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Lecture 15

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MULTIVARIABLE
CALCULUS
RUTGERS UNIVERSITY- NEW BRUNSWICK
Intro to Vectors C Vectors
in Plane) Topic 1
Vector
Have both a speed Cmagnitude) and a on a
cordinate system
given f the
vactor.
3th
AB
Can multiply by constants (called "scalars")
can chan the length of
ge Can reverse direction of vector
and
3
V
Note: Scalar multiples are
PcL
and
Adding Subtracting vectors
Note: order doesn't matter.
Ex
V-w
x vector V
with an initiat P
the origin a terminal point
at PCv, v2) is called a position yes or
of PCy,,v2) and is denoted.
v,
cung vector can be translated to position vector.
Slope
(I, 2
Posit
vector Length CMagnitudey
2/s
Slope V
slope
or
Working with position vector:
a2
For Sc
"C" CV Cv, Gv.
Ecc
2a
a 4- b K-4,
MULTIVARIABLE CALCULUS RUTGERS UNIVERSITY- NEW BRUNSWICK Intro to Vectors C Vectors in Plane) Topic 1 Vector Have both a speed Cmagnitude) and a on a cordinate system given f the vactor. 3th AB Can multiply by constants (called "scalars") can chan the length of ge Can reverse direction of vector and 3 V Note: Scalar multiples are PcL and Adding Subtracting vectors Note: order doesn't matter. Ex V-w x vector V with an initiat P the origin a terminal point at PCv, v2) is called a position yes or of PCy,,v2) and is denoted. v, cung vector can be translated to position vector. Slope (I, 2 Posit vector Length CMagnitudey 2/s Slope V slope or Working with position vector: a2 For Sc "C" CV Cv, Gv. Ecc 2a a 4- b K-4,Unit. Vector: A vector with a length 1
To find a unit vector from an Position vector, divide b
tude
we can define a vector as
I
Magnitue (ll V
ll)> unit vector
ca)
Note: a unit vector usually indicates
the direction
VII- 35 T 2, Now W
Ex: A unit vector in
oppo3ite direction. of
Standard Basis Vectors:
x direction i I,o
direction unit vector ca) Note: a unit vector usually indicates the direction VII- 35 T 2, Now W Ex: A unit vector in oppo3ite direction. of Standard Basis Vectors: x direction i I,o direction
o, 13
PO2
Note: Too planes are l if normals are
Angle bleu Plomes s Angle bluo normals
not solar mult
Perpendicular)
6-3 2 O
69
arccos
CDs
Note: To find the angle
bluo a line t a plane
CPlane p
COS O
Ample blio line aplane z qo cost In.V
Note: tmuersedion of line d Plane: "Plug ne into plane.
a rotat, etc
P s,o, t)
Ez: Find for line of intersedion
Lines need a Point a directonwabr
2 42 3
tlft
Find Plane uf PCR, 4,s) Contain the line 20-2. 2 21, d 3t, Z. 3 st. PO 3,-4,s) Flame n n must be in the plane we need n -to find v PQ 3,2,1> o, 13 PO2 Note: Too planes are l if normals are Angle bleu Plomes s Angle bluo normals not solar mult Perpendicular) 6-3 2 O 69 arccos CDs Note: To find the angle bluo a line t a plane CPlane p COS O Ample blio line aplane z qo cost In.V Note: tmuersedion of line d Plane: "Plug ne into plane. a rotat, etc P s,o, t) Ez: Find for line of intersedion Lines need a Point a directonwabr 2 42 3 tlftAK nstance blu faint Pane
Disdama bliw parallel planes.
d -d
K2,-3
Distance blur skewo lines
skew lines can ke contameel by parallel planes
y Direcion vedors of lines Vid
13 Common normal n
Limes have roints,.P,
D
b line d Point
u sin 9
o sing
D.
u XV
Ext PCI,-2, 3) Line 2
Z +3
12.1 Vector Function s
Parametric Equation: FCH, 3CH,"t" s the
paremeter for some interval, 'I', on a common domain.
But, vectors are defined by
"r
the terminal points of the vedors create
a curv thru space
ota surface.
For the domain op "t"
A vector Fundfon is a parame
olefined meHon uohere
inal peints of our ftace a eurve in 3-D
CThe pact that ut" has a certain al
Cun.
aves TCU an
orientotion)
nt
Ex
nal Point 3, s, 2
Termu
omain
Find vectors at t 2 t 4
Att 2
AK nstance blu faint Pane Disdama bliw parallel planes. d -d K2,-3 Distance blur skewo lines skew lines can ke contameel by parallel planes y Direcion vedors of lines Vid 13 Common normal n Limes have roints,.P, D b line d Point u sin 9 o sing D. u XV Ext PCI,-2, 3) Line 2 Z +3 12.1 Vector Function s Parametric Equation: FCH, 3CH,"t" s the paremeter for some interval, 'I', on a common domain. But, vectors are defined by "r the terminal points of the vedors create a curv thru space ota surface. For the domain op "t" A vector Fundfon is a parame olefined meHon uohere inal peints of our ftace a eurve in 3-D CThe pact that ut" has a certain al Cun. aves TCU an orientotion) nt Ex nal Point 3, s, 2 Termu omain Find vectors at t 2 t 4 Att 2Scatchi
2y use 1 or more components to get a curve or
surface
Caet rid of t)
For two components, scatenh on a plane.
for three
components, the curve is
on a surface.
use values op "t" to find Points orientation.
4
JE 4-20
Tr Co
cut t
at t 24 r
hof
t
Ex
2-21
t 2-1 3-2, S-22e
Cat tsoy
Prometric ucubn op
e PCI, 2,3) o
x use the ey
least one component value of 't', b make at 2 QC3,0, -1)
Ex
J A cylinder alo
ng z" w/ trace on
Plame
More than 1
Surface is possible....
to ones You are
4,8)
faruliar wi
The component you
don't use Arst gives
Curv. On
dncul surface.
urface,
2 t3 gives curve on
at t-o
at t 32 Pa 2,4,8
z cost
glind er al
Trace
Plane
z t is on the colinder.
at t
at t
C2
Scatchi 2y use 1 or more components to get a curve or surface Caet rid of t) For two components, scatenh on a plane. for three components, the curve is on a surface. use values op "t" to find Points orientation. 4 JE 4-20 Tr Co cut t at t 24 r hof t Ex 2-21 t 2-1 3-2, S-22e Cat tsoy Prometric ucubn op e PCI, 2,3) o x use the ey least one component value of 't', b make at 2 QC3,0, -1) Ex J A cylinder alo ng z" w/ trace on Plame More than 1 Surface is possible.... to ones You are 4,8) faruliar wi The component you don't use Arst gives Curv. On dncul surface. urface, 2 t3 gives curve on at t-o at t 32 Pa 2,4,8 z cost glind er al Trace Plane z t is on the colinder. at t at t C2Derivatives of Vector Functions
Fox)
elim (t st CTangen vector b a space curve)
At
CInstantanious velod
Ex
tit tsA Direction vector of a
nt line
r it 2ti +3t2A to the space curve at
for
Ex
r (t
COS
+t
st, 2 sin ty osts2s
Ez
sketch K4
2 Find
3- Find tangent vedtor at fo.T
FCH 4 cost 2 sint
cos t simi t a
stint z
Costs
Ct) K-4 sint, 2 cost
initial raint at the point of tampen
P.o.T. C2, N3)
Tamgemt vedor also gives orientation
of vector. Functions.
Ex rat) 2* Find Tangent veetor at t-o
Unit To
Vectors
TCt)
so TC)
Tangent Lines Needs po.T
To
veutor Cat Pg.
Find Era op tangent hme at ts2
t
4t
PO.T
nes
t i z
Line
3 9
Derivatives of Vector Functions Fox) elim (t st CTangen vector b a space curve) At CInstantanious velod Ex tit tsA Direction vector of a nt line r it 2ti +3t2A to the space curve at for Ex r (t COS +t st, 2 sin ty osts2s Ez sketch K4 2 Find 3- Find tangent vedtor at fo.T FCH 4 cost 2 sint cos t simi t a stint z Costs Ct) K-4 sint, 2 cost initial raint at the point of tampen P.o.T. C2, N3) Tamgemt vedor also gives orientation of vector. Functions. Ex rat) 2* Find Tangent veetor at t-o Unit To Vectors TCt) so TC) Tangent Lines Needs po.T To veutor Cat Pg. Find Era op tangent hme at ts2 t 4t PO.T nes t i z Line 3 9Ex
Taty et cost, ét sinut, sin t et t o
nt, sin t tetoost
e cost-e
Line
I-t z at
3 4
cost cut Integration hy parts
dt
sint
So, cost dt z t cost- Sint dt t t ces t +c
r (t
I an
3/2
tan.
t a, as ,a
4-t t In sect
2,1, -2
sect e
Ex Taty et cost, ét sinut, sin t et t o nt, sin t tetoost e cost-e Line I-t z at 3 4 cost cut Integration hy parts dt sint So, cost dt z t cost- Sint dt t t ces t +c r (t I an 3/2 tan. t a, as ,a 4-t t In sect 2,1, -2 sect eAre Length f Arc length
meteriegation
Length From P. E
dt for teca
For 3-D
tt I. Can
Tive arc length: telaib1
So,
CHas "t")
entth Sundt
K4 cost, 3, -4 sin t
s out st
Ex
F (t) et cost, e sint, e
o s t s
cast er sint, etsint et cost,et
2t
2t.
e
Ex: Parameter atton From previous erample
For 'smooth curves' we can reparameterige by Aro Length
ft 1. we can "walk the curve
2. Some Formulas are easier.
%s have a magnitude 1. Vector
uut lagend)
a. rics) will al
Ex
Find Are zength Fnction (Not 'L',
I, 2, 3
a is where
t begins
Now, 50t
CRefiaee't" with
"u")
SCt
K2 cost-t sint, sint t tcos ty
tt
sin t t sin t +2f
tt
St t
St
Constant is
s in the
denominudory
tan
Vst e tan a
do
t tan
dts sec 9
t Sec ottano
seco tame
Are Length f Arc length meteriegation Length From P. E dt for teca For 3-D tt I. Can Tive arc length: telaib1 So, CHas "t") entth Sundt K4 cost, 3, -4 sin t s out st Ex F (t) et cost, e sint, e o s t s cast er sint, etsint et cost,et 2t 2t. e Ex: Parameter atton From previous erample For 'smooth curves' we can reparameterige by Aro Length ft 1. we can "walk the curve 2. Some Formulas are easier. %s have a magnitude 1. Vector uut lagend) a. rics) will al Ex Find Are zength Fnction (Not 'L', I, 2, 3 a is where t begins Now, 50t CRefiaee't" with "u") SCt K2 cost-t sint, sint t tcos ty tt sin t t sin t +2f tt St t St Constant is s in the denominudory tan Vst e tan a do t tan dts sec 9 t Sec ottano seco tameCurvature: A mecsure op. A curve's failure to be a line.
more curvy the fine Rundion, the larger
the curvature.
is Arc-kength Compared to heading CY)
CHow T is chancing respech to dT
TCS)
ds
T unit ta
dent
N t Nommal
Note: T & TLN
g unit B
r X r
Note BETXN
Torsion
T (r X r
r. fl measure of
a Curve's Failure to be
ntaineel
n a plane.
r X r
dB
At cry paint on a curve, there will be a circle
that Fits that curve best.
the curve
ust touches curve erC kiss curve have to
At the point: The circle and the have
Same aurvature
circle.
Norme
the cember of The Plame.
that contains
plane)
Must contain N
points
mal to the
osculating plane.
is auued the Radius of
Gurvature
8 is
The radius of osculating rde The plane N B is called the normal plane'
that conams
vectors orthogonal to T
CT is norm E Find N,3, k,f, of osoulading plane og Normat
Plane t for TCty Koost, sint, t
sint cost
TCt
cost Sint o
-cost, sint, o -cost,
cast sint o
Point: a
cuuting plane: contains TAN,8 is the normal
PC
we are dealing with the e, ot plane,
so this shit doesnt matter
Curvature: A mecsure op. A curve's failure to be a line. more curvy the fine Rundion, the larger the curvature. is Arc-kength Compared to heading CY) CHow T is chancing respech to dT TCS) ds T unit ta dent N t Nommal Note: T & TLN g unit B r X r Note BETXN Torsion T (r X r r. fl measure of a Curve's Failure to be ntaineel n a plane. r X r dB At cry paint on a curve, there will be a circle that Fits that curve best. the curve ust touches curve erC kiss curve have to At the point: The circle and the have Same aurvature circle. Norme the cember of The Plame. that contains plane) Must contain N points mal to the osculating plane. is auued the Radius of Gurvature 8 is The radius of osculating rde The plane N B is called the normal plane' that conams vectors orthogonal to T CT is norm E Find N,3, k,f, of osoulading plane og Normat Plane t for TCty Koost, sint, t sint cost TCt cost Sint o -cost, sint, o -cost, cast sint o Point: a cuuting plane: contains TAN,8 is the normal PC we are dealing with the e, ot plane, so this shit doesnt matterEa: Normal plane contuint 8 f NJ T
normal
P Co
Normal Plane
Cr
toc
Tot) Ket t, et sint, ety
T, N, k
Ex
K Cost-etsint, et sint et cost, et
(t)
E e cost-ef sinties
t et cost, e
TC
t) t sint, sint+ cost,
K-sin t-cost, cost- sint o
T (t)
2 S
t, cost sint, oS
Sint-Cos
TCt)
Nlt)
Sint-cost, Cost-sint, o
J Sint-cost, cost sint, o
3 e
Ez Find of Oscaulatin Plame t Curvature. To t,1,* e tso
Stop
useless TG
TCH
0,2
T Ct)
Will do it later
curvature
have polynomial.
Ea: Normal plane contuint 8 f NJ T normal P Co Normal Plane Cr toc Tot) Ket t, et sint, ety T, N, k Ex K Cost-etsint, et sint et cost, et (t) E e cost-ef sinties t et cost, e TC t) t sint, sint+ cost, K-sin t-cost, cost- sint o T (t) 2 S t, cost sint, oS Sint-Cos TCt) Nlt) Sint-cost, Cost-sint, o J Sint-cost, cost sint, o 3 e Ez Find of Oscaulatin Plame t Curvature. To t,1,* e tso Stop useless TG TCH 0,2 T Ct) Will do it later curvature have polynomial.Ex find curvature of y
mple
t, sin t,os
3t
6t
6t
6t lf we want to find the marimum curvature,
the Cal
36t Kt
O is defined
Intro to Mullivariable Ranctions
to graph a Runction.gou must have 1 mension more than the th
Independent variable
C4-D)
To graph the domain op A funcHon.
tt of Individual variables
Must have the same demension as D: I- o
FCX 8,2) i D 3-D
(4-D graph)
(3 3-D domain.
2.2 3G
e The output z
Rand
Ex
Z S 2.
To graph domain, Must have
an azis for each depent variable
Ex find curvature of y mple t, sin t,os 3t 6t 6t 6t lf we want to find the marimum curvature, the Cal 36t Kt O is defined Intro to Mullivariable Ranctions to graph a Runction.gou must have 1 mension more than the th Independent variable C4-D) To graph the domain op A funcHon. tt of Individual variables Must have the same demension as D: I- o FCX 8,2) i D 3-D (4-D graph) (3 3-D domain. 2.2 3G e The output z Rand Ex Z S 2. To graph domain, Must have an azis for each depent variablegraph/shode averything but that
ac
So, J
4- D grap
3-D domain
Inside of a sphere w
radius 3 centered
e
Z 3
4 -2C
How To Graph
2. Try to get a surface you know.
3 use a computer.
et r,y,z
opening Towa.
2", shifted on "2"
Ex
36
t365
22
4Z
Level rves:
The shape we get when a plane intersects
our surface at different
levels along ris op the dependent variables.
the
graph/shode averything but that ac So, J 4- D grap 3-D domain Inside of a sphere w radius 3 centered e Z 3 4 -2C How To Graph 2. Try to get a surface you know. 3 use a computer. et r,y,z opening Towa. 2", shifted on "2" Ex 36 t365 22 4Z Level rves: The shape we get when a plane intersects our surface at different levels along ris op the dependent variables. theLimits f Cartinuu
of Multivariable Rundion.
i-Variable: This is a curve CAcertain
Path).
tu a directions
ofpazach the point.
Por 2-variable we have a surface
there are-so-Paths alont
the surface that approach our point.
To prove mit exists --we must prove that al
paths we approach some paint Cwe come
use sueege theorem or we can prove that the
2edis mit does not e
showing that aloy towa
-faths ruse get a different value as
doe approach
the same point.
show that the Limit DNEU
Along
G Tro
eling along the Surfewce. Directly overbunder
2ed
Along
over "r" axis
So, limit DNE e Co, o)
#2 these don't work, chose other paths
Be certain the Point Carb is adually on jour Auth
Ti to. substitute, dgrees "num
d denominofor are.
equal
cu Au acts use eithe xao or tao as
ne path
Ex
Note, Prove. DNE)
L'hopital)
TIm
So, o se, limit DNE Co,
CAS Y
Note: Prove DNEJ
Ex
so, o so limit ONE Coue
Limits f Cartinuu of Multivariable Rundion. i-Variable: This is a curve CAcertain Path). tu a directions ofpazach the point. Por 2-variable we have a surface there are-so-Paths alont the surface that approach our point. To prove mit exists --we must prove that al paths we approach some paint Cwe come use sueege theorem or we can prove that the 2edis mit does not e showing that aloy towa -faths ruse get a different value as doe approach the same point. show that the Limit DNEU Along G Tro eling along the Surfewce. Directly overbunder 2ed Along over "r" axis So, limit DNE e Co, o) #2 these don't work, chose other paths Be certain the Point Carb is adually on jour Auth Ti to. substitute, dgrees "num d denominofor are. equal cu Au acts use eithe xao or tao as ne path Ex Note, Prove. DNE) L'hopital) TIm So, o se, limit DNE Co, CAS Y Note: Prove DNEJ Ex so, o so limit ONE Coueso
so, I so limit DNE C
CLO)
Ex
Show
For 3-variables, Paths are now parametri
"t"
Atuwags
choose ..en Jancis as one Sour Pethsi
Alary storae curve c t, t, 2 at
ONE
Since o
o)
Some, Qurve C
Since o 3
mut DNE
Ex
Ex
Ex: lim
firm Cos o sinag by
so so, I so limit DNE C CLO) Ex Show For 3-variables, Paths are now parametri "t" Atuwags choose ..en Jancis as one Sour Pethsi Alary storae curve c t, t, 2 at ONE Since o o) Some, Qurve C Since o 3 mut DNE Ex Ex Ex: lim firm Cos o sinag byEx
at
Ex
-muthea both sides by
positive,
Tim
the seutge theorem lim
Note: Composition so continuity holds
Derivatives.
Multivariable Rundions
what does the derivative op a mulkivariable Rundtons
look like?
C20.ny). Js..a surface in
the slope of tangent to
-Too ambjuous
the a point
because oo tangents.
find the slope of the Md ne to a surface at a
int, we must give the tongent line a oirection...
diredionoe derivadives come Later,
por now, we restrict
aw r deriyatives To the ur-directi
or In the "J
direction
To find the sefe of
ent lime in Tx-direction.
we mus-
contain the tongent line iri a Plame parallelta the RE:plune
Coondams
2-axis)
requires-- to be held canstant. 3 plane l exe).
Makes aertci tangent ne
is in plane n xappme.
Far- direct
On
bold x" constint forces tangent line to be jn
a Plame. to YZ-plane.
The idea treating ariable as a constant f thereb
line is in the direction of the others
nsuring that
the taryemd
variable is called a Partial Derivatives'
Ex at Ex -muthea both sides by positive, Tim the seutge theorem lim Note: Composition so continuity holds Derivatives. Multivariable Rundions what does the derivative op a mulkivariable Rundtons look like? C20.ny). Js..a surface in the slope of tangent to -Too ambjuous the a point because oo tangents. find the slope of the Md ne to a surface at a int, we must give the tongent line a oirection... diredionoe derivadives come Later, por now, we restrict aw r deriyatives To the ur-directi or In the "J direction To find the sefe of ent lime in Tx-direction. we mus- contain the tongent line iri a Plame parallelta the RE:plune Coondams 2-axis) requires-- to be held canstant. 3 plane l exe). Makes aertci tangent ne is in plane n xappme. Far- direct On bold x" constint forces tangent line to be jn a Plame. to YZ-plane. The idea treating ariable as a constant f thereb line is in the direction of the others nsuring that the taryemd variable is called a Partial Derivatives'Notriti
dE: Holds "y" constant, s
it gives slope ef tangenti lineto
the surpace w/ respect to "x
OR Holds constant, so it gives slope of
the surface respect to
Note:
Ex
Cy sin C2.w)
1melicit Derivatives
is the implicits defined variable, not "8"
or J
Notriti dE: Holds "y" constant, s it gives slope ef tangenti lineto the surpace w/ respect to "x OR Holds constant, so it gives slope of the surface respect to Note: Ex Cy sin C2.w) 1melicit Derivatives is the implicits defined variable, not "8" or JEX
dx
d2
cos t
dyr
Higher Derivatives:
3,2
Ex
or
dy dz
dx dy
2 sin y.cos 2
sin
OC
Note
2, product rule.
20-2
Sin
EX dx d2 cos t dyr Higher Derivatives: 3,2 Ex or dy dz dx dy 2 sin y.cos 2 sin OC Note 2, product rule. 20-2 SinFor any Runction that is continuous on
a n, mi-reel partial
derivatives are. equal CTst make sure
gou have the same
ettars)
Differentials.
tangent line
Fromm Calc 1
Increment
increment (Ay)
differential
ntfaldy)
30y a
the charase in Cheikt
from p to
gives the change in heght
from
P to a point
the tangent
line.
as Ar gets realy small CA
then A1.
Se
dac.
R3 at S
Gren
a surface
AZ
dz
plane
Az is the actual ch
rom "P to
da is the chu
in hegt.
from 'P' to a point
on-
E' t J' are independent variables, az
da,
Az Ar tAy, TF we are in the "x-direction we are zz -Plame
d2.
AZ.
dz
AZ
dz t dz
Therefore
AZ
42
dz.
érenhol
Ex: Find Diff1 of a se
For any Runction that is continuous on a n, mi-reel partial derivatives are. equal CTst make sure gou have the same ettars) Differentials. tangent line Fromm Calc 1 Increment increment (Ay) differential ntfaldy) 30y a the charase in Cheikt from p to gives the change in heght from P to a point the tangent line. as Ar gets realy small CA then A1. Se dac. R3 at S Gren a surface AZ dz plane Az is the actual ch rom "P to da is the chu in hegt. from 'P' to a point on- E' t J' are independent variables, az da, Az Ar tAy, TF we are in the "x-direction we are zz -Plame d2. AZ. dz AZ dz t dz Therefore AZ 42 dz. érenhol Ex: Find Diff1 of a seFind d
22e
Er fressure of a certain aas con be decreased.
f
Find dP
d P
20.2 L
Volume goes from. 20L
Temperature from 30ole-s295 h
pind approximate change in pressure.
TF2
ds
A
duy
GA -w)
Ez Tens
Jr op. A Cable G. it's lowest point is given by
LOL d
dL
a
Chain Rule For Multivariable. Runctian
From Calk
I
dx
at
Note: Even though .3 variables-
C22 Iheie is-Rhsy ane
Tndependent -variable "t"
is a campesilien. C Cu
rule)
"x"works as the intermeduate variable.-t ves a
path from the dependent variable to the
independent vaDable LSU
Ex
t are intermediete variables
-f 't" is the
independent variable..
Find d 22e Er fressure of a certain aas con be decreased. f Find dP d P 20.2 L Volume goes from. 20L Temperature from 30ole-s295 h pind approximate change in pressure. TF2 ds A duy GA -w) Ez Tens Jr op. A Cable G. it's lowest point is given by LOL d dL a Chain Rule For Multivariable. Runctian From Calk I dx at Note: Even though .3 variables- C22 Iheie is-Rhsy ane Tndependent -variable "t" is a campesilien. C Cu rule) "x"works as the intermeduate variable.-t ves a path from the dependent variable to the independent vaDable LSU Ex t are intermediete variables -f 't" is the independent variable..dt
intermediate
Ruretion.
able
Cos
dt
22 t ces halt
d t
21 sinh (t)
sinh
dt
-Diagram...
Int
Ind
dyldu
dr
duo
dr. Tou
dv
Ex
du
du
dependent
able
forgot but
of the idea
Cosh, Ct
You
dt intermediate Ruretion. able Cos dt 22 t ces halt d t 21 sinh (t) sinh dt -Diagram... Int Ind dyldu dr duo dr. Tou dv Ex du du dependent able forgot but of the idea Cosh, Ct Youdu
dv
duo
v sin.
2V
02
Cos
dv
rd V sin Cu)
O
du
tan Cy 2)
V d sin C
due
2
dw
2xze
Cos Cu
ds
3 Crisi
2C
otw
Let w FC3 where for, is a puncton such that
implicht.
s dur
o is the olny independent vacia
d je
Implicit differentiation
Or
d
Ex
Find dy
dr.
2-2
32
Z
du dv duo v sin. 2V 02 Cos dv rd V sin Cu) O du tan Cy 2) V d sin C due 2 dw 2xze Cos Cu ds 3 Crisi 2C otw Let w FC3 where for, is a puncton such that implicht. s dur o is the olny independent vacia d je Implicit differentiation Or d Ex Find dy dr. 2-2 32 ZDirectional D
vatives.
To aive direction. we must have a
vecton Cunit vedar) on the plane
of the independent variable.
P are directional derivative
Idea: hings we need meed-L xy-elane.
T
need PTC int Far specific
rate of chani
Je
Mate line thru P'll C create plane. to be-plane that
makes a curve on the surface in plane
Pa is a secant line in the
plane
As R p, the seca
approaches the tangent
P
So, both
42 As mus
so.
Now, Pa
+hurj
PGE Ar
hu
hu, t AJ
So, Ar
Ario.
A
For a unt ctor
Ex End the direction du derivative of fcr, a 3-2
Q.rc jn. the -dirertion. op the vector that makes an
Need L
.point CI, 2),
mistake
t Sin Q
x Ea op pe op ta.
line to
Fex,y a
3urface buif in direction
3N3
2 3 6
Da PCI, 2)
6
Gradu ent
we call this
wet rod vector
the gradient vslz,yy sri J
Da is a ope, a scalar not a vector
vp is a vector. (part of Da
Gradient relates to "Grode Cclimb of the Sur
For a
Specific
grade, must have a point t
Directional D vatives. To aive direction. we must have a vecton Cunit vedar) on the plane of the independent variable. P are directional derivative Idea: hings we need meed-L xy-elane. T need PTC int Far specific rate of chani Je Mate line thru P'll C create plane. to be-plane that makes a curve on the surface in plane Pa is a secant line in the plane As R p, the seca approaches the tangent P So, both 42 As mus so. Now, Pa +hurj PGE Ar hu hu, t AJ So, Ar Ario. A For a unt ctor Ex End the direction du derivative of fcr, a 3-2 Q.rc jn. the -dirertion. op the vector that makes an Need L .point CI, 2), mistake t Sin Q x Ea op pe op ta. line to Fex,y a 3urface buif in direction 3N3 2 3 6 Da PCI, 2) 6 Gradu ent we call this wet rod vector the gradient vslz,yy sri J Da is a ope, a scalar not a vector vp is a vector. (part of Da Gradient relates to "Grode Cclimb of the Sur For a Specific grade, must have a point tthoperties VE
Daf ao for ang
Da FCX, s) has its. mar value of ILszfer,s)
this
will haffen when. -L. C. F, C a scalar
C directian
Same
Mar. value of Cos o
l en Q -so, gle between
..vf
is To
CParallell
And other
C .give s a It tales
a Fradion of VE Da
be
less steed
3y DA ACxug) hes its Min value
So, VF
gives the vector for the steepest -grade op a surpace.
a
19 is not ll en
think, of as fuming DaF-from the
directen of steepest climb
vF
The metab-
vector for Max
slope of surface Q and point-
22S
CT
vs C1, 2,3)
6 4
In the direction of 3,4
thoperties VE Daf ao for ang Da FCX, s) has its. mar value of ILszfer,s) this will haffen when. -L. C. F, C a scalar C directian Same Mar. value of Cos o l en Q -so, gle between ..vf is To CParallell And other C .give s a It tales a Fradion of VE Da be less steed 3y DA ACxug) hes its Min value So, VF gives the vector for the steepest -grade op a surpace. a 19 is not ll en think, of as fuming DaF-from the directen of steepest climb vF The metab- vector for Max slope of surface Q and point- 22S CT vs C1, 2,3) 6 4 In the direction of 3,4of line from PC2,o) Te Gly 2)
I know how to do it, so J do it later.
Answer
direction
m PC4,2,2) f 6, -2,6
of line. Fra
-1 km how to do it. it later
Ans doer:
EE A part of a certain hillside can be modeled by
or PO 32,3 at the bottom op the
canyon Corigi) are a pack op er harses are about
U chemar you! Finch the direction "F the
fastest path up the hill t
the headi
92
2x43 eading
Tan
an
From North
go 20.s s 9.
rent planes
Normal, me JS
Goal: To find a tangent plane to surface at a
Consider
surface
3-D
ave level Curve.
what this
means
gives a sleple tangent vector of level curves.
2 when a dot product o, e two vectors
are orthogonal
ay So, v F gives us the
PN
norm
al to a level
a point
Curve
of line from PC2,o) Te Gly 2) I know how to do it, so J do it later. Answer direction m PC4,2,2) f 6, -2,6 of line. Fra -1 km how to do it. it later Ans doer: EE A part of a certain hillside can be modeled by or PO 32,3 at the bottom op the canyon Corigi) are a pack op er harses are about U chemar you! Finch the direction "F the fastest path up the hill t the headi 92 2x43 eading Tan an From North go 20.s s 9. rent planes Normal, me JS Goal: To find a tangent plane to surface at a Consider surface 3-D ave level Curve. what this means gives a sleple tangent vector of level curves. 2 when a dot product o, e two vectors are orthogonal ay So, v F gives us the PN norm al to a level a point CurveThe Fastest
level curve
vp Cr,1) is the norm
to a level, curve FCX,s)a C
vFC a z) is the rmal to a level urve
FCr,s,zy:C
Tandent time planes f Norm al lines, both need
paint normal vector
Ex: Find the line f N
ne.
C POS,3)
How to checut Pretend that 3 a Temel curve to
some
surface in Re
evel
curve to fox,J) Heons
Ives mal to
Normal vector to
ang curve.
The PCs, 3 gives a specific keelar
normal to a speciPrc level curve
pe of normal, N
4 3
Checiprocal of MN)
Slope
SoC
specific normal
el curve
vector to that
srecific pind Normal vector to -22
2, 6
Norm
vector to a family of level
surfaces for F
z)
Panes
Normal Lines: Normal Lines need narmol vedor (T)
Z Zo
Ex
4D
ejcenes
Normal
line
The Fastest level curve vp Cr,1) is the norm to a level, cu

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