how to print barcode in crystal report in c#.net Similarly, the partial derivativeof/with respectto y is definedas in Software

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To get an intuitive graspof partial derivatives,recognizethat a function that depends on you aremountainclimbingand have two variablesis a surface ratherthan a curveSuppose to access a functionf that yields elevationas a function of longitude(the east-west oriented
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'The fbrm dt'ldx was devisedby Leibnitz, whereas is attributedto Lagrange y' Note that Newton used the so-calleddot notation: i Todavthe dot notationis usuallv usedfbr time derivatives
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r axis) and latitude(the north-south oriented axis)If you stopat a particularpoint (,rn, -r' -r'e), the slopeto the eastwould be 0f(xu,y,) I 0r,andthe slopeto the noth would be 8/(xo, y6)/3"r 1912 Differentiotion in Engineering ond Science
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The differentiationof a function has so rnany engineering and scientific applications that you wererequiredto takedifferentialcalculusin your first yearat collegeMany specificexamplesof suchapplications could be given in all fields of engineering and science Differentiationis commonplace engineering science in because much of our work involves so and characterizing changes variables both time and space fact,many of the laws and of tlre in In other generalizations that figure so prominentlyin our work are basedon the predictable ways in which changemanifestsitself in the physicalworld A prime exampleis Newton's second law, which is not couchedin termsof thepositionof an objectbut ratherin its change with respectto time Aside from suchtemporalexamples, numerouslaws involving the spatialbehaviorof variablesare expressed terms of derivatives in Among the most colnmon of thesearc the cortstitutive /au's that define how potentialsor gradientsinfluencephysicalprocesses For example,Fourier's law oJ heat conductionquantifiesthe observation that heatflows from regions of high to low temperature For the one-dimensional case,this can be expressed mathematically as
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Q: - K, d T ,' ax
( 1 9r ) r
where q (r) : heat flux (Wm2), k : coefficientof thermalconductivitytW(m K)1, f : (K), andx : distance (m) Thus,the derivative or gradient,providesa measure temperature , of theintensityofthespatial temperaturechange,whichdrivesthetransferofheat(Fig 192)
FIGURE92 I "downhill" Grophlcol grooient depiciion o temperoture of Becouse moves heol fromhighlo low lemperoture, flo* in {o)is fromleftiorlght However, to theorientotion Corteiion ihu due of the coordinoies, slope negotive thiscose is for Thus, negolive o grodient leods c positive to flowThisis iheorigin lhe"minus in Fourier's of XeotcJnduction reverse of low The sign coseis depicied {b),wherethepositrve in lo grodlenl leods o negotive flow fromright left heot to
Directionof heatflow
DIFFERENTIATION NUMERICAL
TABTE| 9 I
Low Fourier's lor,v
Theone-dimensionol of someconstitutive commonly in forms lows used e n g i n e e r i no n d s c i e n c e g Equotion
q : - K *,
PhysicolAreo
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Grodient
-[:mperoture
Flux
Heotfiux
Proportionolily
Thermoi Conductivily
Drru:iu]
lcw Fick's D'Arcy's ow low Ohm's Newion's viscosily low flooke'sow
dc ar ,dh
Mossdiffusion Flowthrough porous medicr flow Currenl Fluids Eosticily
Cor,e,rorio1
Mc Iu
Heod
For'vflux
Currenflux t
ic Hydrou Conductivi\
Elecfricol ConductivilY
dv _ dr
| -t^
, ax
Velocity
Sheor Sfress Siress
Dynomic Viscosily Young s Modulus
EL-! L
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Sirnilar laws prcvide workablemodelsin nranyotherareasof engineering science, and elecincluding the modeling of lluid dynurmics, masstransf'er chernicalreactionkinetics (Table191)The ability to accurately is derivativesan tricity, and solid mechanics estimate importantfacet of our capabilityto work effectivelyin theseareas Beyond direct engineering and scientificapplications, numericaldifferentiationalso is importantin a variety of generalmathernatical contextsincluding otherareas numerjcal of methodsFor example,recall that in Chap 6 the secantmethod was basedon a finitedifferenceapproximation the derivativeln addition,probablythe mostimportant of appliWe cation of numericaldifferentiationinvolvesthe solutionof differentialequations have alreadyseenan examplein the fbrm of Euler's methodin Chap 1 In Chap22,wewillinvestigatehow numerical differentiationprovides the basis for solving boundary-value problemsof ordinary differentialequations These arejust a few of the applicationsof differentiation that you might faceregularly you in the pursuitof your profession When the functionsto be analyzed simple, willnorare mally chooseto evaluate when them analytically However,it is oftendifficult or impossible the function is complicated addition,the underlyingfunction is often unknown deIn and pointsFor both these you rnust fined only by measurement discrete at cases, have ability the valuesfor derivatives, next to obtain approximate usingnumericaltechniques described as
r92 HIGHACCURACY DIFFERENTIATION FORMUTAS
We have already introduced the notion of numerical differentiation in Chap4 Recall tha we employedTaylor series expansions derivefinite-difference of to approximations derivaforward,backward,andcentered tivesIn Chap4 we developed difference approximations of tirst and higher derivatives Remernber that,at besttheseestimates errors were had that O(h2)-that is, their enors were proportionalto the squareof the stepsizeThislevel of
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