Study guide
Calculus Chapter 2
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This chapter includes notes on Vector Differentiation, Curvature, Torsion and Trajectories. Each section includes multiple worked examples. Notes were taken on an iPad
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CHAPTER 2CALCULUS
VECTORS
, Vector
Differentiation
Suppose fat [ =
Ntt ,
ylt ) ] is the position
vector of a particle that moves with time t .
PET
I *
O
}
DE the
Velocity tangential to curve
. =
at
DE
[ =o and
Faf speed
.
=
For example ,
consider :
x2ty2 =/
change to parametric form
ff{
e⇒f
Fo
wit
fsint
sint
cost ( cast )
}
x [ (t )
= = ,
Sint
y
=
: cost )
dd¥
= .
°
⇐ ¥
:{:{toII
⇐
←
*
↳ when to FIT F
t=E ←
on
*
t=t
:{ YI
? , f.
,3- Dimensional Cases ##]
4%
a
of
Helix
=
Consider I A) =
( cost , sintut )
,
•
3g
x Z
y
EYE
so as it farms a circle ,
e-
it moves upward *
.
vector tangent to the helix :
d£=(
.
sint ,
cost ,i
) } 3
works in
dimensions as well
, Logarithmic LpiralDD
Consider Z =
debt where x. B are E ¢
€ mine tangential
vector :
IBI
so as is increased
daft =
xeist .
B =
pz } so is
arg CB )
⇒ spiral
.
Now let 13
=
Bit ipa
So z =
a @(
iBa)t
Bit
IZI =
1×11 eat / |eiP2t/
T.EE#oEcaspttisinBt
'3 't
Kitt
=
Idle =
distance between
origin and particle
so as & increases when B. TO
@ pz
IZIT ¥¥
when 13 . , ↳
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