Study Guide
Block 2: Vector C a l c u l u s
U n i t 3 : Space Curve and t h e Bi-Normal Vector ( O p t i o n a l
The e x e r c i s e s i n t h i s u n i t e x t e n d t h e c o n c e p t of t a n g e n t i a l and
normal v e c t o r s i n t h e p l a n e i n t o a 3-dimensional analog.
The
r e s u l t s d i s c u s s e d h e r e a r e n o t needed i n any of o u r l a t e r work, and
it i s f o r t h i s r e a s o n t h a t w e have made t h i s u n i t o p t i o n a l .
On t h e
o t h e r hand, t h e m a t e r i a l p r e s e n t e d h e r e does have p r a c t i c a l a p p l i c a t i o n , and much of i t s e r v e s a s an i n t r o d u c t i o n t o D i f f e r e n t i a l
Geometry.
I n a d d i t i o n , by t r y i n g t o s o l v e t h e e x e r c i s e s i n t h i s
u n i t , you w i l l g e t a d d i t i o n a l review and r e i n f o r c e m e n t o f t h e major
p r i n c i p l e s of t h e p r e v i o u s u n i t .
1.
Exercises :
a.
Mimic o u r p r e v i o u s d e f i n i t i o n s t o show what i t means f o r t h e s p a c e
curve
6(t)
t o be a c o n t i n u o u s f u n c t i o n o f t.
-+
e q u a t i o n i s g i v e n i n t h e form, R ( t ) = x ( t ) f
t h a t ifR i s a c o n t i n u o u s f u n c t i o n of
a r e c o n t i n u o u s f u n c t i o n s o f t.
b.
In particular, i f the
+ y ( t ) 3+
z ( t ) < , show
t a s soon a s x ( t ) , y ( t ) , and z ( t )
(1) Again, mimicking t h e 2-dimensional c a s e , f i n d formulas f o r
v e i o c i t ~ ,s p e e d , and a c c e l e r a t i o n o f a p a r t i c l e t h a t moves a l o n g
-+
t h e c u r v e C a c c o r d i n g t o t h e e q u a t i o n , R = x ( t )f
-+
+
c.
-+
I n p a r t i c u l a r , f i n d v , a, and
sin 3t j+ t
(2)
z.
(21
+
y ( t ) ?+ z ( t ) % .
i f C i s g i v e n by
5
= cos 3 t
( 3 ) I n the example above, show t h a t
had t o happen,
2
With t h e c u r v e C a s d e f i n e d i n b . ( 2 )
above, f i n d a u n i t t a n g e n t
1
= 0 and e x p l a i n why t h i s
v e c t o r t o t h e curve.
3.
-+
L e t t i n g T denote t h e u n i t tangent v e c t o r
, c u r v e , dP
az --
K
8,
8
where
=.
and d i r e c t i o n a s d?
dii
z,
show
t h a t f o r a space
i s a u n i t v e c t o r which h a s t h e same s e n s e
I n p a r t i c u l a r , d e t e r m i n e r and
8
f o r t h e curve
given i n b. ( 2 ) above.
e.
I n t h e same way t h a t w e need t h r e e u n i t v e c t o r s i n C a r t e s i a n coort
j+k),
w e now need a t h i r d u n i t v e c t o r
t o go a l o n g w i t h 9 and 8. To-+this end w e d e f i n e a binormal v e c t o r ,
-+
-+
-+
-+
dB
B , by B = T X N.
Show t h a t
can .be w r i t t e n i n t h e form, rb, and
i n t e r p r e t t h i s geometrically.
d i n a t e s f o r 3-space
(1,
3 , and
,
Study Guide
Block 2: Vector Calculus
Unit 3: Space Curve and t h e Bi-Normal Vector (Optional)
+.
8
,
.
1v x a1
a.
Show t h a t t h e formula
b.
Given t h e curve 3 = t:
which i s perpendicular t o
K
=
a p p l i e s t o 3-space a l s o .
T
+ 2 t 2 3 + t3z,
8
f i n d t h e equation of t h e plane
a t t h e p o i n t corresponding t o t = 1.
c.
Compute t h e c u r v a t u r e of t h e above curve a t t = 1.
d.
Express
5
and
8
f o r t h i s same curve a t t = 1.
Our space geometry now i n c l u d e s t h e formulas:
d 8 = ~ 8 . Complete t h i s l i s t by computing :d
=.
dli
=
I,d$
=
r8,
and
2.3.4
i s given i n t h e form 8 = %(s)where s denotes a r c l e n g t h .
2+
d38
+
+
Compute z,
and
i n t e r m s of 3, N, and B.
ds
ds
A curve
a.
b.
ds -
-
From a. compute
t o f i n d t h e formula f o r .r.
.
a.
Use b. t o f i n d formulas f o r
8 = x ( s ) f + y ( s ) )+ z(s)Z.
and T i f
3
is expressed i n t h e form
Assume 3 i s given a s a f u n c t i o n of time, t , r a t h e r than of a r c l e n g t h
s. Mimic t h e d i s c u s s i o n on E x e r c i s e 2.2.4 t o show t h a t
(continued on n e x t page)
2.3.2
K
Study Guide
Block 3: Vector C a l c u l u s
U n i t 3 : Space Curve and t h e Bi-Normal Vector ( O p t i o n a l )
2.3.5
b.
continued
+
Compute j - t l f o r t h e c u r v e R =
tl +
2-t
2t 3
+
3+
t k a t t = 1.
MIT OpenCourseWare
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Resource: Calculus Revisited: Multivariable Calculus
Prof. Herbert Gross
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