This document clearly describes, with detailed notes and examples, how to evaluate/solve the following:
~ Vector differentiation
~ Curvature
~ Torsion
~ Trajectories and orthogonal trajectories
as taught by the University of the Witwatersrand.
As a student, I am always searching for a gr...
2. 1 .
✓ECT0RIFFERENTIATl0
REMINDER OF :
THIS IS AN EXAMPLE OF A VECTOR
PARAMETRIC EQUATION OF A CURVE :
VALUE
VARIABLE
FUNCTION
rct)
OF A SINGLE SCALAR
(1) POSITION VECTOR :
POSITION OF A CURVE :
EMPHASISE r IS A FUNCTION OF t
(f)
t ONE
tf
r DEPENDS ON
f
-
:
= pw SCALAR SINGLE VARIABLE !
0 THINK OF t AS TIME AND -
REPRESENTS
SYMBOL :DISPLACEMENT THE TRAJECTORY THAT A PARTICLE OBJECT
VECTOR OF A CURVE
WILL FOLLOW AS TIME MOVES ALONG .
OR
I r→
or i. WILL HAVE TWO COMPONENTS WHICH ARE EACH A FUNCTION OF t GIVING X co-ORDINATE CORD / NATE
-
AND
-
Y
-
-
.
( ✗ (t) ; yct))
OF POSITION VECTOR OF PARTICLE AT
IN TWO DIMENSIONS ( 2D) : r =
anytime .
e. THREE COORDINATES / COMPONENT •
IN THREE DIMENSIONS ( 3D ) : r =
( Xlt) ; yct) ; 2- (t) )
(2) VELOCITY VECTOR :
VELOCITY OF A CURVE :
✓
OR
(f) =
dt
dr (CAN ALSO WRITE R' ( PRIME) )
{ DIFFERENTIATION OF EACH
CO ORDINATE /POSITION
- VECTORS }
it) Ict)
EACH
>
DERIVATIVE OF
ORDINATE •
G ly
- -
PRIME of t)
co -
>
dn
-
-
1- N TWO ( 2D) :
x'Ct ) ; y'Lt)
DIMENSIONS dt
dr
(t) ; yet) ;
'
Ect )
IN THREE DIMENSIONS ( 3D ) :
dt
✗
PICTURE)
CURVE :
t
Ñ%)"°N"="°R°FCURVEAT€=
" :
7 Asti ? g. #
pan , Ast T :
dr
z WILL THE
BECOME
dt
TANGENT VECTOR OF THE CURVE AT
THAT POINT IN TIME
dt
VELOCITY IS TELLING
•
US INSTANEOUS
DIRECTION OF
CURVE AT THAT
TIME !
•
( ;)
(
CO-ORDINATE / POINTS TO POSITION /
VECTOR DISPLACEMENT VECTOR
AT PARTICULAR TIME
EXAMPLES :
(1) r ( t) ( cost ; stint ) f. can see curve in 2D !)
DRAW :
Ynd? • As tT( anytime) ,
randy /
T
dt
I ( cost ; Sint) WILL BE ON THE
- t= €2 UNIT CIRCLE !
y
A 2 CALCULATE VELOCITY VECTOR :
g.
pic7URh
T dr
C- sint ; cost )
t=ñ
Ict)d
↳ ÷÷÷÷÷
3
.at??to?a:Yiii-??=.imiiiiii.siiii-)pomisn.-id.t:t-
ARROW / VECTOR WHOSE BASE /STARTING
POSITION /POINT
o :( o ;D
y -
I
-
Ñq > X
CORRESPONDING
TO DISPLACEMENT
COMES BACK TO
N VECTOR AT THAT
STARTING POINT !
TIME !
7 MOVES
ANTI-CLOCKWISE !
I → DUE TO DOT PRODUCT
-
* POSITION CAN BE ANYWHERE LI ,
,
g
THE CURVE !
-
ALONG
dr
✓
1- =
3¥ I rct)
dt ( DISPLACEMENT
VECTOR )
POINTS IN
ANTI CLOCKWISE
-
DIRECTION OF MOTION
•
, ✗
-
co-ORDINATE
a
§ >
z
(2) rlt) =
( cost ;sint ; t) (i. HAVE A PARAMETRIC CURVE IN 3D
.
)
1 GRAPH CURVE /DRAW : 3D CURVE :
(1) FIRST : Look AT THE CURVE FROM ABOVE
MY • LOOKING ALONG 2- AXIS (VERTICAL)
|
-
L i. LOOKING AT X -
Y PLANE !
"
r → GOING ANTI-CLOCKWISE
o :* Around um >
at
-
-
CIRCLE !
,
i
(2) FROM THE SIDE : TO GET THE 3RD DIMENSION
LOOK AT HOW -
COORDINATES CHANGE
TAKE 2D CURVE AND FLIP IT :
^
A Z
qt=4ñ i. EACH CO ORDINATE
-
INCREASES
1 STEADILY
GRAPH CAN BE
^ AT A CERTAIN CONSTANT
EXTENDED
☐ news
IN
.
.nu
BOTH
pn.ge :@ , ,µt,
t can BE tore / -
ve
I
ddffd.IE?neds:0n
⑥ a ;o ;D
✓
L
Sy
to
3C
Copy COMING OUT
✓
OF BOARD HORIZONTAL ALONG
.
BOARD
6
oo WHEN EXTEND ABOVE CURVE ONE WILL GET
,
A HELIX ( 7T¥ ;¥÷e§?:)
• SHAPE OF A SCREW DRILL TYPE OF
THING ALONG OUTER EDGE .
GOING AROUND IN CIRCLES BUT MP /DOWN WITH z -
COORDINATE
WITH A CONSTANT PITCH / INCLINE /SLANT
2 VEL CITY :
dr
dt
=
C-sinti cost ; 1)
AS
0
TWO METHODS FOR REPRESENTING 2D CURVES / PLANE CURVES
.
PARAMETRIC
!
•
CURVES
(1) COMPLEX NUMBER METHOD
(2) POLAR EQUATIONS METHOD
KID
-
COMPLEX CURVES ARE CURVES IN 2D THAT ARE WRITTEN :
FUNCTION DEFINING
CO -
ORDINATE
Y
-
co-ORDINATE AS
AS 2 OF COMPLEX
COMPLEX PART I
NUMBER
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