Automatic Derivation Of Formulas Using Reforcement Learning
15?/08?/2018 formula derivation transformation is abstracted as a mapping of ... In the derivation of mathematical formulas in order for the formula.
Differentiation Formulas Integration Formulas
Differentiation Formulas d dx k = 0. (1) d dx. [f(x) ± g(x)] = f (x) ± g (x). (2) d dx. [k · f(x)] = k · f (x). (3) d dx. [f(x)g(x)] = f(x)g (x) + g(x)f (x)
120803 Derivation of the formulas of annuities and perpetuities
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difference notation; Approximating to derivatives; Interpolation: Everett's formula; ... Speigel M.R.
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Visit http://tutorial.math.lamar.edu for a complete set of Calculus I & II notes. © 2005 Paul Dawkins. Derivatives. Basic Properties/Formulas/Rules.
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3.MatrixAlgebra
Unitmatrices
zero,i.e.,(I)ij=Products
(AB)ij=lå k=1A ikBkjIngeneralAB6=BA.
Transposematrices
Inversematrices
(A1)ij=transposeofcofactorofAij jAjDeterminants
jAj=å
i,j,k,... ijk...A1iA2jA3k...22matrices
IfA=ab
cd then, jAj=adbcAT=ac
bd A 1=1 jAj db caProductrules
(AB...N)T=NT...BTAT jOrthogonalmatrices
matrixQ, Q1=QT,jQj=1,QTisalsoorthogonal.
5Solvingsetsoflinearsimultaneousequations
x=ATb.Hermitianmatrices
Eigenvaluesandeigenvectors
Theneigenvalues
)=jAIj.IfAisHermitianthentheeigenvalues matrixA.TrA=å
i i,alsojAj=Õ ii.IfSisasymmetricmatrix,
U TSU= andS=UUT. correspondingeigenvalue.Commutators
[A,B]ABBA [A,B]=[B,A] [A,B]y=[By,Ay] [A+B,C]=[A,C]+[B,C] [AB,C]=A[B,C]+[A,C]B [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0Hermitianalgebra
b y=(b 1,b2,...)
MatrixformOperatorformBra-ketform
HermiticitybAc=(Ab)cZ
O=Z (O )h jOjiEigenvalues,
realAui=(i)uiO i=(i) iOjii=ijiiOrthogonalityuiuj=0Z
i j=0hijji=0(i6=j)Completenessb=å
iu i(uib) i i Z i ij iihijiRayleigh-Ritz
Lowesteigenvalue
0bAbbb0Z
O Z h jOj i h j i 6Paulispinmatrices
x=01 10 ,y=0i i0 ,z=10 01 xy=iz,yz=ix,zx=iy,xx=yy=zz=I4.VectorCalculus
Notation
polarcoordinates =(r,,');incaseswithradialsymmetry=(r). areindependentfunctionsofx,y,z. 6 6 6 6 6 6 6 7 7 7 7 7 7 7 5 grad =r,divA=rA,curlA=rAIdentities
grad(1+2)grad1+grad2div(A1+A2)divA1+divA2
grad(12)1grad2+2grad1
curl(A1+A2)curlA1+curlA2 div(A)divA+(grad)A,curl(A)curlA+(grad)A
div(A1A2)A2curlA1A1curlA2 div(curlA)0,curl(grad )0 7Grad,Div,CurlandtheLaplacian
Conversionto
Cartesian
z=rcosDivergence
1 1 +1rsinCurlrA
ijk A xAyAz 1 bb'1 bz A A'Az 1 r2sinbr1rsinb1 rb'ArrArA'sin
Laplacian
r 2 1 +1 1 r +1r2sin sin 1 r2sin2Transformationofintegrals
S=asurfacearea
=avolumecontainedbyaspeciedsurface b t=theunittangenttoCatthepointP b n=theunitoutwardpointingnormalA=somevectorfunction
dL=thevectorelementofcurve(=btdL) dS=thevectorelementofsurface(=b ndS) ThenZ CAbtdL=Z
C AdL andwhenA=r Z C (r )dL=Z C dGauss'sTheorem(DivergenceTheorem)
WhenSdenesaclosedregionhavingavolume
Z (rA)d=Z S (Ab n)dS=Z S AdS also Z (r)d=Z SdSZ (rA)d=Z S (b nA)dS 8Stokes'sTheorem
WhenCisclosedandboundstheopensurfaceS,Z
S (rA)dS=Z C AdL also Z S (b nr )dS=Z CdLGreen'sTheorem
Z S rdS=Z r( r)d =Z r2+(r )(r)dGreen'sSecondTheorem
Z ( r2r2 )d=Z S [ (r)(r )]dS5.ComplexVariables
Complexnumbers
Thecomplexnumberz=x+iy=r(cos
realquantityristhemodulusofzandtheangle r(cos isin)=rei;zz=jzj2=x2+y2DeMoivre'stheorem
(cos +isin)n=ein=cosn+isinnPowerseriesforcomplexvariables.
e z=1+z+z22!++znn!+convergentforallnitez
sinz=zz33!+z55!convergentforallnitez
cosz=1z22!+z44!convergentforallnitez
ln(1+z)=zz22+z33principalvalueofln(1+z)
tan1z=zz3 3+z55 (1+z)n=1+nz+n(n1)2!z2+n(n1)(n2)3!z3+
96.TrigonometricFormulae
cos2A+sin2A=1sec2Atan2A=1cosec2Acot2A=11tan2A.
2 2 tan(AB)=tanAtanB1tanAtanBsinAcosB=sin(A+B)+sin(AB)2
sinA+sinB=2sinA+B2cosAB2
sinAsinB=2cosA+B2sinAB2
cosA+cosB=2cosA+B2cosAB2
cosAcosB=2sinA+B2sinAB2cos2A=1+cos2A
2 sin2A=1cos2A 2 cos3A=3cosA+cos3A 4 sin3A=3sinAsin3A 4 a a2=b2+c22bccosA
a=bcosC+ccosB cosA=b2+c2a2 2bc tanAB2=aba+bcotC2
area=1 sina sinA=sinbsinB=sincsinC cosa=cosbcosc+sinbsinccosA cosA=cosBcosC+sinBsinCcosa 107.HyperbolicFunctions
coshx=12(ex+ex)=1+x22!+x44!+validforallx
sinhx=12(exex)=x+x33!+x55!+validforallx
coshix=cosxcosix=coshx sinhix=isinxsinix=isinhx tanhx=sinhx coshxsechx=1coshx cothx=coshx sinhxcosechx=1sinhx cosh2xsinh2x=1Forlargepositivex:
coshxsinhx!ex 2 tanhx!1Forlargenegativex:
coshxsinhx!ex 2 tanhx!1Relationsofthefunctions
sinhx=sinh(x)sechx=sech(x) coshx=cosh(x)cosechx=cosech(x) tanhx=tanh(x)cothx=coth(x) sinhx=2tanh(x=2)1tanh2(x=2)=tanhxq
1tanh2xcoshx=1+tanh2(x=2)1tanh2(x=2)=1q
1tanh2x
tanhx=q1sech2xsechx=q1tanh2x
cothx=q cosech2x+1cosechx=qcoth2x1 sinh(x=2)=r coshx12cosh(x=2)=r
coshx+1 2 tanh(x=2)=coshx1 sinhx=sinhxcoshx+1 sinh(2x)=2sinhxcoshxtanh(2x)=2tanhx1+tanh2x
tanh(3x)=3tanhx+tanh3x1+3tanh2x
11 sinh(xy)=sinhxcoshycoshxsinhy cosh(xy)=coshxcoshysinhxsinhy tanh(xy)=tanhxtanhy1tanhxtanhy
sinhx+sinhy=2sinh1 sinhxsinhy=2cosh1 sinhxcoshx=1tanh(x=2)1tanh(x=2)=ex
tanhxtanhy=sinh(xy) coshxcoshy cothxcothy=sinh(xy) sinhxsinhyInversefunctions
sinh1x a=ln x+p x2+a2 a! for1Afor0 cosech1x a=ln0 @a x+s a2 x2+11 Aforx6=0
8.Limits
n cxn!0asn!1ifjxj<1(anyxedc) x n=n!!0asn!1(anyxedx) (1+x=n)n!exasn!1,xlnx!0asx!0 Iff(a)=g(a)=0thenlimx!af(x)
g(x)=f0(a)g0(a)(l'Hopital'srule) 12 9.Differentiation
(uv)0=u0v+uv0,u v 0=u0vuv0v2
wherenCrn r =n! r!(nr)! d dx(sinx)=cosxddx(sinhx)=coshx d dx(cosx)=sinxddx(coshx)=sinhx d dx(tanx)=sec2xddx(tanhx)=sech2x d dx(secx)=secxtanxddx(sechx)=sechxtanhx d dx(cotx)=cosec2xddx(cothx)=cosech2x d 10.Integration
Standardforms
Z x ndx=xn+1 n+1+cforn6=1 Z 1 xdx=lnx+cZ lnxdx=x(lnx1)+c Z e axdx=1 aeax+cZ xeaxdx=eaxxa1a2 +c Z xlnxdx=x2 2 lnx12 +c Z 1 a2+x2dx=1atan1xa +c Z1 a2x2dx=1atanh1xa +c=12alna+xax +cforx2a2 Z x (x2a2)ndx=12(n1)1(x2a2)n1+cforn6=1 Z x x2a2dx=12ln(x2a2)+c Z1 pa2x2dx=sin1xa +c Z 1 px2a2dx=ln x+px2a2 +c Z x px2a2dx=px2a2+c Z p a2x2dx=12h xpa2x2+a2sin1xai +c 13 Z1 01 (1+x)xpdx=cosecpforp<1 Z 1 0cos(x2)dx=Z
1 0sin(x2)dx=1
2r 2 Z 1 1exp(x2=2
2)dx=p2
Z 1 1xnexp(x2=2
2)dx=8
:135(n1) n+1p2 0forn2andeven
forn1andoddZ sinxdx=cosx+cZ sinhxdx=coshx+c Z cosxdx=sinx+cZ coshxdx=sinhx+c Zquotesdbs_dbs47.pdfusesText_47
Aforx6=0
8.Limits
n cxn!0asn!1ifjxj<1(anyxedc) x n=n!!0asn!1(anyxedx) (1+x=n)n!exasn!1,xlnx!0asx!0Iff(a)=g(a)=0thenlimx!af(x)
g(x)=f0(a)g0(a)(l'Hopital'srule) 129.Differentiation
(uv)0=u0v+uv0,u v0=u0vuv0v2
wherenCrn r =n! r!(nr)! d dx(sinx)=cosxddx(sinhx)=coshx d dx(cosx)=sinxddx(coshx)=sinhx d dx(tanx)=sec2xddx(tanhx)=sech2x d dx(secx)=secxtanxddx(sechx)=sechxtanhx d dx(cotx)=cosec2xddx(cothx)=cosech2x d10.Integration
Standardforms
Z x ndx=xn+1 n+1+cforn6=1 Z 1 xdx=lnx+cZ lnxdx=x(lnx1)+c Z e axdx=1 aeax+cZ xeaxdx=eaxxa1a2 +c Z xlnxdx=x2 2 lnx12 +c Z 1 a2+x2dx=1atan1xa +c Z1 a2x2dx=1atanh1xa +c=12alna+xax +cforx20cos(x2)dx=Z
10sin(x2)dx=1
2r 2 Z 11exp(x2=2
2)dx=p2
Z 11xnexp(x2=2
2)dx=8
:135(n1) n+1p20forn2andeven
forn1andoddZ sinxdx=cosx+cZ sinhxdx=coshx+c Z cosxdx=sinx+cZ coshxdx=sinhx+c Zquotesdbs_dbs47.pdfusesText_47[PDF] Maths Devoir 1
[PDF] Maths Devoir 1 / 2NDE CNED
[PDF] Maths devoir 1 CNED exercice 4
[PDF] maths devoir 1 seconde exercice 4
[PDF] Maths Devoir 11 3ème Exercice 2
[PDF] Maths devoir 11 suite
[PDF] maths devoir 12 CNED
[PDF] Maths devoir 2 cned seconde
[PDF] Maths Devoir 4 (Exercice 3 et 4) Cned 3eme !
[PDF] MATHS Devoir 4 de quatrième au cned
[PDF] Maths devoir 4 exercice 3
[PDF] Maths Devoir 4 exercices 2 et 4 et5
[PDF] maths devoir 6 de 3eme cned
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