Study Guide
CK 2:
VECTOR CALCULUS
Study Guide
Block 2 : V e c t o r Calculus
Study Guide Block 2: Vector Calculus Unit 1: Differentiation of Vector Functions 1.
Lecture 2.010 , ,
a .
Study Guide
Block 2: Vector Calculus
Unit 1: D i f f e r e n t i a t i o n of Vector Functions
2.
Read Supplementary Notes, Chapter 3.
3.
Read Thomas, S e c t i o n s 1 4 . 1 and 14.2.
a.
If
b.
If l i m f(x) =
x+a
c.
If
z
f
+
+
i s any c o n s t a n t v e c t o r , prove l i m c = C.
x+a
2
then l i m
x*a
[z
.
2.
f(x)l =
i s a d i f f e r e n t i a b l e f u n c t i o n of x and i f h ( x ) =
z f (x),
+
then h i s a l s o a d i f f e r e n t i a b l e f u n c t i o n of x and h' (x) = c
'?,
.
. '
...
f ' (x) .
(Note: t h e aim of t h i s e x e r c i s e i s t o have you s e e t h a t t h e prod u c t r u l e f o r d i f f e r e n t i a t i n g a c r o s s product i s s i m i l a r t o t h e
u s u a l product r u l e e x c e p t t h a t t h e o r d e r of t h e f a c t o r s i s c r u c i a l ;
+
i.e., 3' ( t ) x ij(t) = - g ( t ) x 3' ( t ).)
Let
f ( t ) = t~ + t2j+ ( 2 t + 1)i:and l e t ~ ( t= )t31+ 3 t 3 + (t2+ 112.
g ( t ) = f ( t ) X G ( t ) . Compute g ( t ) and then d i f f e r e n t i a t e t h i s
Let
r e s u l t t o obtain
Find
G'
( t ) and
3'
%'
(t).
(t) and then compute [ 3 ( t l X
G'
(t)]
+ ~ 2 ' ( t )X
How do t h e answers t o a. and b. compare?
b
[f (t) x
Compute
(t)]
Are t h e answers t o b.
a.
+
[G(t)
x
$1
( t )1.
and d. t h e same?
Why?
Mimic t h e corresponding s c a l a r procedure t o prove t h a t i f
d i f f e r e n t i a b l e f u n c t i o n of t a t t = tl then
AR =
dR
A t
t=tL
+ j: A t ,
where l i m $ =
At + O
(continued on n e x t page)
6.
%
is a
G(t)].
1
Study Guide
Block 2 : Vector C a l c u l u s
U n i t 1: D i f f e r e n t i a t i o n of Vector F u n c t i o n s
2.1.3
continued
,
b.
..
U s e a . t o prove t h e f o l l o w i n g v e r s i o n of t h e c h a i n r u l e :
I f $ i s a d i f f e r e n t i a b l e f u n c t i o n o f x and x i s a d i f f e r e n t i a b l e
&
f u n c t i o n of u, t h e n
d6
zi
c.
i s a l s o 6 d i f f e r e n t i a b l e f u n c t i o n o f u and
d% dx
(x)
(xi)
=
+
Let R = (t
+
1):
t
2-t
3 -+
4
t k and l e t t = u
.
d6
Compute a3 i n two
+
d i f f e r e n t ways, f i r s t by r e p l a c i n g t by u4 i n t h e formula f o r R,
+
-J
+
and t h e n by t h e c h a i n r u l e .
Show t h a t t h e two answers t h u s ob-
tained a r e equal.
2.1.4(L)
A p a r t i c l e moves a l o n g t h e c u r v e C i n t h e xy-plane where t h e
e q u a t i o n of motion i s g i v e n p a r a m e t r i c a l l y by x = x ( t ) and
y = y ( t ), where b o t h x and y a r e d i f f e r e n t i a b l e f u n c t i o n s o f t.
Let
6 ( t ) denote
t h e v e c t o r from t h e o r i g i n t o t h e p o i n t on C which
c o r r e s p o n d s t o t.
dR
a g r e e s w i t h t h e p h y s i c a l n o t i o n of speed a l o n g a curve.
a.
Show why
b.
What can w e deduce a b o u t t h e p a t h i f R i s 1
-b
?l(t2)
implies
-
1, i . e . ,
i f %(tl=
)
t h a t tl = t 2 ?
A p a r t i c l e moves a l o n g t h e c u r v e a c c o r d i n g t o t h e e q u a t i o n of
motion x = et and y = e
2t
.
a.
F i n d t h e v e l o c i t y and t h e a c c e l e r a t i o n o f
t h e p a r t i c l e a s f u n c t i o n s of t (where, o f c o u r s e , t d e n o t e s t i m e ) .
b.
A t t = 0 , what i s t h e v e l o c i t y ,
s p e e d , and a c c e l e r a t i o n o f t h e
particle?
c.
What i s t h e s l o p e o f t h e p a t h a t t = O?
2.1.6 (L)
a.
Assume t h a t
f
i s a d i f f e r e n t i a b l e f u n c t i o n of t.
- z(t),
ting f (t)
then $ ( t )
Z1(t)
show t h a t i f t h e magnitude o f
5 0.
By d i f f e r e n t i a -
f (t) is
constant
( c o n t i n u e d on n e x t page)
.-
-
2.1.3
.
,
.
Study Guide
Block 2: Vector Calculus
Unit 1: D i f f e r e n t i a t i o n of Vector Functions
2.1.6(L)
.
continued
b. Apply t h e r e s u l t of a. t o s e e what can b e concluded about t h e
motion of a p a r t i c l e i f t h e p a r t i c l e moves w i t h c o n s t a n t speed.
c. Prove t h a t i f a p a r t i c l e moves i n a c i r c u l a r p a t h w i t h c o n s t a n t
speed, t h e a c c e l e r a t i o n i s always along t h e r a d i u s drawn t o t h e
I
given p o i n t .
2.1.7
A p a r t i c l e moves s o t h a t i t s equation of motion i n v e c t o r form i s given by a . Show t h a t t h e p a r t i c l e moves w i t h c o n s t a n t speed.
+
b. Compute v and g, and v e r i f y t h a t
2 = 0 ( a s i t should be when
t h e speed i s c o n s t a n t ) .
.
c. S i n c e t h e magnitude of t h e speed i s c o n s t a n t , must t h e magnitude
of t h e a c c e l e r a t i o n a l s o be c o n s t a n t ? Explain.,
8
r
8
8
1
d. Find t h e v e l o c i t y and t h e a c c e l e r a t i o n of t h e p a r t i c l e when t = 5.
5
components, show
a. By w r i t i n g a l l f u n c t i o n s i n terms of
and
t h a t i f $' ( t ) = 3 ( t ) then t h e s e t of a l l f u n c t i o n s whose deriva+ +
t i v e w i t h r e s p e c t t o t i s f , i s given by { $ ( t ) + c:c i s an a r b i t r a r y vector constant).
b. A p a r t i c l e moves i n t h e xy-plane i n such a way t h a t i t s a c c e l e r a t i o n a t any t i m e t i s given by
Moreover, a t t = 0 , both
% and
c . Where i s t h e p a r t i c l e a t t =
i?
are
8.
Find
3
a s a f u n c t i o n of t.
1
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