Review Vector Functions - (9.1)
1. Vector-Valued Functions
A vector-valued function in 2-D:
r Ÿt fŸt , gŸt fŸt i gŸt j, a t t t b.
xŸt fŸt and yŸt gŸt are called parametric equations. The set of points
x, y fŸt , gŸt ; a t t t b
describe a curve C in the xy "plane.
A vector-valued function in 3-D:
rŸt fŸt , gŸt , hŸt fŸt i gŸt j hŸt k. a t t t b.
xŸt fŸt , yŸt gŸt and zŸt hŸt are called parametric equations. The set of points
x, y, z fŸt , gŸt , hŸs , a t t t b
describes a space curve C.
Example r 1 Ÿt t, t 2 , for "1 t t t 1; r 2 Ÿt t, t 2 , 1 , for "1 t t t 1;
r 3 Ÿt t, t 2 , t , for "1 t t t 1
1
2
1
-1
t
0
1
-1
0
1
y
1
-1
0
1
-1
t
y
1
-1
r Ÿt ¡t, t 2 ¢, "1 t t t 1
r 2 Ÿt t, t 2 , 1
r 3 Ÿt t, t 2 , t
y x2, " 1 t x t 1
y x2, z 1
y x2, z x
Example r Ÿt 1
1
2 sinŸt ,
2 sinŸt , 2 cosŸt , for 0 t t t = ; for = t t t =.
2
2
2
1
1.4
1.2
1
0.8 t 0.6
0.2
0.4
0
-1
r Ÿt y x, z 2 sinŸt ,
y
1
2 sinŸt , 2 cosŸt , solid - 0, =/2 , dash - =/2, =
2 2 " x 2 , solid - 0 t x t 1
y x, z " 2 2 " x 2 , dash - "1 t x t 1;
Example r Ÿt 2 cosŸt , 3 sinŸt , t , 0 t t t 2=
8
6
4
-3
2
-2
0
-1
y t
1
1
2
-2
-1
2
3
1 x2 1 y2 1
9
4
solid - 0 t x t 2=, dash - 2= t x t 3=
r Ÿt 2 cosŸt , 3 sinŸt , t ,
Vector-valued functions for special curves:
a. A line segment from the point P 1 x 1 , y 1 , z 1 to the point P 2 x 2 , y 2 , z 2 :
rŸt x 1 , y 1 , z 1 t x 2 " x 1 , y 2 " y 1 , z 2 " z 1 , 0 t t t 1
2
A line passes through the points P 1 x 1 , y 1 , z 1 and P 2 x 2 , y 2 , z 2 :
rŸt x 1 , y 1 , z 1 t x 2 " x 1 , y 2 " y 1 , z 2 " z 1 , " . t .
b. A circle with center h, k and radius r, Ÿx " h 2 Ÿy " k 2 r 2 in the plane z z 0
rŸt h r cos t, k r sin t, z 0 , 0 t t t 2=
c. An ellipse
Ÿx " h 2
Ÿy " k 2
1 in the plane z z 0 :
a2
b2
rŸt h a cos t, k b sin t, z 0 , 0 t t t 2=
2
y2
d. hyperbola x 2 " 2 1 in the plane z z 0 :
a
b
rŸt h a cosh t, k b sinh t, z 0
Example Find two vector functions r Ÿt for each given the curve C:
a.
4x 2 y 2 9z 2 1
b.
y 2x
x 2 y 2 25
z 25 " x 2
a. Ÿi x t, y 2t, z o 1 " Ÿ4t 2 4t 2 o 1 " 8t 2
C1 : r 1 Ÿt t, 2t, 1 " 8t 2 ; and C 2 : r 2 Ÿt t, 2t, " 1 " 8t 2
Ÿii x 1 cosŸt , y cosŸt , z 13 sinŸt 2
1 cosŸt , 1 cosŸt , 1 sinŸt C:
rŸt 3
2 2
2
b. Ÿi x t, z 25 " t 2 , y o 25 " t 2
C1 : r 1 Ÿt t, 25 " t 2 , 25 " t 2 ; and C 2 : r 2 Ÿt t, " 25 " t 2 , 25 " t 2
Ÿii x 5 sinŸt , y 5 cosŸt , z 25 " 25 sin 2 Ÿt 25 cos 2 Ÿt C: rŸt 5 sinŸt , 5 cosŸt , 25 cos 2 Ÿt 2. Derivative of a Vector Function:
Let fŸt , gŸt , and hŸt be differentiable. If r Ÿt fŸt , gŸt , hŸt , then
U
U
U
U
r Ÿt f Ÿt , g Ÿt , h Ÿt .
Chain Rule: if r is a differentiable vector function and s uŸt is differentiable scalar function, then
dr dr ds r U Ÿs u U Ÿt .
ds dt
dt
Tangent line: the parametric equations of the tangent line to the curve rŸt at the time t t 0
U
Ÿt L
rŸt 0 t r Ÿt 0 .
U
UU
Example Let r Ÿt t 2 , lnŸt , tan "1 Ÿt . Find r Ÿt , r Ÿt and the tangent line at the point where
t 1.
3
UU
2t
1
2t, 1 ,
, r Ÿt 2, " 12 , "
t 1 t2
t
Ÿ1 t 2 r U Ÿ1 2, 1, 1 , rŸ0 1, 0, =
4
2
U
tangent line: LŸt rŸ1 t r Ÿ1 1, 0, = t 2, 1, 1 1 2t, t, = 1 t
2
2
4
4
r U Ÿt Example Let r Ÿs t 1.
r U Ÿs s cosŸ2s , s sinŸ2s , tanŸ2s and sŸt t 2 " t 1. Find dr and dr when
dt
dt
1 cosŸ2s " 2 s sinŸ2s , sinŸ2s 2s cosŸ2s , 2 sec 2 Ÿ2s , s U Ÿt 2t " 1
2 s
dr r U Ÿs s U Ÿt Ÿ2t " 1 dt
1 cosŸ2s " 2 s sinŸ2s , sinŸ2s 2s cosŸ2s , 2 sec 2 Ÿ2s 2 s
When t 1, sŸ1 1.
dr | t1 Ÿ1 1 cosŸ2 " 2 sinŸ2 , sinŸ2 2 cosŸ2 , 2 sec 2 Ÿ2 2
dt
3. Integrals of Vector Functions:
Let fŸt , gŸt , and hŸt be integrable. Then
; rŸt dt ; fŸt dt , ; gŸt dt , ; hŸt dt
b
b
b
,
b
; a rŸt dt ; a fŸt dt , ; a gŸt dt , ; a hŸt dt
Length L of a space curve C : r Ÿt fŸt , gŸt , hŸt , a t t t b :
L
b
;a
2
2
2
¡f U Ÿt ¢ ¡g U Ÿt ¢ ¡h U Ÿt ¢ dt 1
UU
1
1 t2
Example Let r Ÿt t 2 ,
r U Ÿt r U Ÿt dt
1
i tj t 2
k dt
1 t2
i tj t 2
k dt Ÿtan "1 Ÿt | 10 i 1 lnŸ1 t 2 | 10j Ÿt " tan "1 Ÿt | 10 |
2
=
1
, lnŸ2 , 1 " =
4 2
4
Example Evaluate the integral ;
;0
b
;a
1
0
t 1 , e "2t , r Ÿ0 U
r Ÿt 1
i
2
U
" 2k and r Ÿ0 i j k. Find r Ÿt .
; r UU Ÿt dt ; t 2 dt, ;
t 1 dt,
; e "2t dt
1 t 3 , 2 Ÿt 1 3/2 , " 1 e "2t C
1
3
3
2
1 1, 1 , 3
1 ¡1, 1, 1¢, C
r U Ÿ0 0, 2 , " 1 C
3
2
3 2
U
rŸt ; r Ÿt dt ; 1 t 3 dt, ; 2 Ÿt 1 3/2 dt, ; " 1 e "2t dt 1, 1 , 3 t
2
3
3
3 2
5/2
1
2
2
1
1
3
4
"2t
t ,
e
,
t C2
1,
Ÿt 1 ,
12
3 5
4
3 2
4
rŸ0 0,
rŸt 4 , 1 2 1 , " 4 , " 9
2 1 , 0, " 2 , C
0C
15 4
2
2
15
4
1 t 4 t 1 , 4 Ÿt 1 5/2 1 t " 4 , 1 e "2t 3 t " 9
2 15
3
2
12
15 4
4
Example Find the length of the curve C: r Ÿt t, t cosŸt , t sinŸt , 0 t t t =.
U
r Ÿt 1, cosŸt " t sinŸt , sinŸt t cosŸt ,
r Ur 1 ŸcosŸt " t sinŸt 2 ŸsinŸt t cosŸt 2 2 t 2
L
5
=
;0
r Ÿt UrŸt dt =
;0
2 t 2 1 = Ÿ2 = 2 " 1 ln 2 ln = Ÿ2 = 2 2
2