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# MAT 237 Exam Guide Part 1

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University of Toronto St. George

Mathematics

MAT237Y1

Robert Brym

Fall

Description

MAT s3t
I Euclidean Spaces Vectors
Sa
13 a vector space.
Cauchy Schwa 2's Inequal
for any x,uER
Line in Space (l)
teIR
Ex y Find paramedic
equation at the Line through
A i, 3), Co, and describe the set
of
points on the
seamerot AB
OB-OA
scalar rolodion liya s-t
tEN
Ab
is V telo, il
AB is an image.
f Do, IR- R'
t, S-t
Plane in Space
T: R IR
Ex Find a paramadric equation of the plane through
P.
P Co
Describe the set of points in the parallelogram
ninth PP., PP, as adjacent edges.
Co
P P
2, 1) Co
Cs.telR)
OP- OP S
o, t t.
33 St
Pora image f Co. Co. CR-R
t)
Stt.
St.
MAT s3t I Euclidean Spaces Vectors Sa 13 a vector space. Cauchy Schwa 2's Inequal for any x,uER Line in Space (l) teIR Ex y Find paramedic equation at the Line through A i, 3), Co, and describe the set of points on the seamerot AB OB-OA scalar rolodion liya s-t tEN Ab is V telo, il AB is an image. f Do, IR- R' t, S-t Plane in Space T: R IR Ex Find a paramadric equation of the plane through P. P Co Describe the set of points in the parallelogram ninth PP., PP, as adjacent edges. Co P P 2, 1) Co Cs.telR) OP- OP S o, t t. 33 St Pora image f Co. Co. CR-R t) Stt. St.MAT
TE
NO
a k-dimensional linear manifold HinR is the
s a point
s a une
k is a plane
[a,b] x [c,dl
la, bl
an unit S box
[a,bl
an unit n box
X-Stt
f IRT R defined by (x,y -3 13x-3y
Euclid Spoke Vector Spo
Inner Preduct
ned on it
Standard Inner Productio
properties symmetry 4:2.
3 StHct Positiv t
The Triangle Inequality
for an
!All COSE
Cauchy-Schnarz
Distance
E Prove d
Cross PDoduct ONLY
(o
MAT TE NO a k-dimensional linear manifold HinR is the s a point s a une k is a plane [a,b] x [c,dl la, bl an unit S box [a,bl an unit n box X-Stt f IRT R defined by (x,y -3 13x-3y Euclid Spoke Vector Spo Inner Preduct ned on it Standard Inner Productio properties symmetry 4:2. 3 StHct Positiv t The Triangle Inequality for an !All COSE Cauchy-Schnarz Distance E Prove d Cross PDoduct ONLY (oS t
MAT
V2u
V+ W
Lagrang
Identitu
sin
loses TU)
Yll sin 9
sing
1.s Subsets of Euclidean Space
Det a set ScR is sad to
be open if z ES, 3r> o
st. Vy ER
satisfying ll yllcr also belongs to S
y are points)
Ex y a) S
3 open
bo S (x, y EIR o o st. Vy ER satisfying ll yllcr also belongs to S y are points) Ex y a) S 3 open bo S (x, y EIR o K
Ex y Prove that the segueme zka ha.TE) converges
to Co, o)
O, O
choose K
ECE)
IHMa Let ScR
d to
st.
Prove that ci, ES
Construct
YR20 define
2h-> Cl, DES boundary point
CI S)
the
following are equivalent
I is at a we have f ay
eventu boundary monotone sequence in R is
convergent (limut is supi
o
XI, TX
Xt, X2
increasing, bounded above by Converged to 1.
that ix i con
find rts limrt
shon boundary
wart to X
k
for kal
2 assume
prove that for k xk xun s s
MAT Completeness is complete (no hole, every sequence in has a limit n is not complete La sequence n Rn converges to L if YE20 zk LIN E whenever k> K Ex y Prove that the segueme zka ha.TE) converges to Co, o) O, O choose K ECE) IHMa Let ScR d to st. Prove that ci, ES Construct YR20 define 2h-> Cl, DES boundary point CI S) the following are equivalent I is at a we have f ay eventu boundary monotone sequence in R is convergent (limut is supi o XI, TX Xt, X2 increasing, bounded above by Converged to 1. that ix i con find rts limrt shon boundary wart to X k for kal 2 assume prove that for k xk xun s sMAT
nereasing, bounded from above by
Conv
lim x
L by imit the ten
The Nested Lateral THMa
Let I. Ca., b], I2. Caz, b.),... be sequence.
closed, bounded intervals in R
Suppose that
then point contained in all the intervals
idea.
a. a
as b.
ak increases, bounded
converges
bk decreases, bounded
Ex
If is a sequence in
STHctly increasing rteN then is a subsequence
is subsequence of G
IHM If a sequence in Rn converges to geR.
then an Subsequence of TXn Converges to e
M: Every bounded sequence in R (or R) has
a convergent subse
have a convergent
Exl Does
ce If yes, find one
Sub
NS
ounded
Sin
has convergent sub
-take k 4n+ l
n An
O tor onlu subsequence converges
NOTE Another noy to express
completeness
nhenever kj
then converges
uch an
ixin is called Cauchy Sequence.
St.
every bounded sequence in
R has
a convergent
sub eaten
MAT nereasing, bounded from above by Conv lim x L by imit the ten The Nested Lateral THMa Let I. Ca., b], I2. Caz, b.),... be sequence. closed, bounded intervals in R Suppose that then point contained in all the intervals idea. a. a as b. ak increases, bounded converges bk decreases, bounded Ex If is a sequence in STHctly increasing rteN then is a subsequence is subsequence of G IHM If a sequence in Rn converges to geR. then an Subsequence of TXn Converges to e M: Every bounded sequence in R (or R) has a convergent subse have a convergent Exl Does ce If yes, find one Sub NS ounded Sin has convergent sub -take k 4n+ l n An O tor onlu subsequence converges NOTE Another noy to express completeness nhenever kj then converges uch an ixin is called Cauchy Sequence. St. every bounded sequence in R has a convergent sub eatenMAT 3
Co -Compactness
A subset S of Ra is compact if it is closed bounded
S is bounded if R o St. ScBCR, Q)
Ex V a.
OS X
SCB rs, o) S is bounded
S is closed it contains all its boundary
points
S is bounded, SCB 0,o)
Q: irrationals
H SCRT, then the following are equivolert
S is compact
every sequence of points in S has
a congruent
Subs
a) b
Suppose S is compact
S every sequence izk CS
has a con
s
Main Q2suit s)
and the limit is in S by
since S closed. holds.
b) a
Suppose S is not compact
if S is ost closed 3 ES but S
and S
so by fai
if S not bounded 3 st.
so foi
not compact
take
Yes, choose any atternating
g2rt to Ca,b) CS
Converr
S Li where Li is a line segmen Joining
Coo) with
(T Sin
Sin7.
u sequence in S
two cases
0 nite G terms of is in L
there is a con
S
since Li clased Fi
otherwise choose zki in L
Co,o) E3 u Bolzano's compact
MAT 3 Co -Compactness A subset S of Ra is compact if it is closed bounded S is bounded if R o St. ScBCR, Q) Ex V a. OS X SCB rs, o) S is bounded S is closed it contains all its boundary points S is bounded, SCB 0,o) Q: irrationals H SCRT, then the following are equivolert S is compact every sequence of points in S has a congruent Subs a) b Suppose S is compact S every sequence izk CS has a con s Main Q2suit s) and the limit is in S by since S closed. holds. b) a Suppose S is not compact if S is ost closed 3 ES but S and S so by fai if S not bounded 3 st. so foi not compact take Yes, choose any atternating g2rt to Ca,b) CS Converr S Li where Li is a line segmen Joining Coo) with (T Sin Sin7. u sequence in S two cases 0 nite G terms of is in L there is a con S since Li clased Fi otherwise choose zki in L Co,o) E3 u Bolzano's compactMAT 37
IHME Continuous functions map compact sets
to compacts sets
SCR compact f: S- Rm continuous
20ES is compact
On
f: R- Rim ScR" NS compact f: S- RM is
Continuous f S) if
x E 13 compact.
It is true that continuous f map closed to
closed or bounded to bounded
Only combination is preserved.
Ex ax S R ,closed unbounded
(S) Co, IT .not closed!
b SE CI,S) bounded Inst closed
(S): -00,-1) unbounded
Extreme value THM
Suppose SCR is compact and f s-R is
continuous, then has an absolute minimum
value and absolute maximuun volue in S.
astnes
both in Ex
are courte examples
a) min
b) no max at all.
Ex Does the function f
exe Sinu have
an absolute maxlmin. on SE wheNe
f s cot as a composition
le
and product of cont functions
S is not compact.
assumption of E.VT. not satisfied.
Ex Same for foxy
and
MAT 37 IHME Continuous functions map compact sets to compacts sets SCR compact f: S- Rm continuous 20ES is compact On f: R- Rim ScR" NS compact f: S- RM is Continuous f S) if x E 13 compact. It is true that continuous f map closed to closed or bounded to bounded Only combination is preserved. Ex ax S R ,closed unbounded (S) Co, IT .not closed! b SE CI,S) bounded Inst closed (S): -00,-1) unbounded Extreme value THM Suppose SCR is compact and f s-R is continuous, then has an absolute minimum value and absolute maximuun volue in S. astnes both in Ex are courte examples a) min b) no max at all. Ex Does the function f exe Sinu have an absolute maxlmin. on SE wheNe f s cot as a composition le and product of cont functions S is not compact. assumption of E.VT. not satisfied. Ex Same for foxy andMAT 337
Cont. on
fix, so
on
Uel
S compact
Yes, be S contains all its boundory points
o f cont. on S2
No, discontinuous
at the point Co,o)
along the path
x 0, y- o L z oo
along the z o u o L- O
No, assumption at E T. not satisfied
NOTE finite U, n of open (closed) sets is open
closed Mespectively?
infinite collection (union) of open sets is open.
but imnite collection of closed sets is not
necassaMlu closed.
ESL ZESA for some n.
but S is open 3BCE, 2 CS: st. Bec Sa,
but SNES is open.
1.7 _Connectness
ScR is disconnected if S., S, CR st. SaS US,
(S disconnected if it is the union st S non-em
subsets S, and S, neither nhich intersects the
closure the other one
A pour (S, S.) is called a disconnection of S.
Ex a S xy of s S connected
0,0
LS is disconnected.
Sins
and 3 ns.
MAT 337 Cont. on fix, so on Uel S compact Yes, be S contains all its boundory points o f cont. on S2 No, discontinuous at the point Co,o) along the path x 0, y- o L z oo along the z o u o L- O No, assumption at E T. not satisfied NOTE finite U, n of open (closed) sets is open closed Mespectively? infinite collection (union) of open sets is open. but imnite collection of closed sets is not necassaMlu closed. ESL ZESA for some n. but S is open 3BCE, 2 CS: st. Bec Sa, but SNES is open. 1.7 _Connectness ScR is disconnected if S., S, CR st. SaS US, (S disconnected if it is the union st S non-em subsets S, and S, neither nhich intersects the closure the other one A pour (S, S.) is called a disconnection of S. Ex a S xy of s S connected 0,0 LS is disconnected. Sins and 3 ns.MAT
34
S is connected.
NOTE connected sets in R are imtervals. Particularly
I lo, IT is connected.
DEA Suppose I is disconnected
impossible
ceI
US 1
Never, R is comp
Ex Il CTHMI)
rave thot the entire space R is connected
pt Suppose R is disconne
ond S, S.) is
a disconnection.
Let ge S, be S and consider the line
segment S
Joinin
a b
Now let A
teIR a t to- a)ES,
(Assns
S... S dis
disconnection of Co,
A and B ane be LOU is connected
Prove that the only subsdts of Rnnhich
are both open and closed one and R
PE Supose ACIR is both open clovod.
BER A is also open a closed
A is non-empty and Ost all of R
CA forms disconnection for R
Contradicting THM above
NOE For other sets S than R if Possible to
join any paints ou palutgenal curve
Det. Open set ACR is colled arcwise conneaed
if any Points b an be joined by a
al curve
Uunion
of a finite
polygon
of ordered line seamaris LUL.U. ULke
ying wholly in A.
UL
MAT 34 S is connected. NOTE connected sets in R are imtervals. Particularly I lo, IT is connected. DEA Suppose I is disconnected impossible ceI US 1 Never, R is comp Ex Il CTHMI) rave thot the entire space R is connected pt Suppose R is disconne ond S, S.) is a disconnection. Let ge S, be S and consider the line segment S Joinin a b Now let A teIR a t to- a)ES, (Assns S... S dis disconnection of Co, A and B ane be LOU is connected Prove that the only subsdts of Rnnhich are both open and closed one and R PE Supose ACIR is both open clovod. BER A is also open a closed A is non-empty and Ost all of R CA forms disconnection for R Contradicting THM above NOE For other sets S than R if Possible to join any paints ou palutgenal curve Det. Open set ACR is colled arcwise conneaed if any Points b an be joined by a al curve Uunion of a finite polygon of ordered line seamaris LUL.U. ULke ying wholly in A. ULMAT
Practical Ilustratan
Find a polygonal curve
from CI, to (JI) to (5,4).
L LIUL
S) to (SI)
LISI)
CIT SI Su
OSS
L2 from CS, 1) to (5, 4)
(3+ 3 S
3S2
Charge scale
S, St
osts
DeiTne
St, S-St
ost.
L LIUL
S is arcwise connected
Fot any other set S
Sis comeaod if any S Points x
He s can be joined
a continuous curve wholly lying in S.
NOTE If ScRn any set
than
S arcwis2 connected connected
Ex/
The set S is connected
but S is not archise connected.
Let us join ace S at t Coo), ak e S. Qkt Co,o)
ke I,
t (ak-g
Let a
(Bud no finite union can do that.
If S
compact opan st
C S, Sa S. is open
and archise connected
HML continuous functons map connected sets
to connected sets.
ie IF f: S IR is cominuous and S is
xe S is
connected
MAT Practical Ilustratan Find a polygonal curve from CI, to (JI) to (5,4). L LIUL S) to (SI) LISI) CIT SI Su OSS L2 from CS, 1) to (5, 4) (3+ 3 S 3S2 Charge scale S, St osts DeiTne St, S-St ost. L LIUL S is arcwise connected Fot any other set S Sis comeaod if any S Points x He s can be joined a continuous curve wholly lying in S. NOTE If ScRn any set than S arcwis2 connected connected Ex/ The set S is connected but S is not archise connected. Let us join ace S at t Coo), ak e S. Qkt Co,o) ke I, t (ak-g Let a (Bud no finite union can do that. If S compact opan st C S, Sa S. is open and archise connected HML continuous functons map connected sets to connected sets. ie IF f: S IR is cominuous and S is xe S is connectedIHM (the Inte mediata Value Thaotem)
Suppose f: SCR-a R is continuous on S and
VCS is connected. If bCV and fig stefc
Cor fub) sts fla)) then JCEV st. foc at
V) is an interval Coconected), it comtoins
,a), f b) it comtains the entire interval
between them
and Va. (x, y): x
Is there a point st f s l.
V is connected Card compact)
Look for points bev st fca
f
by
Let as Co C) f
O, O
Let Co
34
Yes
Ex Let lit
Stt, -t I), P
a) Find the shoMtest distance from tol.
by Find an uation of The plane comtaining
l and p
Find the point on l closest to P,
sin ay idea
P. (s, o, -1)
doh
Lt)
ength
POP
a (-1,2,-1
Given
u.LTU.
PE TU.
A,B, C
R A, (20,-
an Xn
zeR
IHM (the Inte mediata Value Thaotem) Suppose f: SCR-a R is continuous on S and VCS is connected. If bCV and fig stefc Cor fub) sts fla)) then JCEV st. foc at V) is an interval Coconected), it comtoins ,a), f b) it comtains the entire interval between them and Va. (x, y): x Is there a point st f s l. V is connected Card compact) Look for points bev st fca f by Let as Co C) f O, O Let Co 34 Yes Ex Let lit Stt, -t I), P a) Find the shoMtest distance from tol. by Find an uation of The plane comtaining l and p Find the point on l closest to P, sin ay idea P. (s, o, -1) doh Lt) ength POP a (-1,2,-1 Given u.LTU. PE TU. A,B, C R A, (20,- an Xn zeRMAT
Ex Find of the planes through the point
P, CI, I, I, orthogonal to the line
ect
St. t, 2-st, 5-t)
Given: P, C
I,
let
Ex (base lecture)
lt
(s+t, -t -1)
teR
PE TL
P. (J,0,-1)
PoP (I-s, 3- O
20-1)
Find the point Pel closest to P..
through P
intersection Tu, l
aside:
S box
3 box v
W. W. W
a sts b
(b)
Ex a) circle
x y a
Ct) Cocost asint
O sts STt.
cost
y as Int
NOTE
or z (acos tort), a sin orty) te Do, RJ
MAT Ex Find of the planes through the point P, CI, I, I, orthogonal to the line ect St. t, 2-st, 5-t) Given: P, C I, let Ex (base lecture) lt (s+t, -t -1) teR PE TL P. (J,0,-1) PoP (I-s, 3- O 20-1) Find the point Pel closest to P.. through P intersection Tu, l aside: S box 3 box v W. W. W a sts b (b) Ex a) circle x y a Ct) Cocost asint O sts STt. cost y as Int NOTE or z (acos tort), a sin orty) te Do, RJMAT
O sts b
Let C be x 1-4
osy
Ost
t", t
parametrization of
c is not unigue
Let us sint
Sint COS
t
ccesat, sint) Ost
t tangent vector to c.
Ex/ A particle is following the Poth
C Ct) a CScast, sint, t)
3TU.
lies off along the targert line at t
find the the Porticle a ta
(5,0,1)
SCOS
Po el
Sin
-for s 0, at Co. not a start (teo)
the point at interest is
for ss Tt
MAT O sts b Let C be x 1-4 osy Ost t", t parametrization of c is not unigue Let us sint Sint COS t ccesat, sint) Ost t tangent vector to c. Ex/ A particle is following the Poth C Ct) a CScast, sint, t) 3TU. lies off along the targert line at t find the the Porticle a ta (5,0,1) SCOS Po el Sin -for s 0, at Co. not a start (teo) the point at interest is for ss TtMAT
Derivatives
Let f: ScR- R, S is open
The partial dervative off at geS Murt.
vaHade
TAT
3xif 2
Exa
NO
Ex
SST
by symmetry 4zsinx
ILLUSTRATION f R R (z fox,
forty
section
(ab)
in saction Syab
sloped a carb
n S
normal to tangent plane
ation
(a,b)(x-a)
a,b
X-ray
Let R-ash,
h (h, h
a by
a,b
camed a gradient
ca by h.
plane
MAT Derivatives Let f: ScR- R, S is open The partial dervative off at geS Murt. vaHade TAT 3xif 2 Exa NO Ex SST by symmetry 4zsinx ILLUSTRATION f R R (z fox, forty section (ab) in saction Syab sloped a carb n S normal to tangent plane ation (a,b)(x-a) a,b X-ray Let R-ash, h (h, h a by a,b camed a gradient ca by h. planeMAT37
Ex Let
a Sha that f
s not continuous at Coo).
by Shaw co,o) 3t (o,o) exist
sin a) along x o
f not cominuows, since limit dne.
o,o) O
simi
partial deivatives exist.
Recall from MAT 137 f:R- R
lim
X-a
7C-a
The tangent line at a,fay
l Ca
linear approximation of fo at tha neighbourhood of a
In another notation, hex-a, x-ath
Cayh
linn
at
Denote: Echo: fathy-foay-fra h
where
o as h- o
Det Let f be ned on an epen set SCHR
then f is oallad differentiable at a point
ges if there is a vector ges st.
ath
lim
Nata near approximation in
R
in othe case, at a
lim
f SCRT- R is differentiable at ges CS spen).
Cay- g-h
What is
Let z Chr.
th,
ch
a, thi, a
h- o
similarly ci
NuIE f is differentiable at a. then all axafe)
exist and
S) W
torget huperpane to graph fat a.
MAT37 Ex Let a Sha that f s not continuous at Coo). by Shaw co,o) 3t (o,o) exist sin a) along x o f not cominuows, since limit dne. o,o) O simi partial deivatives exist. Recall from MAT 137 f:R- R lim X-a 7C-a The tangent line at a,fay l Ca linear approximation of fo at tha neighbourhood of a In another notation, hex-a, x-ath Cayh linn at Denote: Echo: fathy-foay-fra h where o as h- o Det Let f be ned on an epen set SCHR then f is oallad differentiable at a point ges if there is a vector ges st. ath lim Nata near approximation in R in othe case, at a lim f SCRT- R is differentiable at ges CS spen). Cay- g-h What is Let z Chr. th, ch a, thi, a h- o similarly ci NuIE f is differentiable at a. then all axafe) exist and S) W torget huperpane to graph fat a.MAT 27
Ex y use the det, to ve fy whether the following
functions are tiable.
Ccont.
L'Hop
lim
lim
hih
d.n.e
lim
f s not differentiable
IHMI f f is differenviable at
a
then f is continuous
as h
is Continuous
L Converse not true. Ex y b) is
counterexample.
IHM Lat f: SCR R. Suppose that fo CCS)
has continuous partials of 1st order
f is differentiable co S
in Ex a) portials continuous f is differentiable
but n b) partials are cott continuous.
20t
Lim
C- O
does not imply that f is not difeMentiable
L Converse of THM is not true
Ex Counter- example)
aside.
f is of class
C
f k-th partial
h )sin
exist and are
lum
hi hi
continuous CNCS).
banded
f is differentiable
MAT 27 Ex y use the det, to ve fy whether the following functions are tiable. Ccont. L'Hop lim lim hih d.n.e lim f s not differentiable IHMI f f is differenviable at a then f is continuous as h is Continuous L Converse not true. Ex y b) is counterexample. IHM Lat f: SCR R. Suppose that fo CCS) has continuous partials of 1st order f is differentiable co S in Ex a) portials continuous f is differentiable but n b) partials are cott continuous. 20t Lim C- O does not imply that f is not difeMentiable L Converse of THM is not true Ex Counter- example) aside. f is of class C f k-th partial h )sin exist and are lum hi hi continuous CNCS). banded f is differentiableMATT T
partial exists, continuous
S x sin(
COS
cos dane
not comtinuaus
ax,
differential
Cort f's
Portals
exist
The Differential
error
hi dxi
dW
dx.
dx.
Ex use differential to evaluata NSCs. as 77)
sln Fath
O.
0.02
IN- R differentiable.
Ex
dx
DINeehooal Det native
O)
CO +tei)
axi
Let u be an unit vectoT
MATT T partial exists, continuous S x sin( COS cos dane not comtinuaus ax, differential Cort f's Portals exist The Differential error hi dxi dW dx. dx. Ex use differential to evaluata NSCs. as 77) sln Fath O. 0.02 IN- R differentiable. Ex dx DINeehooal Det native O) CO +tei) axi Let u be an unit vectoTMAT
THM If f: ScR R differentiable at aES OS is peer),
then the ditectional derivatives f all exist and
they given
tu
30
(a +t
(f differentiable
g ty
Ex V Evaluate the di Nectional derivative ei
Czy, z) y z at C1, s, in tho direction
,3) the line from CUN, to C 3,5, o)
is differentiable on R
NOTE Suppose term distibution in space
T x, y, z
rate change of temp. olkm
Q: In what direction is the largat charge sff
A where avf has largest value
largest in the chNection
of a groduent
Temp (cold front)
Suppose the temp. ground level at y is given
SO-X
O.
You live at
a) What change temp. in the north-east disection?
Snorth y-axis)
b) In what direction the charge is the largest?
How much
(1,1)
sln. a) N-E
4)
t 0.
by in the directon of
T
-0.4
4)
Led CoRT be a curve parametrized by
Let f R
MAT THM If f: ScR R differentiable at aES OS is peer), then the ditectional derivatives f all exist and they given tu 30 (a +t (f differentiable g ty Ex V Evaluate the di Nectional derivative ei Czy, z) y z at C1, s, in tho direction ,3) the line from CUN, to C 3,5, o) is differentiable on R NOTE Suppose term distibution in space T x, y, z rate change of temp. olkm Q: In what direction is the largat charge sff A where avf has largest value largest in the chNection of a groduent Temp (cold front) Suppose the temp. ground level at y is given SO-X O. You live at a) What change temp. in the north-east disection? Snorth y-axis) b) In what direction the charge is the largest? How much (1,1) sln. a) N-E 4) t 0. by in the directon of T -0.4 4) Led CoRT be a curve parametrized by Let f RMAT 37
2 dxa
todt
Ct
Ex Fnd the rate charge of fCzy
the ellipse go Cost, sint) atthe point
ty
30-s cost simt)
sin st
LNOTE DiNectional derivative can be written as
auf def g tu along the line.
a tu. (a ,a
un
A smooth surface in R ctanged plate charges sonfinoulp
o F differentiable (zaf Suppose that cpan. by
s any smooth cut
that lies On
the surface utrz
throuan PES
VE is orthegonal to Sat P
NOTE tangent Pane
Exsp Find the tongeni plane to the surface
E)
MAT 37 2 dxa todt Ct Ex Fnd the rate charge of fCzy the ellipse go Cost, sint) atthe point ty 30-s cost simt) sin st LNOTE DiNectional derivative can be written as auf def g tu along the line. a tu. (a ,a un A smooth surface in R ctanged plate charges sonfinoulp o F differentiable (zaf Suppose that cpan. by s any smooth cut that lies On the surface utrz throuan PES VE is orthegonal to Sat P NOTE tangent Pane Exsp Find the tongeni plane to the surface E)MAT
Ex y You are stouing at the Pimt 00,430) on
a hill that how the shope the graph of
Where x u z in the positive X-axis points east
NN, nill you start oscand or descend
at whof rate
b) going io nhor di Nections) you nill not experience any
change in attitude exploin.
sln
oo)
OO
NOTE the greatest increase is in the direction
GfL o.8.o.o
500 Co.oo4.
CO
for
NOTE nLS
evel surface of f of 3 variables
LLC level curve)
4 The Mean Value THME
MVT Let SciR containing a,b ES and the
f defined on S, cont. ot each point
uppose L (including a 2 t differentiable on L
except possibly
a, b, then EL
st
Let h b-a
L a +th o st
Let pet) flat th
osts
p cent. on Co, UL, duff on Co
and
Ca +t
By Calc 1 MVT fue CoD set
Lot C
a uh
What about global prop
on S
Det. A set SCP is caled CONVEX set if
s convex 2 arcwise connected.
Exy Show that
XE
S Con
by S
3izER 13 not conve
a Let g,b ES, Canu two points)
Let L
osts
MAT Ex y You are stouing at the Pimt 00,430) on a hill that how the shope the graph of Where x u z in the positive X-axis points east NN, nill you start oscand or descend at whof rate b) going io nhor di Nections) you nill not experience any change in attitude exploin. sln oo) OO NOTE the greatest increase is in the direction GfL o.8.o.o 500 Co.oo4. CO for NOTE nLS evel surface of f of 3 variables LLC level curve) 4 The Mean Value THME MVT Let SciR containing a,b ES and the f defined on S, cont. ot each point uppose L (including a 2 t differentiable on L except possibly a, b, then EL st Let h b-a L a +th o st Let pet) flat th osts p cent. on Co, UL, duff on Co and Ca +t By Calc 1 MVT fue CoD set Lot C a uh What about global prop on S Det. A set SCP is caled CONVEX set if s convex 2 arcwise connected. Exy Show that XE S Con by S 3izER 13 not conve a Let g,b ES, Canu two points) Let L ostsMAT
t b
tb
tlb
CI-t a
CI-t) a
IHMI Suppose f is differentiable on an OPEN, CONVEX
set S and Ilvfi SM ES, then
Cb)
MILD-g
f(g).(b-g)
Mb
b-a
Ex
is convex Copen)
f is di
On
OUTH
IHM S Supose fcR- R is differentiable on an OPEN
CONVE
set S. and 2) e o ES, then
constant on S.
pf by THM
Va. b ES
Ca
Constant
THM 2 f is differentiable on open, connected
set
S and vf o VzES trien f constant
Ex Let S
ne on S
Lax Pfcz) g ES but f is not constant
on
S. Does it contradict THM 3
me on S st g f but
Find s
ha-
-g is not constant on S
hef-S is Constant
on S.
sin ax No S is Dot comedted
ii) Let g (2 f(x)
h f9f ISI constant
Let f SCR
Linear map f R
a21 a22
MAT t b tb tlb CI-t a CI-t) a IHMI Suppose f is differentiable on an OPEN, CONVEX set S and Ilvfi SM ES, then Cb) MILD-g f(g).(b-g) Mb b-a Ex is convex Copen) f is di On OUTH IHM S Supose fcR- R is differentiable on an OPEN CONVE set S. and 2) e o ES, then constant on S. pf by THM Va. b ES Ca Constant THM 2 f is differentiable on open, connected set S and vf o VzES trien f constant Ex Let S ne on S Lax Pfcz) g ES but f is not constant on S. Does it contradict THM 3 me on S st g f but Find s ha- -g is not constant on S hef-S is Constant on S. sin ax No S is Dot comedted ii) Let g (2 f(x) h f9f ISI constant Let f SCR Linear map f R a21 a22MAT 37
Dzi A mapf: ScR (Sig open) is said to
ba differentiable at geS if there is a mattfx
St
cay-L. h
cath
lim
o h
NOTATION: D
NOTE If m
v, I n matrix Dfog fcgy
1-th row of Lis
Def f ScR R differentiable at geS CAS isopen)
each co
f, fa,...fm is drfferentiade
a a and D is a moftfx nhose jth ton is
ExY
b)
af
Suppose there is another matix L'EL with
the property
then setting h sw (s- o hao)
We howe
Sy
Conttodichon,since.
sto Vector-Valmed Eunctions:
R-R differentiade at geR and
RA is diffeMentiable at goa then H f
s differentiable at a and
pt. H differerdiable each componens differentiate
t
Lo this is chain
MAT 37 Dzi A mapf: ScR (Sig open) is said to ba differentiable at geS if there is a mattfx St cay-L. h cath lim o h NOTATION: D NOTE If m v, I n matrix Dfog fcgy 1-th row of Lis Def f ScR R differentiable at geS CAS isopen) each co f, fa,...fm is drfferentiade a a and D is a moftfx nhose jth ton is ExY b) af Suppose there is another matix L'EL with the property then setting h sw (s- o hao) We howe Sy Conttodichon,since. sto Vector-Valmed Eunctions: R-R differentiade at geR and RA is diffeMentiable at goa then H f s differentiable at a and pt. H differerdiable each componens differentiate t Lo this is chainMAT
Let got
ti
Evaluate Dof
a) using chain rule.
by di Neatly
Se
Sze
Ex Given
V u
SX.
es 4e
3 17
aza art.
holds
at
zer
zea o xe
o yer
Sye
MAT Let got ti Evaluate Dof a) using chain rule. by di Neatly Se Sze Ex Given V u SX. es 4e 3 17 aza art. holds at zer zea o xe o yer SyeMAT
Ex Supase. R is differentiate., we u,V
Find the value of kif anp
w satisfies .PP. E. partial derivative
gestion
2 V
NOTE by chain rule
u,V)
Ex Ey Consider the paar map P R R'
a) Express. ,26 in terms
by Is it possible to express 3L n terms of #,3
under what condition?
24
cos 6
(sing
13 ner
b) Yes, its a linear systems with unknowns
there's a solutien definmon of koejficient
matt-HX :t 0
sino
tsing.
ICOS9
We can save for 5 if rato
R--R differentiable at aeg
n example above
hsin9
sing
bian of a map
NOTE
MAT Ex Supase. R is differentiate., we u,V Find the value of kif anp w satisfies .PP. E. partial derivative gestion 2 V NOTE by chain rule u,V) Ex Ey Consider the paar map P R R' a) Express. ,26 in terms by Is it possible to express 3L n terms of #,3 under what condition? 24 cos 6 (sing 13 ner b) Yes, its a linear systems with unknowns there's a solutien definmon of koejficient matt-HX :t 0 sino tsing. ICOS9 We can save for 5 if rato R--R differentiable at aeg n example above hsin9 sing bian of a map NOTEMAT
NOTE norm a linear map (mainx) is detined by
IN An sup LA L t is the smallest consont c
o
t. A
b.
THM suppose f SER
where S is opan, convex.
Vab ES
MI b-a
fos f is dedined explicitly
yo) are given implicittu
Let z
a function
Not alo
assure that itis, then
io general
by chain rule 2E
2E du
mokes sense in nenahbournood at Cx..u.)
L y or function at
consider (a
u is a functon
3-2
st,
chon
paricular casa LET
suppoa R- R is class c' on an open disk
DCN, a
a to F zaupao defines u as a function
of int
Does define y as a functien
eigh
NOTE
32
patamettizati
st
Q: F(my,z) o be solved for z as a functon sf x
f yes
MAT NOTE norm a linear map (mainx) is detined by IN An sup LA L t is the smallest consont c o t. A b. THM suppose f SER where S is opan, convex. Vab ES MI b-a fos f is dedined explicitly yo) are given implicittu Let z a function Not alo assure that itis, then io general by chain rule 2E 2E du mokes sense in nenahbournood at Cx..u.) L y or function at consider (a u is a functon 3-2 st, chon paricular casa LET suppoa R- R is class c' on an open disk DCN, a a to F zaupao defines u as a function of int Does define y as a functien eigh NOTE 32 patamettizati st Q: F(my,z) o be solved for z as a functon sf x f yesMAT
If Fo o can be soved fot z n terms of ziy
so that z fuxu
22.
Ex Let zz(z
in the netah
C-1, 3,1), z is a function
zmy so z fix, y). In Nhat direction
is
changing
most rapidly at the point
S
sla in the direction of vz)
5,3
Euncion THM (one equation)
neigh

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