,
and
are the components of the vector r(t) , then f, g, and h are
real valued functions called the component functions of
r and we can write:
The limit of a vector function r is defined by taking the limits of its component function as follows.
If
,
then:
Provided the limit of the component functions exist.
A vector function r is continuous at a if
There is a close connection between continuous vector functions and space curves. Suppose that f, g, and h are continuous real-valued functions on an interval I. Then the set C of all points (x ,y ,z) in space, where
and t varies throughout the interval I, is called a space curve. The equations in the part above are know as parametric equations of C and t is called a parameter.
Contents [VF.1] [VF.2] [VF.3] [VF.4]
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