Advanced Engineering Mathematics, 7th ed.

913 Pages·2010·23.54 MB·English

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Guide to Notation
L[f] Laplacetransformof f
L[f](s) Laplacetransformof f evaluatedats
L−1[F] inverseLaplacetransformof F
H(t) Heavisidefunction
f ∗g often denotes a convolution with respect to an integral transform, such as the Laplace
transformortheFouriertransform
δ(t) deltafunction
<a,b,c> vectorwithcomponentsa,b,c
ai+bj+ck standardformofavectorin3-space
(cid:3)V(cid:3) norm(magnitude,length)ofavectorV
F·G dotproductofvectorsFandG
F×G crossproductofFandG
Rn n-space,consistingofn-vectors<x ,x ,···,x >
1 2 n
[a ] matrixwhosei, j-elementisa .IfthematrixisdenotedA,thisi, j elementmayalsobe
ij ij
denoted A
ij
O n×m zeromatrix
nm
I n×nidentitymatrix
n
At transposeofA
A reduced(rowechelon)formofA
R
ran.k(A) rankofamatrixA
.
[A.B] augmentedmatrix
A−1 inverseofthematrixA
|A|ordet(A) determinantofA
p (λ) characteristicpolynomialofA
A
(cid:4) oftendenotesthefundamentalmatrixofasystemX(cid:4)=AX
T oftendenotesatangentvector
N oftendenotesanormalvector
n oftendenotesaunitnormalvector
κ curvature
∇ deloperator
∇ϕorgradϕ gradientofϕ
(cid:2)Duϕ(P) directionalderivativeofϕinthedirectionofuat P
(cid:2) f dx+gdy+hdz lineinte(cid:2)gral
C
F(cid:3)·dR(cid:3) ano(cid:3)thernotationfor f dx+gdy+hdzwithF= fi+gj+hk
C C
C(cid:2)1 C2 ··· Cn joinofcurvesC1,C2,···,Cn
f(x,y,z)ds lineintegralof f overC withrespecttoarclength
C
1
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2 GuidetoNotation
∂(f,g)
Jacobianof f andgwithrespecttou andv
∂(u,v)
(cid:2) (cid:2)
f(x,y,z)dσ surfaceintegralof f over(cid:8)
(cid:8)
f(x −), f(x +) leftandrightlimits,respectively,of f(x)atx
0 0 0
F[f]or fˆ Fouriertransformof f
F[f](ω)or Fˆ(ω) Fouriertransformof f evaluatedatω
F−1 inverseFouriertransform
F [f]or fˆ Fouriercosinetransformof f
C C
F−1or fˆ−1 inverseFouriercosinetransform
FC[f]orCfˆ Fouriersinetransformof f
S S
F−1or fˆ−1 inverseFouriersinetransform
S S
D[u] discrete N -pointFouriertransform(DFT)ofasequenceu
j
ˆ
f windowedFouriertransform
win
χ oftendenotesthecharacteristicfunctionofaninterval I
I
σ (t) oftendenotesthe NthCesàrosumofaFourierseries
N
Z(t) inthecontextoffiltering,denotesafilterfunction
P (x) nthLegendrepolynomial
n
(cid:12)(x) gammafunction
B(x,y) betafunction
J Besselfunctionofthefirstkindoforderν
ν
γ dependingoncontext,maydenoteEuler’sconstant
Y Besselfunctionofthesecondkindoforderν
ν
I ,K modifiedBesselfunctionsofthefirstandsecondkinds,respectively,oforderzero
0 0
∇2u Laplacianofu
Re(z) realpartofacomplexnumberz
Im(z) imaginarypartofacomplexnumberz
z complexconjugateofz
|z| magnitude(alsonormormodulus)ofz
(cid:2)arg(z) argumentofz
(cid:4) f(z)dz integralofacomplexfunction f(z)overacurveC
C
f(z)dz integralof f overaclosedcurveC
C
Res(f,z ) residueof f(z)atz
0 0
f :D→D∗ f isamappingfrom Dto D∗
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A D V A N C E D
E N G I N E E R I N G
M A T H E M A T I C S
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AdvancedEngineeringMathematics (cid:2)c 2012,2007CengageLearning
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A D V A N C E D
E N G I N E E R I N G
M A T H E M A T I C S
7th Edition
PETER V. O’NEIL
The University of Alabama
at Birmingham
Australia·Brazil·Japan·Korea·Mexico·Singapore·Spain·UnitedKingdom·UnitedStates
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
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