Synoptic Meteorology I: Finite Differences September Partial Derivatives (or, Why Do We Care About Finite Differences?


 Ellen Flynn
 1 years ago
 Views:
Transcription
1 Synoptic Meteorology I: Finite Differences September 2014 Prtil Derivtives (or, Why Do We Cre About Finite Differences?) With the exception of the idel gs lw, the equtions tht govern the evolution of fundmentl tmospheric properties such s wind, pressure, nd temperture (known s the primitive equtions) re fundmentlly relint upon prtil derivtives. Indeed, mny thermodynmic nd kinemtic properties of the tmosphere re typiclly expressed in terms of prtil derivtives. We will explore mny specific exmples of such equtions throughout both this nd next semester. Mthemticlly speking, the prtil derivtive of some generic field f with respect to some generic vrible x cn be expressed s: f = lim ( 0 In other words, x is equl to the vlue of f Thus, for smll (or finite) vlues of, we cn pproximte f how do we compute from vilble tmospheric dt? s pproches (but does not equl) zero. by f. Tht begs the question: To do so, we use wht re known s finite differences to pproximte the vlue of f over some finite. Applied to isoplethed nlyses of meteorologicl fields, finite differences enble us to evlute the sign nd/or mgnitude of given quntity tht depends upon one or more prtil derivtives. Applied to gridded dt, such s is used nd produced by numericl wether prediction models, finite differences re one mens by which the primitive equtions cn be solved so s to obtin numericl wether forecst. In the following, we wish to describe how finite difference pproximtions re obtined, the degree to which ech is n pproximtion, nd begin to describe how they cn be pplied to the tmosphere. Developing Finite Difference Approximtions First, let us consider generic continuous function f(x), grphicl exmple of which is depicted below in Figure 1. This function doesn t necessrily represent meteorologicl field, but it doesn t not necessrily represent one either; it is simply generic function. Along the curve given by f(x), there re three points of interest: x, x +1, nd x 1. The function f(x) hs the vlues f(x ), f(x +1 ), nd f(x 1 ) t these three points, respectively. The distnce between x nd x 1 is equl to the distnce between x nd x +1, nd we cn denote this distnce s. Finite Difference Approximtions, Pge 1
2 The Tylor series expnsion of f(x) bout x = b, where b is some generic point, is given by: f ''( b) 2 f '''( b) 3 f ( x) = f ( b) + f '( b)( x b) + ( x b) + ( x b) +... (2) In other words, f(x) is equl to the vlue of f(x) t x = b plus series of higherorder terms, ech of which hs different derivtive (primes), exponent on x  b, nd fctoril (!) order. Figure 1. Grphicl depiction of generic function f(x) evluted t three points. Plese see the text for further detils. Let us consider the cse where x = x +1 nd b = x. The distnce x b, or x +1 x, is equl to. Conversely, let us consider the cse where x = x 1 nd b = x. The distnce x b, or x +1 x, is equl to . Mking use of this informtion, we cn expnd (2) for ech of these two cses: f ''( x ) 2 f '''( x ) 3 f ( x + 1 ) = f ( x ) + f '( x ) + ( ) + ( ) +... (3) f ''( x ) 2 f '''( x ) 3 f ( x ) = f ( x ) f '( x ) + ( ) ( ) +... (4) Note the similr ppernce of (3) nd (4) prt from the leding negtive signs on the first nd third order terms in (4). These rise becuse x b =  here, s noted bove. f From (3) nd (4), we re interested in the vlue of f (x ). This is equivlent to. We cn use (3) nd (4) to obtin n expression for this term; we simply need to subtrct (4) from (3). Doing so, we obtin the following: Finite Difference Approximtions, Pge 2
3 2 f '''( x ) 3 f ( x + 1 ) f ( x = 2 f '( x ) + ( ) +...(odd order terms) (5) Note how the zeroth nd second order terms in (3) nd (4) cncel out in this opertion. If we rerrnge (5) nd solve for f (x ), we obtin: f ( x+ f ( x f '''( x ) 2 f '( x ) = ( ) +... (6) 2 At this point, we wish to neglect ll terms higher thn the first order term from (6). Doing so, we re left with: f '( x ) = f ( x + 1 ) f ( x 2 ) (7) Eqution (7) is wht is known s centered finite difference. It provides mens of clculting t x = x by tking the vlue of f t x = x +1, subtrcting from it the vlue of f t x = x 1, nd dividing the result by the distnce between the two points (2). Note tht x here nd in lter exmples cn be ny vrible; it does not hve to represent the xxis or the estwest direction. Eqution (7) is equivlent if x is replced by y, z, p, or ny number of other vribles. There exist other wys for us to use (3) nd (4) to get expressions for f (x ). For instnce, we cn solve (3) for this term. If we do so, we obtin: f ( x+ f ( x ) f ''( x ) f '''( x ) 2 f '( x ) = ( ) ( )... (8) Neglecting ll terms higher thn the first order term in (8), we obtin: f '( x ) = f ( x+ f ( x ) (9) Eqution (9) is wht is known s forwrd finite difference. It provides mens of clculting t x = x by tking the vlue of f t x = x +1, subtrcting from it the vlue of f t x = x, nd dividing the result by the distnce between the two points (). Alterntively, we cn solve (4) for f (x ). If we do so, we obtin: f ( x ) f ( x f ''( x ) f '''( x ) 2 f '( x ) = + ( ) ( ) +... (10) Finite Difference Approximtions, Pge 3
4 Neglecting ll terms higher thn the first order term in (10), we obtin: f '( x ) = f ( x ) f ( x (1 Eqution (1 is wht is known s bckwrd finite difference. It provides mens of clculting t x = x by tking the vlue of f t x = x, subtrcting from it the vlue of f t x = x 1, nd dividing the result by the distnce between the two points (). Finite Differences s Approximtions Note tht we do not necessrily need to neglect the higherorder terms in obtining ny of the bove expressions for f (x ); we hve done so here primrily for simplicity. If we were to retin the higher order terms, we would end up with more ccurte pproximtions for f (x ). This highlights key point: ll finite differences re pproximtions. All finite differences re ssocited with wht is known s trunction error, which is determined by the power of on the first term tht is neglected in obtining the finite difference pproximtion. For instnce, consider our centered finite difference given by Eqution (7). In obtining (7), the first term tht we neglected in (6) included () 2 term. As result, we sy this finite difference is secondorder ccurte. By contrst, consider our forwrd nd bckwrd finite differences, given by Equtions (9) nd (1, respectively. In obtining ech eqution, the first terms tht we neglected in (8) nd (10) included () term. As result, we sy tht these finite differences re firstorder ccurte. The higher the order of ccurcy, the more ccurte the finite difference. In synoptic meteorology, where exct vlues for prtil derivtives re often not necessry, we typiclly utilize the centered finite difference. Forwrd nd bckwrd finite differences re rrely utilized except long the edges of the dt, where the 1 nd +1 points my not exist. Higherorder finite differences, typiclly fourth or sixthorder ccurte, re necessry for numericl wether prediction models given chos theory, which sttes tht very smll differences in dt cn led to very lrge forecst differences. A Finite Difference Approximtion for Second Derivtives While the first prtil derivtive of some field provides mesure of its slope, sometimes we re interested in evluting the second prtil derivtive of some field. Recll from clculus tht the second prtil derivtive of field provides mesure of its concvity; positive second prtil derivtives infer tht field is concve up (or convex), while negtive second prtil derivtives infer tht field is concve down. We cn obtin finite difference pproximtion for the second prtil derivtive by dding (3) nd (4). Doing so, we obtin: Finite Difference Approximtions, Pge 4
5 f ''( x ) 2 f ( x + 1 ) + f ( x = 2 f ( x ) + 2 ( ) +... (12) If we solve (12) for f (x ), we obtin: f ''( x ) = f ( x + 1 ) + f ( x 2 f ( x ) ( ) 2 (13) 2 f Eqution (13) provides fourthorder ccurte mens of evluting, or f ''( x ) 2, by dding the vlue of f t x +1 to the vlue of f t x 1, subtrcting two times the vlue of f t x, nd dividing the result by the squre of the distnce between points () 2. Just s for the finite difference pproximtion for the first prtil derivtive, x here nd in lter exmples cn be ny vrible; it does not hve to represent the xxis or the estwest direction. Eqution (13) is equivlent if x is replced by y, z, p, or ny number of other vribles. Likewise, just s for the finite difference pproximte for the first prtil derivtive, higherorder ccurte finite difference pproximtions for the second prtil derivtive re possible if dditionl terms re not truncted. Applying Finite Differences: An Exmple One of the most importnt ttributes of the wind is its bility to trnsport. The trnsport of some quntity by the wind is known s dvection. We re most often interested in its horizontl trnsport, or horizontl dvection, where the horizontl surfce cn be tken to be Erth s surfce, constnt height surfce, n isobric surfce, or even n isentropic surfce. For convenience, we sometimes refer to horizontl dvection simply s dvection. In synoptic meteorology, we re prticulrly interested in temperture dvection, referring to the horizontl trnsport of energy (recll tht temperture is simply mesure of the verge kinetic energy of the ir) by the wind. Ptterns of cold ir dvection nd wrm ir dvection reflect the (horizontl) motion of ir msses nd, s we will see next semester, ply crucil role in forcing verticl motions, cn bring bout chnges in the mplitude of troughs nd ridges, nd cn influence cyclone nd nticyclone development. Mthemticlly, temperture dvection is expressed s the product of the pproprite component of the wind whether estwest (u) or northsouth (v) nd the locl chnge of temperture in some direction estwest (x) or northsouth (y) where: dvection = u v (14) y Finite Difference Approximtions, Pge 5
6 In vector nottion, (14) cn be written s: dvection = v T (15) The units of temperture dvection re the units of wind m s 1 multiplied by the units of temperture either C or K divided by distnce units m. As result, temperture dvection hs units of C s 1 or K s 1 ; in other words, how temperture is chnging loclly over some finite mount of time t. We cn evlute (14) from chrts of wether dt using our centered finite difference pproximtion developed bove. Consider the hypotheticl nlysis presented in Figure 2. We re interested in computing the horizontl temperture dvection t the point mrked by the closed circle nd wind observtion. We hve lredy completed n isotherm nlysis using temperture dt from this point s well s the other loctions tht surround it. We thus hve everything we need to compute horizontl temperture dvection. Figure 2. Hypotheticl surfce temperture observtions ( F, red numbers), isotherm nlysis (every 5 F, blck lines), nd single wind observtion (10 kt = 5.15 m s 1 out of the northwest). Depicted for reference re horizontl scles nd the north nd est crdinl directions. Dt re plotted on mp constructed using the Merctor mp projection. To compute horizontl temperture dvection, we must first set up our x nd yxes. Fortuntely, since we re told tht the dt re plotted on Merctor mp projection, the positive xxis points to the right, or due est, while the positive yxis points up, or due north. Since our centered finite difference pproximtion is only vlid over finite distnces here, nd y we must set up smll grid centered on the loction of our wind observtion. This is done so tht we cn estimte the temperture t points x+1, x1, y+1, nd y1 in other words, the terms tht Finite Difference Approximtions, Pge 6
7 enter into the numertor of our centered finite differences. The result of doing so is given in Figure 3. Figure 3. As in Figure 2, except with finite grid drwn in centered on our wind observtion. Both in this exmple nd in prctice, the distnce is tken to be equl to the distnce y. In this cse, using the distnce references on the edges of the mp, both nd y re 50 km (or 50,000 m). Next, we use our isotherm nlysis to estimte the vlue of temperture t points x+1, x1, y+1, nd y1. We must do so becuse we do not hve n exct temperture observtion t ny of these loctions. Visully doing so, we stte tht the temperture t x+1 is 72 F, t x1 is 67 F, t y+1 is 67 F, nd t y1 is 73 F. This enbles us to compute the finite difference pproximtions to our prtil derivtives, where: dvection = u v y T = u T 2 T v T 2 y 72 F 67 F 67 F 73 F = u v m m x+ 1 x y+ 1 y (16) Note tht, per Figure 3 s cption, we know tht = y = 50,000 m, such tht 2 = 2 y = 100,000 m. Now, we need to know the vlues of u nd v, the zonl (estwest) nd meridionl (northsouth) wind components, respectively. To obtin these vlues, we need to use bit of trigonometry. Recll tht in meteorologicl convention, from the north = 0 /360, from the est = 90, from the south = 180, nd from the west = 270. If the wind direction (in degrees) is known, then the u nd v components of the wind cn be obtined using the following equtions: Finite Difference Approximtions, Pge 7
8 u = v sin π * wdir (17) 180 π v = v cos * wdir (18) 180 In both (17) nd (18), v is the mgnitude of the wind vector v. In pplied terms, v is simply equl to the wind speed. The π/180 fctor in both the sin nd cos sttements converts the wind direction from degrees to rdins. Returning to our exmple given by Figure 2, we know tht the wind speed is equl to 10 kt = 5.15 m s 1. We lso know tht our wind is out of (or from) the northwest. Expressed in degrees, from the northwest = 315 (e.g., hlfwy between 270 /west nd 360 /north). If we substitute these vlues into (17) nd (18), we obtin: v u π sin *315 = = ms ms 5.15 π cos *315 = = ms ms 5.15 (19) (20) A bit of snity check is in order before proceeding. The positive xxis is to the est, while the positive yxis is to the north. Our wind is blowing from the north nd west nd, thus, to the south nd est. Our wind thus blows in the positive x but negtive y directions. Since u is long the xxis (estwest) nd v is long the yxis (northsouth), we would expect tht u should be positive nd v should be negtive for northwest wind nd, indeed, we find tht this is true. If we plug (19) nd (20) into (16) nd run through the clcultions, we obtin: F 67 F m 67 F 73 F ( 3.64ms ) = dvection = ms Fs ( m In other words, due solely to horizontl dvection, the temperture t the loction of our wind observtion is cooling by F every second. If we multiply this by 3,600 (the number of seconds in one hour) or 84,600 (the number of seconds in one dy), we cn convert this to F h 1 or F dy 1, respectively. Doing so, we obtin vlues of F h 1 nd F dy 1. In other words, due solely to horizontl dvection, the temperture t the loction of our wind observtion is cooling by 1.44 F every hour nd F every dy. Before we proceed further, it is gin time for nother snity check. In Figure 2, we see tht the wind is blowing towrd the sttion from where it is colder. As result, we would expect the wind Finite Difference Approximtions, Pge 8
9 to be dvecting (or trnsporting) colder ir towrd the observtion sttion. Our clcultion suggests tht this is true due to dvection, the temperture t the observtion sttion is cooling. The bove clcultion process represents firly complex mens of evluting horizontl temperture dvection. By contrst, our snity check hints t nother, fr less complex mens of doing so. Insted of using Crtesin (x,y) coordintes, s we did before, we my use nturl coordinte system to ssess horizontl temperture dvection. Recll tht in the nturl coordinte system, the pproprite coordintes become s, or long (stremwise) the wind, nd n, or norml to the wind. For the exmple given in Figure 2, the positive sxis points to the southest, in the direction tht the wind is blowing, nd the positive nxis points to the northest, or 90 to the left of the positive sxis. Figure 4 below provides grphicl depiction of the nturl coordinte system pplied to the exmple from Figures 2 nd 3. Figure 4. As in Figure 3, except with the finite grid drwn in the nturl (rther thn Crtesin) coordinte system. Both in this exmple nd in prctice, the distnce s is tken to be equl to the distnce n. In this cse, using the distnce references on the finite grid, both s nd n re 50 km (or 50,000 m). In the nturl coordinte system, dvection is expressed mthemticlly s: Ts+ 1 Ts 1 dvection = v = V = V (22) s s 2 s In the bove, V is the wind speed, equivlent to the mgnitude of the velocity vector ( v ). Here, we hve used secondorder ccurte centered finite difference pproximtion to obtin the reltionship t the fr end of Eqution (22). Note tht we no longer need to brek down our wind Finite Difference Approximtions, Pge 9
10 into its u nd v components, nor del with chnge in temperture in both the x nd y directions. We only need to know the wind speed 10 kt or 5.15 m s 1 in this exmple nd the chnge in temperture long the sxis. Becuse the wind is ligned with the sxis, the wind component perpendiculr to the xis is zero, nd thus we do not need to know the chnge in temperture long the nxis. Evluting from Figure 4, we estimte tht T s+1, or the temperture t the grid point long the positive sxis, is 73.5 F nd tht T s1, or the temperture t the grid point long the negtive s xis, is 66 F. Plugging these vlues into (22), we obtin: ( 5.15ms ) 73.5 F 66 F 2 dvection = = Fs (23) ( 50000m) Note tht this result is very nerly identicl to tht in Eqution (2, s we expect using the sme dt. Tht this is true provides snity check upon our result. The two re not exctly equl to ech other becuse of the inherent pproximte nture to ech of our two nlyses, nmely in obtining the vlues of T t ech of our grid points. With the bove in mind, we cn stte severl generl rules relted to temperture dvection: Where wind blows from cold ir towrd wrm ir, cold ir dvection is occurring. Where wind blows from wrm ir towrd cold ir, wrm ir dvection is occurring. When the chnge in temperture over fixed distnce is lrge, the mgnitude of the dvection will be lrge. When the chnge in temperture over fixed distnce is smll, the mgnitude of the dvection will be smll. When the wind blows prllel to the isotherms, no horizontl temperture dvection occurs. When the wind blows perpendiculr to the isotherms, horizontl temperture dvection is mximized. Horizontl temperture dvection is lrger when the wind component tht blows perpendiculr to the isotherms is lrger. Horizontl temperture dvection is smller when the wind component tht blows perpendiculr to the isotherms is smller. Horizontl temperture dvection is one of mny processes tht cn chnge temperture! For Further Reding Any collegelevel Clculus textbook will contin extensive informtion regrding the mthemticl definition of limits, prtil derivtives, nd Tylor functions nd series. Sections nd of MidLtitude Atmospheric Dynmics by J. Mrtin provides similr informtion from the perspective of their ppliction to the tmospheric sciences. Finite Difference Approximtions, Pge 10
Review of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationVorticity. curvature: shear: fluid elements moving in a straight line but at different speeds. t 1 t 2. ATM60, ShuHua Chen
Vorticity We hve previously discussed the ngulr velocity s mesure of rottion of body. This is suitble quntity for body tht retins its shpe but fluid cn distort nd we must consider two components to rottion:
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationCHM Physical Chemistry I Chapter 1  Supplementary Material
CHM 3410  Physicl Chemistry I Chpter 1  Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 596), nd "Mthemticl Bckground " (pp 109111). 1. Derivtion
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 200910 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
More informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationSection 14.3 Arc Length and Curvature
Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationPhysics 201 Lab 3: Measurement of Earth s local gravitational field I Data Acquisition and Preliminary Analysis Dr. Timothy C. Black Summer I, 2018
Physics 201 Lb 3: Mesurement of Erth s locl grvittionl field I Dt Acquisition nd Preliminry Anlysis Dr. Timothy C. Blck Summer I, 2018 Theoreticl Discussion Grvity is one of the four known fundmentl forces.
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationx = b a N. (131) The set of points used to subdivide the range [a, b] (see Fig. 13.1) is
Jnury 28, 2002 13. The Integrl The concept of integrtion, nd the motivtion for developing this concept, were described in the previous chpter. Now we must define the integrl, crefully nd completely. According
More informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
More informationPhysics 9 Fall 2011 Homework 2  Solutions Friday September 2, 2011
Physics 9 Fll 0 Homework  s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you
More informationMath 231E, Lecture 33. Parametric Calculus
Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationf(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as f(x) dx = lim f(x i ) x; i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationdt. However, we might also be curious about dy
Section 0. The Clculus of Prmetric Curves Even though curve defined prmetricly my not be function, we cn still consider concepts such s rtes of chnge. However, the concepts will need specil tretment. For
More informationMAT 168: Calculus II with Analytic Geometry. James V. Lambers
MAT 68: Clculus II with Anlytic Geometry Jmes V. Lmbers Februry 7, Contents Integrls 5. Introduction............................ 5.. Differentil Clculus nd Quotient Formuls...... 5.. Integrl Clculus nd
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationChapter 1. Basic Concepts
Socrtes Dilecticl Process: The Þrst step is the seprtion of subject into its elements. After this, by deþning nd discovering more bout its prts, one better comprehends the entire subject Socrtes (469399)
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationMath 32B Discussion Session Session 7 Notes August 28, 2018
Mth 32B iscussion ession ession 7 Notes August 28, 28 In tody s discussion we ll tlk bout surfce integrls both of sclr functions nd of vector fields nd we ll try to relte these to the mny other integrls
More informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
More informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
More information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
More informationBernoulli Numbers Jeff Morton
Bernoulli Numbers Jeff Morton. We re interested in the opertor e t k d k t k, which is to sy k tk. Applying this to some function f E to get e t f d k k tk d k f f + d k k tk dk f, we note tht since f
More informationI do slope intercept form With my shades on MartinGay, Developmental Mathematics
AATA Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #1745 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
More information(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35
7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More informationLine Integrals. Chapter Definition
hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xyplne. It
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More information1. GaussJacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. GussJcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationECO 317 Economics of Uncertainty Fall Term 2007 Notes for lectures 4. Stochastic Dominance
Generl structure ECO 37 Economics of Uncertinty Fll Term 007 Notes for lectures 4. Stochstic Dominnce Here we suppose tht the consequences re welth mounts denoted by W, which cn tke on ny vlue between
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry and basic calculus
ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationDensity of Energy Stored in the Electric Field
Density of Energy Stored in the Electric Field Deprtment of Physics, Cornell University c Tomás A. Aris October 14, 01 Figure 1: Digrm of Crtesin vortices from René Descrtes Principi philosophie, published
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationf a L Most reasonable functions are continuous, as seen in the following theorem:
Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More informationConservation Law. Chapter Goal. 5.2 Theory
Chpter 5 Conservtion Lw 5.1 Gol Our long term gol is to understnd how mny mthemticl models re derived. We study how certin quntity chnges with time in given region (sptil domin). We first derive the very
More informationSummary Information and Formulae MTH109 College Algebra
Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)
More information31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes
Numericl Integrtion 3. Introduction In this Section we will present some methods tht cn be used to pproximte integrls. Attention will be pid to how we ensure tht such pproximtions cn be gurnteed to be
More informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
More informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More information