Calculus Chapter 2

CHAPTER 2
CALCULUS
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 . , ↳