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This lecture explores vector-valued functions and curves in three-dimensional space (r^3). It delves into the concept of vector-valued functions, their domains, and how they define curves in space. The lecture provides examples of finding domains, describing curves, and sketching curves associated with vector-valued functions. It also includes a review of polar coordinates in two dimensions and their relationship to rectangular coordinates.
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WEEK 3 LECTURE 2 Example
o D and Q 4 2 2 intersectthe^ plane a y t 2 6 Solution PG o (^) D It QC't L Ez (^) of line^ point^ PC 1 o^ D direction ve doit b to v POI^4 y z o^ Z^ D V 3 2 I a b^ c I III^ t.r.net We will^ find the spefic Value of t (^) when this line intersects^ the^ plane Xt^ y t 2 6
It 3T^ C 2E^ It^ t^6 y z 2 Zf^ G^ 2T^4 t Finally t (^2) correspond to^ the^ point on^ the^ line
1 3
7 RC (^7) 4, Y Z Z 4 (^2) It z 3 CHAPTER
RE Calculus (^) III functions f R R Now we^ will^ discuss^ functions which attack to (^) a real number^ at a vendor^ k 7 Vectorvalued
set fat^ get in componenfutations f (^) g h (^) R R
Constrains first component F^ I t i^ zo t^ z^ a a (^) second component en (^) LEED no constrains (^) here third component fft 5 t Zo^ t All (^) together we obtain that^ the domain^ ofLCE^ is 1 5 REMARII (^) Aug vector (^) valued (^) function set L fat (^) get htt
(^1123) that is traced out by the tip of^ tee moving vector ret aHitf getthe t (^) t T Equation of^ awe C fyz fqft Caparameter
Example Describe the^ curve^ determined^ by the vector valued function
L z^ t^4 3T^ Itt Solution FEI^ FEI htt curve C^ x^2 t y (^4) 3T g of^ a line C Z^ I t (^) t 2 P 2, l 1,407 4 t I (^) QU n^ o y fo p 2,4 1 y i Exemp le (^) sketch the^ curve^ associated^ to (^) thevector valued (^) function set E t't t^ E ittj t.ie Solution (^) cure C 3 x^ t yet't t any
scalar
Kmart kzo it's^ a^ distance
by
be positive (^) if movingfrom^ Ox counterclockwise negative if^ moving
ox clockwise Ex n i r R t (^) I f FI n f opc e^ n^ t I 4
PEE EI or (^) Pera If 4 REMARK ft (^) PG
4 47 Relation between^ Cx^ y
r VFL
f cost X reost 1 suit (^) y rsn
