. . Ilgw to rea$ matlgmatl*$.lygrbols it Engllqh

. . Ilgw to rea$ matlgmatl*$.lygrbols it Engllqh

III
UATEI{LTIKA
. !{ATHSMATICS A}ID PHYSICS
A FTZIKA
]. . Ilgw to rea$matlgmatl*$.lygrbols
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42
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43
2. Somenoteson readingand writing mathematicaland physical expressions
a) Mathematicalexpressionsareusuallypronouncedin the shortestway possible
is to be unequivocal
the contextwill makethemclear.Only if the expression
because
preferto usethe longerform.
or someconfusionmay occur,will the mathematician
Thus,e.g.
'eks/
may meanfx(i.e.,f multtpliedby x),f(x) (i.e., thefunction of x), f* (i.e.,f
"fx['rf
subsuiptx) , FX (i.e.,line FX), or FX (i.e.,vectorFn.
Also: only a slightlydifferentintonationor longerpausemay indicatethe differencein
e x p r e s s i o n s a s , e . g . , a ' -a1n d a ' - ' o r x + ( y + z ) a n d ( x + y ) + 2 .
b) Note the indefinite article in suchphrasesas : the bodyhasa lengthof 5m, at a
distanceof Shn, at a temperatureof 5"C (readas 5 degreesCelsius),at a rate/speedof
80km/h(readas 80 kilometresper hour), at a pressureof l0 Pa, a body of a massof
500kg,at a width of 3mm,an area of 50 squaremetres),etc.
c) Thereis no full-stop after abbreviationsof units, e.g.,Smlong,lOcmwide
d) Theunitsdo not usea plural if usedin writing as abbreviationS:e.9.,an areaof
8x5cm2( but : readas 8 by 5 cetimetressquared).If pronouncedor if written in full,
the unit canbe read in the plural.
e) If usedasan adjective,the numeralis usedin the singular,e.8.,a I}-kilometre
distance,
/) If written in full, the units arewriffen with a lower-caseletter, althoughin
letter,e.9.,5joules x 5J
abbreviationthey usean upper-case
g) Insteadof a full-stopor comma,a spaceis oftenusedto separatehundredsfrom
respectively,
e.g.,5 000 ,
andmillionsfrom hundredsof thousands,
thousands
5 500500
h) Thereis a decimalpoint insteadof a commain decimalnumbers,e.9.,5.357
are
i) If equationsare part of a written text, commasand full-stops,asappropriate,
usedto setoff factsor enda sentence.
j) Thereis no article in expressions
as,e.g.,... in TableI /Equation2 / Figure3 ,etc.
44
Lrytuthemqticaloperytions
tilitltton
fr'aflea]
- ![{t{sl'
to a{{ t ieAJ - a6{tat
rlus ['pfa:J- Pr*
l l v r l * u r Eav.a oqxale tr*lto
5+7*12
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ra},eg
art
io ogual;to
axrlusbequaka
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I lrl itr sorlaal o0o6{tda{
- oilEdJtat
to snbtrrct [rab'trart]
'rat
* ulrus
neaJ
nlme
[
gr*tr1gt*ga
9 - ) * 6
a-brc
lf}}tlp*lPll*31
stsc nlaus tbree sguala oLx
arlauabegualac
[,na]ttpl.t'kel[enJ
]
to arltlpiy
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by, tlnoa
r I a *ultlptlrd
1r.
2 r
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5;3*'t5
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to 01v1dte [ili'vrrn J - i[l1t
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45
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Sraotloas
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S{ra
, om !$.f
oas fourtb
el.gJ rlonky cr tvo}{ tr*, Iq r lltrtall
Jr rtr{y r{l$r'dnl Sfct*vlrr ?r Jsr*
aovatelt fa{ovd, Jo-lt lltatst v6tS{ mS 1, Jo glrnavatrl v no{n{r S{*h,
tJ. nr koncl Jr -r. Jc*1"1.Jneaovatol rckons*lt na J"d&f$S, $txr JrJ $rto
{f}oak0 [teur p{,anaa Jrto ; abrsrils a
sdttaila{ S{aLovlu. lI aaprrytcb
" i'
rloneaor JrLo rotorf [euva],
X/a
tbrea lalvra
['Ulrr
tro ftftbs
['frf$a
four tsntbs
a ovar b
flva ovcr tr*aty*636
2/'
411O
aA
5lat
]
J
'deslral 'fr
b) {aclu$. fraatlons
orhseal J * {sttttln{
rlg
[
I{sto dssrttnn{ [{rfp byvd todka ( {.salirL potnt [ "iwXrsl' ]**ntl
]
ItuLa pfe0" deEotl,nnoutetrtou ac [aato aeplSr a nl$tl. ll{str le dr*rtlnnqr:
te$tou sa 6tou JoilaottlvS, psrit rlrrstlnnfl t*tm
Jihr ollrk.
tl
nougbt
o
s6re
.l
0.1
'ao:_t
f
]
L-ou
- r l c rJs u l
f
polnt ono, nouglt poXat ono
46
polst nnugbt ono
potnt iloubl.c nougbt one
po*nt tbrse tro ont
tro polat onc
ttsl.vr potnt f,lvs
.01
.001
.)el
e.l
te.t
I e'neellsLg] - nqtan"stxcb* analyxa
*. C*lculus nnd qqf,tyjis [.kpl,Ilules
- firnxcs
Iqfalt,$eil]
-I\notlons
f(r) i t(:r), ctc,
filrctloa of x, firnctlon r, f,x - funkc* r
y 1g equaL to thc fimatla:r of, l, -$ lt oqua3. to tht
i, y ls agual to f of - - lr rov:06 sc fiurkcl r
t*ftx)
P*tr:tfoli:tl*f
[,iltfsrensl'al[anJ
to {lffsrsntlata
]
frdlfe'raaELolte
*
x to oerlvr
o{volQvat
[dl'ralv]
3r
Ay
i
*
f
'(
* dtrltov{nd
:- dsri'vovat
illf,famnt,lal y leffo'rsnSElJ
a varlatlon ln y fr"vqerL'tl*eal
'lakrlnsnt]
aa luorausat of $ f
{y
.{*T1,.
i y'i
f
'(x}
i }x y
ro)
S,r(a);So){*i{r
t*lurifJr,g);Dr(u)
rj (r,y)
f, (ro, tro)
*-
i o";
r#r'r)
S, ( !ru)
i
* dlf,*ronsl{.l y
* Y$tr'ls$t y
* $*{rts't$k y
$f y
dertvstlva
tho_ tfxrnt}
[dl'rlvottv)
*
x
f(x)
rhors
to
x,
rooPect
;tth
$rtn*
Y
dorlvaca Y dls r, Ldo Jr. * ft r)
of, f *t \ * }wn{ iln*
dulvat*vs
tbs { ftrgt}
dls r v boSS xo
rtvrca ftr)
tb$ ntb d.erlvallvs of, S * f{.r} rttb reapect
to r- r-tC dsrLvaer ypod-l*:
d to the ntb Y by d: ts tb.* ntb ( r.g.ff*"l_ I
s
d stuarod y by dx squarc{ ; X.D' ls
prosouncod urch loager th*n tn alx rbov*)
au'rtvetlrJ
th* Bart**l {*rtratlvo
[]",5f
p*rol,{ln{
of u * f(rry) dth raaptct tq r
dori,vaca u dla x
partl*} 0n bS partlaL ilr
of ftx' y} rlth
tbs flrst pa:tlal {srlv*tlv*
*
(xo,
parel"{ln{
ptrn{
go)
roapact to r *t
0erlraoa ftrn y) podl.r : v boil6 rs, Io
thc accond p*rtlal {arlratl.vc of u o f(:, y} r
.x anil than r*th
{tarrn flratJ ritb r*apact to
*
pamt{t*{
{srt'vaca
y
iln${
renpect ts
u*f(xry) poitt*raY
p5tl.al. d *Snarad 1 by 3a$1al &y .lx
d two u by d x y
* rot.o*;;
[,lat1'gralrsa]
:}3?g:::*:
to lntogratr
l'lntlgratt
J - lntagrorat
lntegnnd
Lntagraail [ "tntlgrast]
!
lntogrd3.
!.ntogral
['tntlgrel]
f btho lategral Of, ,... frma to b * latrgr6l..
I
-d
f,gnctton
s* a Eob
47
ff
-
doublt lntlgrel
tb
- dr
dr
/oo
f'
tbr (iLrfinlts)
tntwrat
sf fir)
oa a do b
l.la
*'-t
fir)
l1rtt - llaltr
- ]lflf
to, approaober [s'PrcultrJ
teada ['tcal]
to b
c-qual
to
a,ie
tllTg:-tcada
: ! the lldt
of f(r)
t/-
IlrLte
6
Itr [f(r)
r *+ a
a ts bt - l,atrgrJl
f(r)
- ]lnttv
['unrtal
llrr*
fro:
t,:
a
lh
t* r - iatatr{l
roaprtt
rtth
tbn Latcgral of f(r)
r -+
'
dr*Jnf tutlSr{I
- Pro r
tr
I rotlf
3.
D
tbs l.ldt of ftr] nlur S(r] rt r trnsg to r lr
e{nral to c Pluc t
. r * t
+ g(r]]
bl'Mol
rr
- trl€onomtric
Irtrrse.'aonltrl]
'eka,l
['eaj.D. --['aara 'erg
r
]
Erac :
'rks
].
r
['toe
oiirr"-f'raucata'"L"J
'ara)-,
-['taailsnt
I
['t an
'ehs
J
iingcat r
'.Lr
r ; c t n r ['kot
J
'shr
'res't
l
milisrrt
toiang;nf : t
'sl:Lsat '.&"
aocarrt r [
J'
x
'rhal
cot.o.rt r I tar'rl:Igs3
ticococ r
T,rr$t:om.t{t
sla
cos
trn.
aot
lao
cro
'
llnr
-
COII
tgr
ott *
-
r*s 5
00*ls x
'gr!.rh larrelotJ
l, gleelr t*IpLet
A a c
3 p
ry
Atr
f € ,
z g
H q
0i,
I r
fs,
rtf
t1,w
N'v
? s
5
0 o
Ir n-
? g
Ie'
T r
Yrr
fr
alfa
['prr"]
bctr
['ul:te]
gala
t'ecrc:
itelta
['aelte]
I'cpeilean ap'aal}aa{. rylllon.
'''
['sl;ta]
... d,ret*" '"
'
eta
t'llta;
'9t:ta, 'erltal
tbitr
f
'trrteJ
rf
lotr
I
kape
['tocpa]
l.rabitc
['l rude]
u{
[":o: ]
nf
['alu: ]
'asl]
I.s{
[tsal,
or{hron
['edhrerreu'nattroal
p{
['t.r ]
ro
L reul
ctgna
['atgra]
'to:]
tau
Ifo.airJ
f
-!.paLl,aarJ
-aailca
['1lgf1oa,
y3rell,on. ..
L,Juap
? e
t
a
['fatJ
f{
Xr
l'rar l
chd
Y,y
Iinr]
p8{
.O ar
g'*argo3
omgs
48
6. Latin nbbrcviationsis csmmonuss
i.n Sn*eh
s o e ps s i , ,
' '
.
.
.
:r
i
.
srsl|noJ
pep$"
at{.
,
'..
a n$cle{uJdc{
.. .;..;" f .g.,j,:itthlit
le
tantg*
:
tentdl, / s
tdhofi, au,:tor* /
'
t"j*
."
i:,,*,1'
ltrrrili^, i0 xt'",iizJ
' ;1
; .,
:r:,loq:pitJr,in.
: , l O C r jiir;'
O l t r r . lin
'
,
trre:
n A , P I A C O quotrd
n tttrg-nrl.?
;11.,',.iqr.
a-p}eia
.",i,;,,,{*,i:f,r*lti]tn.i
urgurari|va'*lr
-.
'
Ii;5.
r' :'PSBEEC''IPE'"
x$J'
. . , t '
Sa.en.tnvandn n{st$
...'
rukopla
"
1'n Cnjuitrtptl
'
.,.N.,8,
;
."
,
.' . ,: :.nota bens
t -noto ryell
'nout'r',
'ultnl],4i
','',r*,is,['nouir
Htn:,.:tl r r
mt'aIt
. . , . .
nota.b*n*,
rvlS$t$ nl"
v$iwrl
v Ettpvnn$
, prdcl"
"
,:ptist **ri.p*
'
"tak, dou$ka
cn* yis
""
'eer 'pouoarriPtJ
, ['pli:
'oi'g.
e t"or totifr,
et al.
Ir'U
and others
r
a
A
I
J
a ostah:i
49
7. [xercises:I -6
1. *:L'
a ) 3 * + ) x i 1 O * 8 : a ; a t * 1 ) * ; ? Q+ ? o i 4 9 + 9 * . i 9 9 + 1 * t
1 2 9+ ) ? * i , ? 1 + ' ? 1 ' I ' r * b r * x + l l r ' + y , I + y
1 8 * i 1 1* ) l i A 0 - l S r : i
U )1 9 - ? o i 2 ) - 9 = i 9 t
l 0 t 0 * 8 5 * ; ) r 0 0 O- ) , O 0 O * ; s * r
r r - I, q - b
t } O - l 0 O * i
c ) 5 . 5 . i . 3 . 8 o , ? . ? a r 4 . 1 ? r * r1 ) . 1 0 * r 6 " 9 * , s b * s r 4 r * $ r
d) 9;.) i: r 5:t '' ; ?l:f
a 3 b *rrX ;Z*Y
3*b
: r ? ? 1 9 * , ) 5 : 5 * , 1 0 0 : 1 0' . , . 4 8 ; 1 2 r , ? I l l )
* r
s ) ?/1, */7, 1/2, \la, 1/1Ar 51100f '11000, 6let, a/1, Ift
Vb, b3/s, e/y, Ir/2r'l/x, tla, 1/ t'x , c+d,/c*d,
f)
O . l ; . l I , 0 0 2 ; 0 , 0 0 ) i . 2 1 1 4 ; 5 . , | ; ? , 9 9 i 1 0 . 5 ; J S 0 . e 5;
g)
?3, 23, a?, a-2,
"DaE,
(a*t)-lr
"),
t*2 t y?) :r ! , az * b?, (a + blS, (a * bln.
.-),
1/.a\
""*o,
r-lr r-1 , ^1/',
aE."l
- a,-[,
str/aB-l
a-1/',
ir,
( aE)n
1 88,
r-1/7,1.2'l)1r
i*o,
^l/tr
( g/b)orrE/bn,
- t r 6 ,- 1 8 , ' l / ; 3
. r, V; =
/;,',6, t/*i, iF,
"1/o,
V} ,. t Yila * .n/ra,1ry. * l/lru = "l/o ,
n\fi,b=Vi,
r ai f f i , W * o
Y;,av*l
',
a. +fag.!,bl.fol*g*Iq fl#,l{ :
O i + t -,
:
i r.b i !:b i {ty*a;}
rlb r ydn i t ) it li i-, fi
l r l i n l i - + ; : = * i I i r i Jl ;
3;40 tbs fo}l.
).
;^ji i f
l/;*
.tJ
of, 0rr*h, alpbabet
lrtters
* , 1 3 r f , ,f , o r r A , f , r d , L t y t f
c
c
? c r( , f , u ' o r r r i X , T t - f I
Plv
*.
attsntloa
to stmsa
Trarslatc
,.
lnto
r !,
I ; xl &; a ) b i ld s i
ia';a"ia*ix1 lygi
glv&g
tbolr
f 'tr
Crdch Bailca
:
s , . f , rF r . Y r
i
to dl.fforsatlata,
il!.ff,cr*nt,
i.lfflnnoa,
aditltlonal
a{d, a{{ltlonr
aubtractton
;ubtractr
nnltlPllaatioa
mlttplYr
{.lvlalon
illvlde'
lntagratl,on
lntegralr
latagratrr
'
i I ; *;
ill.ffrrontiatlont
dtffsr*ntlal
Crcob ;
tJrcorTr {$anti.t3rr uonstrnt'
forur}a,'tb*orou,
a) oquatlon, e{I}n}raloa,
dsfl,n*r {leftrnttton
to
vaflabls,
vllUa, Broparty, rolatlon,
b) accor0lnglyn hanc.s, thonolr ;htnca
o) lat us asggls that ,.. i lot t - b i 19t 8 d,anstr 'r.. i lct Surtl.on I
dcaots ... I aecord&g to Eq,.2, J'st a - b i
i SC.} }sldS
at) tbr aquatlot ls val,l{ for r,, i (}} 1o tnrO, lf, *....
'for
,,.
*11. r*Luas of
i
for r,. ; tbe rolati.on appller
A) 3{.1 neSrbe oxprsased as r.. i fs qqn SI}rtAa S1.1, an r., i S{.} uly }a
a8 ., r ; tb* foilloufry
1a tbc f,cr,6 i, r . I ? oan bs ltlttln
rrlttoa
t.
.rgatlcn'uaY
be Put as | .. r
6. 3{mt3}3-1o*,
*s}:ub'
ilo$s$ntr, !$sntr3*t
vfrlcilt*f
a$ss}*n{,, it$lrnl,
rova!,or, rsftdndr oil$dt{n{,
lonenot
v* imasrat*ill1
},o}an, r}oaak, v Sltatollr
,... fnirri.odnociordn{,
illfor*neLd[t
o{voii't,
d*rl.vovat,
d.srlvov{bl,
a:r.a$xa,
llteuattcX,d
,i S*ioo,
^.
od${tct, n{robrno 5, {$l.cno }r umos$t nr
*}{tat,
.:, ,'1*xsgr&,
Latcgror{q{,
'.'.'.llr-nhoS, :,i.nooolt ns tfet{,
od*o*iovat
lato6mvat,
xocnLt, d*rl.rorat,
j-
-
8" How to read equntions
I..
a +,h
gl3
a - b " e - d
a plus b ovsr a s!.nus b ts equal *o c plu$ * .o*#.,$
"nikltr* s
e cubd !.C sgrrAl ts,the lqgartt*n of d.,tn tJra .b*s*
*,
Q. a! = log"dl
:lllfa *
{*} *
* lafa-i-
),
',
rJ
{}
F of s 1o equal to U, square brsckets, par*ntheola, s dl.vtded by c eub I p}$s
2, parenthoslc cloeed, to the poser ts oner n ulnug 1, nlnus 1,
squsre brackets
closed
b) P of z I's equal to b nrrltlplled by the whole quantlty: the quanttty
tvo plue z
. olrer c sub n, to tbe pousr I over n nlnus l, nlnus :.
* . . ,
f-
lvrQl - f.f*)t *, I n{r,- $l - rlftr- flt
the abEolute value of the quantlty
p *ub.$,qf t ons, xtnrr*
r sub J of,,t,tmr
ls less then or equal to the absoiuta oatue of the quantlf
X of t1 d.nrr p
over J, olnus l[ of t, nlnus B over J
.t,
f :
l.
y#rfcEfe)J
{qefor$J;/*rr.ar.".,cJ
:
k la agual to the naxlntn ovor J of the sun frun 1 eqgals one to I rqr:alr n
of
ths Eodulus of att of t, r*rere t I1ee ln the cloaed lnterval
a b anil *orc
N
run$ frQu orn to i
,
., ,
:
t
6.
tb, tt
, * o { ,
t
f[s,foatJ*aoG)]As
r
ItttlFk)Jat
"
t\
l{ntt
aq n baeouos lnftntte
of t}rc lnt*grcl
delta n of a, rlth reepect to e, froo T to t,
and ? of a, rlth rctpect to s, froil
r to t
?.
qt-.
r a - E - f - f' t )
cf f of a ant p, of. q p}ue
ls equal to ttre lntegral of f s
E
"tl]r'.f._c,r
g,raub n olnua r sub c plus 1 of t ta equal to p aub n alnuc r aub e plur
tluea a to the poter t tloec L aub q plr,ra c
'1 t
8.
Lr.g
'
r
r
-.
(-t)"
(4
.(tl
g)"'
* (-r)*'(4
L,
gS{t'r) r -.-.. * 6r g
'
t uub n adJolnt of I ls equal to ulnus 1 to tlre n, tlasa thg nth derlvatlve of
* gub zoro conJugata tlsos gi pluti ntw.ia sns to tlts rl nfir$s l, tls*o t$* n
nlnug flrgt ilorlvatlve of a eub ons e qu&ate ttn*s g, pl.ur .,. plge a gub n
eonJugate tlnes g
9.
rr{*qjg,g&}ftl*l'r.#-l*,lltl|ro'...
?7,
the par:tlal denlvatlve
of F of lanbda cub 1 of t and t, rlth reapect to laubda,
urtrltlplled
by lanbda sub 1 prlw
of t, plus thc partlel
derlvattva
of F ylth
ar8unent* lanbda aub t of t anfl t; wlth $eppsct to t, ln equal ts o
51
*
*
l_o"
dst
[t
r o
* 5@]s
ths sec:onetdsrXvatll,vo of y *ttlr
plus b of s, 1c aqual to sero
11.
'E
f of
rr
le.
,1
-
ff")
as abooluto
r
'
ths qudntiW 1 ""
i
,'
i €{r,:' f, )
sub nk hat, plus brg o of one $ver the ribEolub vsl*n of
fle/'t*
Pr* * 0 { i : 1 " }
la equal to
FB$psct to s, F}us y; tlu*s
r bccscto
lnfLnltc,
r!,th tbo ensr,xsnt of * rq*nl
to gpnna
t:}'-'
{i,r*r !"ft.
D sub n'd.nuc 1 prlue of :r la cqual to the. produet tlo! r eqrol to ccro to n
I ulnus x sub s aquareds parethaals clorcd, to t&c pona rplllmi
of, parcntheals,
nlnua 1
lr.
Ktlil
_
rGrd
*-u
rf* )
I
*
J
dy.
/r'$1..tr1
KoftErsN.il8aqua}toorxor*rtffi.'|r',t1mathr1ntogm}ofXoftei$;,
oysr I d.nua r of r, rlth regpsct to r doqg the eurve of tho mdolus of r nlntl
ons balf,
gaa
.I{"
-
tr
equal to rtu
;
fu*ilu*s
{a>0}
i
tlrc ;;swd partlaS. (dcrlvatlea) of h dtt* r*sp*et to tr plus'*, ts ttlo,*st$*&l :
' tlmo tbs Laplaalan of tha Laplaotair of a, Lr *qunl to lerol
* *l Ws'*tgf;*
-
15.
lr{t}-*J
ct{.
; ' '
-
. 4
!'tu)*"*
(c>t)
Eltral' frs s. rrrrr tof il*tn to ttre k of !$r x to t$s * dlrferd bf r,
fnf1aig1 to c plus L lr$tnt$
. ' " .
gneatcr
t$nsr :.
re
c
ls
t,
tt
tc
rtth ibssst
52