3.1 - Vector-Valued Functions and Space Curves
Definition of a Vector-Valued Function
A vector valued function is a function in the form of:
r(t) = f( t )i + g( t )j
r(t) = f( t )i + g( t )j + h( t )k
Where the component functions f, g, and h, are real-valued functions of the parameter t. They are also written in this form:
r(t) = < f( t )i , g( t )j >
r(t) = < f( t )i , g( t )j , h( t )k >
The parameter t can lie:
Between real numbers: a </ t </ bε
t might take on all real numbers
Domain restrictions that enforce restrictions on t
Graphing Vector-Valued Functions
The graph of a vector-valued function of the form r(t) = f( t )i + g( t )j consists of the set of all ( t, r( t ) ), and the path it traces is called a plane curve
The graph of a vector-valued function of the form r(t) = f( t )i + g( t )j + h( t )k consists of the set of all ( t, r( t ) ), and the path it traces is called a space curve
Any curve has an infinite number of reparametrizations
r(t) = f( t )i + g( t )j can be defined as x = f( t ) and y = f( t )
Limits and Continuity of a Vector-Valued Function
A vector-valued function r approaches the limit L as t approaches a.
lim r( t ) = L
t --> a
Provided:ε0
lim || r( t ) - L || = 0
t --> a
Limit of a Vector-Valued Function
Let f, g, and h be functions of t. Then the limit of the vector valued function r(t) = f( t )i + g( t )j as t approaches a is given by
lim r( t ) = [ lim f( t ) ] i + [ lim g( t ) ] j
t --> a
lim r( t ) = [ lim f( t ) ] i + [ lim g( t ) ] j + [ lim h( t ) ] k
t --> a
Continuity
Let f, g, and h be functions of t. The the vectors r(t) = f( t )i + g( t )j and r(t) = f( t )i + g( t )j + h( t )k are continuous at point t = a if the following three conditions hold
lim r( t ) exists
t --> a
r ( a ) exists
lim r( t ) = r ( a )
t --> a