dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.


 Clarence Blair
 1 years ago
 Views:
Transcription
1 Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd x,y denote dependent vriles. Let I R e n intervl, nd Ω R 2 e domin. Let us consider the system dx = F (t, x, y), dt dy = G(t, x, y), dt where the functions re defined on I Ω, nd re loclly Lipschitz w.r.t. vrile (x, y) Ω. Definition 8.1 (Autonomous system) A system of ODE hving the form (8.1) is clled n utonomous system if the functions F (t, x, y) nd G(t, x, y) re constnt w.r.t. vrile t. Tht is, dx = F (x, y), dt dy = G(x, y), dt Definition 8.2 A point (x 0, ) Ω is sid to e criticl point of the utonomous system (8.2) if F (x 0, ) = G(x 0, ) = 0. (8.3) A criticl point is lso clled n equilirium point, rest point. Definition 8.3 Let (, ) e solution of twodimensionl (plnr) utonomous system (8.2). The trce of (, ) s t vries is curve in the plne. This curve is clled trjectory. Remrk 8.4 (On solutions of utonomous systems) (i) Two different solutions my represent the sme trjectory. For, (8.1) (8.2) (1) If (x 1 (t), y 1 (t)) defined on n intervl J is solution of the utonomous system (8.2), then the pir of functions (x 2 (t), y 2 (t)) defined y (x 2 (t), y 2 (t)) := (x 1 (t s), y 1 (t s)), for t s + J (8.4) is solution on intervl s + J, for every ritrry ut fixed s R. (2) However, trces of oth the solutions on the respective intervls is the sme. If the independent vrile t is interpreted s time, then we note tht the two different solutions visit every point on the trjectory with time lg s. Also see Exmple
2 (ii) The trjectories do not cross. For, (1) Suppose two trjectories γ 1 nd γ 2 cross t point (x 0, ) Ω. Let (x 1 (t), y 1 (t)) defined on intervl I 1 e solution whose trce is γ 1, nd (x 2 (t), y 2 (t)) defined on intervl I 2 e solution whose trce is γ 2. By the ssumption of crossing, there exist t 1 I 1, nd t 2 I 2 such tht (x 1 (t 1 ), y 1 (t 1 )) = (x 0, ) = (x 2 (t 2 ), y 2 (t 2 )). (8.5) Let us define pir of functions (x 3 (t), y 3 (t)) on intervl t 1 t 2 + I 2 y (x 3 (t), y 3 (t)) := (x 2 (t t 1 + t 2 ), y 2 (t t 1 + t 2 )) (8.6) It cn e esily checked, vi Chin rule, tht (x 3 (t), y 3 (t)) is solution of (8.2) on the intervl t 1 t 2 + I 2. Note tht (x 3 (t 1 ), y 3 (t 1 )) = (x 2 (t 1 t 1 + t 2 ), y 2 (t 1 t 1 + t 2 )) = (x 2 (t 2 ), y 2 (t 2 )) = (x 0, ) (8.7) Thus, (x 3 (t), y 3 (t)) nd (x 1 (t), y 1 (t)) re solutions of the sme initil vlue prolem, contrdicting the uniqueness of solutions to IVP. Therefore two trjectories do not cross ech other. (iii) The trjectories fill the domin Ω, since through every point trjectory psses. This is consequence of existence theorem. (iv) Through every point in the phse spce Ω, exctly one trjecory psses. This is consequence of uniqueness of solutions to IVPs. (v) From the lst two remrks, it follows tht the trjectories prtition the phse spce Ω. In fct, defining reltion on Ω y sying tht two points (x 1, y 1 ), (x 2, y 2 ) Ω re relted if (x 1, y 1 ) nd (x 2, y 2 ) lie on the sme trjectory, it is esy to verify tht this reltion is n equivlence reltion nd therey giving rise to prtition of Ω in terms of equivlnece clsses. Ech equivlence clss is trjectory. (vi) Note tht trjectories consisting of single point correspond to criticl points. (vii) types of trjectories: For utonomous systems with two dimensionl phse spce, three types of trjectories re possile. A trjectory consisting of single point (corresponding to equilirium solutions), nd if trjectory hs more thn one point then it could e closed curve (corresponding to periodic solutions), or curve without selfintersection. (viii) For liner utonomous systems, specil clss of systems (8.2) for which F nd G re liner in x, y, note tht sturted solutions re glol, i.e., sturted solutions re defined on the entire rel line R. Hence for liner utonomous systems, we do not mention the intervl on which given solution is defined. The ove remrk is illustrted y the following exmple. Exmple 8.5 dx dt = y, dy dt = 4x. (8.8) Note tht (x 1 (t), y 1 (t)) := (cos 2t, 2 sin 2t) is solution of (8.8). The trjectory pssing through the point (1, 0) R 2 is the ellipse x 2 + y2 4 = 1 trvelled counterclockwise. Consider the solution (x 2 (t), y 2 (t)) := (cos(t π/2), 2 sin 2(t π/2)). This solution stisfies (x 2 ( π 2 ), y 2( π 2 )) = (1, 0), nd hence hs the sme trjectory s (x 1 (t), y 1 (t)). Drw figure
3 Chpter 8 : Stility theory Solving liner plnr systems with constnt coefficients Consider the system of ODE ( ( x x x y = =: A. (8.9) c d) y y) Fundmentl mtrix Definition 8.6 (Fundmentl mtrix) A mtrix vlued function Φ whose columns re solutions of the system of ODE (8.9) is clled solution mtrix. A solution mtrix Φ is clled fundmentl mtrix if the columns of Φ form fundmentl pir of solutions for the system (8.9). A fundmentl mtrix Φ is clled the stndrd fundmentl mtrix if Φ(0) is the identity mtrix. Remrk 8.7 Since the columns of solution mtrixφ re solutions of (8.9), the mtrix vlued function Φ stisfies the system of ODE Φ = AΦ. (8.10) Exercise 8.8 A solution mtrix is fundmentl mtrix if nd only if its determinnt is not zero. Exercise 8.9 Prove tht if Ψ is fundmentl mtrix then ΨC is lso fundmentl mtrix for every constnt invertile mtrix C. Prove tht ll fundmentl mtrices occur this wy. Computtion of fundmentl mtrix By definition of fundmentl mtrix, finding fundmentl pir of solutions to system of ODE (8.9) is equivlent to finding fundmentl mtrix. In view of Exercise 8.9, fundmentl mtrix is not unique ut the stndrd fundmentl mtrix is unique. (1) Oserve tht e is nontrivil solution of (8.9) if nd only if ( ) Tht is, λ is n eigenvlue nd ( ( ( 0, A = λ. (8.11) 0) ) ) is n eigenvector corresponding to λ. (2) Question Is it possile to find fundmentl pir, oth of which re of the form e? Answer Supposing tht φ 1 (t) = e λ 1t nd φ 2 (t) = e λ 2t c re two solutions of d (8.9), φ 1, φ 2 form fundmentl pir if nd only if c d 0, (8.12) since the ove determinnt is the Wronskin of φ 1, φ 2 t t = 0. Tht is, the mtrix A should hve two linerly independent eigenvectors. Note tht this is equivlent to sying tht A is digonlisle.
4 Solving liner plnr systems with constnt coefficients (3) Question Wht if the mtrix A does not hve two linerly independent eigenvectors? This cn hppen when A hs only one eigenvlue of multiplicity two. Inspired y similr sitution in the context ( of ) constnt coefficient( second ) order liner ODE, we re tempted to try φ 1 (t) = e λ1t nd φ 2 (t) = te λ1t s fundmentl pir. But note tht φ 1, φ 2 does not form fundmentl pir since Wronskin t t = 0 will e zero, lso note tht φ 2 is not even solution of the liner system (8.9). Nevertheless, we cn find solution hving the form of φ 1. Therefore, we try vrint of ove suggestion to find nother solution tht together φ 1 constitutes fundmentl pir. Let φ 2 (t) = te λ1t + e λ1t c (8.13) d Then φ 2 (t) solves the system (8.9) if nd only if ( ( c (A λ 1 I) =. (8.14) d) ) One cn esily verify tht y φ 1 (t) = e λ 1t, φ 2 (t) = te λ 1t is fundmentl pir, where, ( ( c, re linerly independent. Thus, φ ) d) 1, φ 2 defined + e λ 1t c d c re relted y the eqution (8.14). d (8.15) (4) In cse the mtrix A does not hve rel eigenvlues, then eigenvlues re complex conjugtes of ech other. In this cse, (λ, v) is n eigen pir if nd only if (λ, v) is lso n eigen pir for A. (8.16) α + iβ Denoting λ = r + iq (note q 0), v =, define γ + iδ φ 1 (t) = e rt α cos qt β sin qt, φ γ cos qt δ sin qt 2 (t) = e rt α sin qt + β cos qt. (8.17) γ sin qt + δ cos qt Verifying tht the pir of functions defined ove constitute relvlued fundmentl pir of solutions is left s n exercise Mtrix exponentils Inspired y the formul for solutions to liner sclr eqution y = y, given y = ce t, where y(0) = c, we would like to sy tht the system (8.9) hs solution given y ( ( x(0) = exp(ta), where =. (8.18) ) ) y(0) First of ll, we must give mening for the symol exp(a) s mtrix, which is clled the exponentl of mtrix B. Then we must verify tht the proposed formul for solution, in (8.18), is indeed solution of (8.9). We do this next. Definition 8.10 (Exponentil of mtrix) If A is 2 2 mtrix, then exponentil of A (denoted y e A, or exp(a), is defined y e A = I + k=1 A k k! (8.19)
5 Chpter 8 : Stility theory 71 We record elow, without proof, some properties of mtrix exponentils. Lemm 8.11 Let A e 2 2 mtrix. (1) The series in (8.19) converges. (2) e 0 = I. (3) (e A ) 1 = e A (4) exp(a + B) = exp(a) exp(b) if the mtrices A nd B stisfy AB = BA. (5) If J = P 1 AP, then exp(j) = P 1 exp(a)p. (6) d dt eta = Ae ta. Theorem 8.12 If A is 2 2 mtrix, then Φ(t) = e ta is fundmentl mtrix of the system (8.9). If (, ) is solution of the system (8.9) with (x(t 0 ), y(t 0 )) = (x 0, ), then (, ) is given y = e (t t0)a x0. (8.20) Remrk 8.13 Thnks to the ove theorem, we do not need to struggle to find fundmentl mtrix s we did erlier. We need to to tke just the exponentil of the mtrix ta nd tht would give very esily fundmentl mtrix. But it is not s simple s it seems to e. In fct, summing up the series for exponentil of mtrix is not esy t ll, even for reltively simpler mtrices. To relise this, solve the exercise following this remrk. Exercise 8.14 Find exponentil mtrix for 2 0 A 1 =, A = 1 1, A = 1 2. (8.21) 2 3 Remrk 8.15 After solving the ove exercise, one would relise tht it is esier to find fundmentl mtrices y finding fundmentl pirs of solutions insted of summing up series! There is n lternte method to clculte exponentil mtrix for ta, vi fundmentl pirs, y oserving tht exponentil mtrix is nothing ut the stndrd fundmentl mtrix. So, find fundmentl mtrix Φ, then exponentil mtrix e ta is given y e ta = [Φ(0)] 1 Φ(t). (8.22) 8.2 Stility of n equilirium point Definition 8.16 (1) A criticl point (x 0, ) of the system (8.2) is sid to e stle if for ech ɛ > 0 there exists δ > 0 such tht every solution (, ) for which there is n s such tht (x(s), y(s)) (x 0, ) < δ, exists for ll t s nd stisfies (, ) (x 0, ) < ɛ, t s. (8.23) (2) A criticl point (x 0, ) of the system (8.2) is sid to e unstle if it is not stle. (3) A criticl point (x 0, ) of the system (8.2) is sid to e symptoticlly stle if (i) the criticl point (x 0, ) is stle, nd, (ii) if there exists δ 0 > 0 (0 < δ 0 < δ) such tht (x(s), y(s)) (x 0, ) < δ 0 for some s = (, ) (x 0, ) s t. (8.24)
6 Phse spce picture Convention We re interested in the ehviour of solutions s t. Therefore, for us lwys t 0, nd hence we indicte directions on trjectories only for incresing t. Remrk 8.17 In other words, criticl point is stle if every trjectory tht comes within δ distnce from the criticl point t some time, stys within n ɛ distnce t ll lter times. Further if the trjectory pproches the criticl point, then the criticl point is symptoticlly stle. 8.3 Phse spce picture Before we strt discussing the nture of trjectories for liner systems, we need to understnd the following questions. (1) Wht re ll 2 2 rel mtrices? (2) Under liner trnsformtion, wht will e the imges of point, closed curve, curve? Let us nlyse some specil cses of liner utonomous system (8.9): Cse 1: Invertile digonl mtrix A 1 = λ 0, with λ 0 µ. (8.25) 0 µ (1) If (, ) is solution of the system (8.9) corresponding to A 1, with (x(0), y(0)) = (x 0, ), then (, ) is given y = e ta 1 x0 e 0 x0. = 0 e µt (8.26) (2) To drw trjectories in the plne, we hve to drw the unique trjectory pssing through ech point. (i) The criticl point (0, 0) is clled nondegenerte since it is isolted (wellseprted from other criticl points, if ny; in fct none!) (ii) Let us fix (x 0, ) = (0, 0) The trjectory through origin consists of only one point, which is origin. (iii) Let us consider the cse of x 0 = 0, ut 0. Then solution is given y (, ) = (0, e µt ). Similrly, in the cse where = 0, ut x 0 0, the solution is given y (, ) = (e x 0, 0). (iv) Let us fix (x 0, ) with x 0 0. In this cse, we cn write µ λ =, (8.27) x 0 hence trjectory through (x 0, ) is given y µ λ x y =, with x 0 x x 0 > 0, y > 0. (8.28) (3) Now drw the trjectories nd indicte the progress of the curve s t increses. The nture of the trjectories vry nd depend on the nture of λ, µ. There re five distinct scenrios: λ = µ < 0, λ < 0 < µ, λ < µ < 0, 0 < µ < λ, 0 < λ = µ. (8.29)
7 Chpter 8 : Stility theory 73 Cse 1B: Noninvertile digonl mtrix λ 0 A 2 =, 0 µ with λµ = 0. (8.30) (1) The criticl point (0, 0) is clled degenerte since there re infinitely mny criticl points of the system in ny neighourhood of the origin. (criticl points re not wellseprted from ech other) In fct, ll points on x xis (respectively y xis) re criticl points if λ = 0 (respectively, µ = 0). (2) If (, ) is solution of the system (8.9) corresponding to A 2, with (x(0), y(0)) = (x 0, ), then (, ) is given y = e ta 2 x0 e 0 x0 = 0 e µt (8.31) (3) To drw trjectories in the plne, we hve to drw the unique trjectory pssing through ech point. (i) Let us fix (x 0, ) = (0, 0) The trjectory through origin consists of only one point, which is origin. (ii) Let us fix (x 0, ) to e ny other criticl point. The trjectory through ny criticl point consists of only one point, which is the criticl point itself. (iii) Let us fix (x 0, ) different from ny criticl point. Then solution is given y (, ) = (e x 0, e µt ). (4) Now drw the trjectories nd indicte the progress of the curve s t increses. The nture of the trjectories vry nd depend on the nture of λ, µ. There re three distinct scenrios: λ < µ = 0, 0 = µ < λ, 0 = µ = λ. (8.32) Cse 2: Mtrix hving the form λi + A 3 = λ 1, λ R (8.33) 0 λ (1) The exponentil mtrix is given y e ta 3 ( ) 0 1 = exp(i) exp t 0 0 e 0 1 t e te = 0 e = e (8.34) (2) If (, ) is solution of the system (8.9) corresponding to A 3, with (x(0), y(0)) = (x 0, ), then (, ) is given y = e ta 3 x0 = e te x0 0 e (8.35) (3) To drw trjectories in the plne, we hve to drw the unique trjectory pssing through ech point.
8 Phse spce picture (i) If λ 0, the criticl point (0, 0) is clled nondegenerte since it is isolted (wellseprted from other criticl points, if ny; in fct none!). If λ = 0, then criticl point (0, 0) is clled degenerte since there re infinitely mny criticl points of the system in ny neighourhood of the origin. (criticl points re not wellseprted from ech other); in this cse, ll points on x xis re criticl points. (ii) The trjectory through ny criticl point consists of only one point, which is the criticl point itself. (iii) Let us fix (x 0, ) different from criticl point. Let us ssume tht x 0 = 0 nd 0. Then = t, nd = log. (8.36) Thus trjectories re given y ( y λx = y log ), with y > 0. (8.37) (iv) In the generl cse of oth x 0 0 nd 0, solution (, ) stisfies = [x 0 + t ], (8.38) nd thus psses through y xis t time t = x 0. nd thus trjectory is lredy descried in the previous point. Recll tht trjectory does not chnge y introducing dely in the time t which solution visits point, nd even hve the sme ehviour s t. (v) The other cse is = 0, ut x 0 0. Then solution is given y (, ) = (e x 0, 0). (4) Now drw the trjectories nd indicte the progress of the curve s t. The nture of the trjectories vry nd depend on the nture of λ. There re three distinct scenrios: 0 < λ, λ < 0, λ = 0. (8.39) Cse 3: Mtrix hving the form A 4 =, λ R (8.40) (1) The exponentil mtrix is given y ( ) e ta 4 0 = exp(ti) exp t (8.41) Denoting J =, we must evlute exp(tj). By evluting vrious powers of 1 0 tj nd sustituting them in the definition of exponentil, it cn e seen tht exp(tj) = (1 (t)2 2! + (t)4 4! (t)6 6! + ) I+ ( t 1! (t)3 3! + (t)5 5! (t)7 7! + ) J (8.42)
9 Chpter 8 : Stility theory 75 On noting tht the series expnsions in the ove eqution correspond to wellknown functions, the ove eqution reduces to cos t sin t exp(tj) = cos ti + sin tj = (8.43) sin t cos t Thus exponentil mtrix ecomes ( ) 0 e ta4 = exp(ti) exp t 0 = e t cos t sin t sin t cos t (8.44) (2) If (, ) is solution of the system (8.9) corresponding to A 3, with (x(0), y(0)) = (x 0, ), then (, ) is given y = e ta 4 x0 = e t cos t sin t x0 (8.45) sin t cos t Tht is, = We cn write the ove eqution in the form = where α = tn 1 ( y0 x 0 ). Also, note e t (x 0 cos t sin t) e t. (8.46) (x 0 sin t + cos t) ( (x0, ) e t cos(t α) (x 0, ) e t sin(t α) ), (8.47) = tn(t α) (8.48) (3) To drw trjectories in the plne, we hve to drw the unique trjectory pssing through ech point. (i) The criticl point (0, 0) is clled nondegenerte since it is isolted (wellseprted from other criticl points, if ny; in fct none!). (ii) The trjectory through origin, eing criticl point, consists of only one point, which is the criticl point itself. (iii) Oserve tht () 2 + () 2 = e 2t (x y 2 0) (8.49) There re two points to e noted from here: (i). If = 0, then solution lies on circle of rdius (x 0, ). (ii). If 0, depending on the sign of, we hve (, ) goes to zero or infinity. In fct y switching to polr coordintes, x = r cos θ, y = r sin θ, r = x 2 + y 2, Trjectories re given y θ = t α. (8.50) r = () 2 + () 2 = e t (x θ+α y2 0 ) = e( (x ) y2 0 ), (8.51) which is nothing ut where c = e ( α ) (x y2 0 ). These curves re spirls. r = ce ( )θ (8.52)
10 Phse spce picture (4) Now drw the trjectories nd indicte the progress of the curve s t. The nture of the trjectories vry nd depend on the nture of,. There re four distinct scenrios: < 0 <, < 0 <, = 0& > 0, = 0& < 0. (8.53) Exercise 8.18 Discuss the phse spce picture of liner utonomous systems corresponding to ech of the mtrices A 1 =, A =, A =, A =, A 5 =, A =, A =, A =, A 9 =, A =, A =, A =, A 13 =, A =. 1 0 Exercise 8.19 Comment on the stility of origin in ech of the ove systems. Phse spce picture for liner utonomous systems corresponding to generl mtrix So fr, we discussed phse spce portrit for systems descried y mtrix of vrious specil forms. The discussion for generl mtrix cse cn e done once we hve nswers to the two questions posed t the eginning of this Section 8.3. The nswers re (1) The specil mtrices we considered re ll the 2 2 rel mtrices. It mens tht ny 2 2 rel mtrix is similr to one of the three types of specil mtrices we considered. (2) Under liner trnsformtion, imges of point, closed curve, nd curve re point, closed curve, nd curve respectively! Liner trnsformtion stnds for line goes to line. Note tht circle will ecome n ellipse under liner trnsformtion, in generl. Let P 1 AP = J. Introducing new set of dependent vriles (v, w) defined y ( v = P w) 1 x. (8.54) y It is esy to verify tht x x x solves y y = A y if nd only if ( v w) solves v = J w v w (8.55) We discussed the phse spce portrit of (v, w) since J is one of the specil mtrices for which we nlysed the phse spce picture. Now returning to the originl dependent vriles (x, y), strting from solution (v(t), w(t)) corresponding to mtrix J, solution (, ) corresponding to mtrix A is given y = P v(t). (8.56) w(t) The trjectories in xy plne re nothing ut imges of trjectories in vw plne corresponding to specil mtrix J, under liner trnsformtion defined y the mtrix P. Note tht the qulittive ehviour of criticl point remins the sme for ll systems tht correspond to mtrices which re similr. This finishes the discussion of trjectories for liner plnr utonomous system.
Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.
Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one
More informationLinear Systems with Constant Coefficients
Liner Systems with Constnt Coefficients 4305 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
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 information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More informationLecture 3: Curves in Calculus. Table of contents
Mth 348 Fll 7 Lecture 3: Curves in Clculus Disclimer. As we hve textook, this lecture note is for guidnce nd supplement only. It should not e relied on when prepring for exms. In this lecture we set up
More informationVectors , (0,0). 5. A vector is commonly denoted by putting an arrow above its symbol, as in the picture above. Here are some 3dimensional vectors:
Vectors 1232018 I ll look t vectors from n lgeric point of view nd geometric point of view. Algericlly, vector is n ordered list of (usully) rel numers. Here re some 2dimensionl vectors: (2, 3), ( )
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
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 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 out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationLecture Solution of a System of Linear Equation
ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville  D. Keffer, 5/9/98 (updted /) Lecture 8  Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationComplex, distinct eigenvalues (Sect. 7.6)
Comple, distinct eigenvlues (Sect 76) Review: Clssifiction of 2 2 digonlizle systems Review: The cse of digonlizle mtrices Rel mtri with pir of comple eigenvlues Phse portrits for 2 2 systems Review: Clssifiction
More informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationMath Lecture 23
Mth 8  Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
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 informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationSpace Curves. Recall the parametric equations of a curve in xyplane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xyplne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationThe Dirichlet Problem in a Two Dimensional Rectangle. Section 13.5
The Dirichlet Prolem in Two Dimensionl Rectngle Section 13.5 1 Dirichlet Prolem in Rectngle In these notes we will pply the method of seprtion of vriles to otin solutions to elliptic prolems in rectngle
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More information378 Relations Solutions for Chapter 16. Section 16.1 Exercises. 3. Let A = {0,1,2,3,4,5}. Write out the relation R that expresses on A.
378 Reltions 16.7 Solutions for Chpter 16 Section 16.1 Exercises 1. Let A = {0,1,2,3,4,5}. Write out the reltion R tht expresses > on A. Then illustrte it with digrm. 2 1 R = { (5,4),(5,3),(5,2),(5,1),(5,0),(4,3),(4,2),(4,1),
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 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 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 informationLinear Inequalities. Work Sheet 1
Work Sheet 1 Liner Inequlities RentHep, cr rentl compny,chrges $ 15 per week plus $ 0.0 per mile to rent one of their crs. Suppose you re limited y how much money you cn spend for the week : You cn spend
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationCoalgebra, Lecture 15: Equations for Deterministic Automata
Colger, Lecture 15: Equtions for Deterministic Automt Julin Slmnc (nd Jurrin Rot) Decemer 19, 2016 In this lecture, we will study the concept of equtions for deterministic utomt. The notes re self contined
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 informationSeptember 13 Homework Solutions
College of Engineering nd Computer Science Mechnicl Engineering Deprtment Mechnicl Engineering 5A Seminr in Engineering Anlysis Fll Ticket: 5966 Instructor: Lrry Cretto Septemer Homework Solutions. Are
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 informationDEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.
398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
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 informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN MARICES FOR ENGINEERING Dr Clum Mcdonld Contents Introduction Definitions Wht is mtri? Rows nd columns of mtri Order of mtri Element of mtri Equlity of mtrices Opertions
More informationSection 4: Integration ECO4112F 2011
Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic
More informationExploring parametric representation with the TI84 Plus CE graphing calculator
Exploring prmetric representtion with the TI84 Plus CE grphing clcultor Richrd Prr Executive Director Rice University School Mthemtics Project rprr@rice.edu Alice Fisher Director of Director of Technology
More informationMATH 260 Final Exam April 30, 2013
MATH 60 Finl Exm April 30, 03 Let Mpn,Rq e the spce of nyn mtrices with rel entries () We know tht (with the opertions of mtrix ddition nd sclr multipliction), M pn, Rq is vector spce Wht is the dimension
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = 2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationset is not closed under matrix [ multiplication, ] and does not form a group.
Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed
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 informationMath 211A Homework. Edward Burkard. = tan (2x + z)
Mth A Homework Ewr Burkr Eercises 5C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z =
More informationP 1 (x 1, y 1 ) is given by,.
MA00 Clculus nd Bsic Liner Alger I Chpter Coordinte Geometr nd Conic Sections Review In the rectngulr/crtesin coordintes sstem, we descrie the loction of points using coordintes. P (, ) P(, ) O The distnce
More informationu(t)dt + i a f(t)dt f(t) dt b f(t) dt (2) With this preliminary step in place, we are ready to define integration on a general curve in C.
Lecture 4 Complex Integrtion MATHGA 2451.001 Complex Vriles 1 Construction 1.1 Integrting complex function over curve in C A nturl wy to construct the integrl of complex function over curve in the complex
More informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
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 informationConsequently, the temperature must be the same at each point in the cross section at x. Let:
HW 2 Comments: L13. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the
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 informationIn Section 5.3 we considered initial value problems for the linear second order equation. y.a/ C ˇy 0.a/ D k 1 (13.1.4)
678 Chpter 13 Boundry Vlue Problems for Second Order Ordinry Differentil Equtions 13.1 TWOPOINT BOUNDARY VALUE PROBLEMS In Section 5.3 we considered initil vlue problems for the liner second order eqution
More informationMTH 505: Number Theory Spring 2017
MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of $ nd $ s two denomintions of coins nd $c
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 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 informationBIFURCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS
BIFRCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible
More information7. Indefinite Integrals
7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationState space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies
Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response
More informationCONIC SECTIONS. Chapter 11
CONIC SECTIONS Chpter. Overview.. Sections of cone Let l e fied verticl line nd m e nother line intersecting it t fied point V nd inclined to it t n ngle α (Fig..). Fig.. Suppose we rotte the line m round
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationChapter 3 Polynomials
Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling
More information308K. 1 Section 3.2. Zelaya Eufemia. 1. Example 1: Multiplication of Matrices: X Y Z R S R S X Y Z. By associativity we have to choices:
8K Zely Eufemi Section 2 Exmple : Multipliction of Mtrices: X Y Z T c e d f 2 R S X Y Z 2 c e d f 2 R S 2 By ssocitivity we hve to choices: OR: X Y Z R S cr ds er fs X cy ez X dy fz 2 R S 2 Suggestion
More informationLecture 2: January 27
CS 684: Algorithmic Gme Theory Spring 217 Lecturer: Év Trdos Lecture 2: Jnury 27 Scrie: Alert Julius Liu 2.1 Logistics Scrie notes must e sumitted within 24 hours of the corresponding lecture for full
More informationpadic Egyptian Fractions
padic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationHW3, Math 307. CSUF. Spring 2007.
HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem
More informationREPRESENTATION THEORY OF PSL 2 (q)
REPRESENTATION THEORY OF PSL (q) YAQIAO LI Following re notes from book [1]. The im is to show the qusirndomness of PSL (q), i.e., the group hs no low dimensionl representtion. 1. Representtion Theory
More information8.6 The Hyperbola. and F 2. is a constant. P F 2. P =k The two fixed points, F 1. , are called the foci of the hyperbola. The line segments F 1
8. The Hperol Some ships nvigte using rdio nvigtion sstem clled LORAN, which is n cronm for LOng RAnge Nvigtion. A ship receives rdio signls from pirs of trnsmitting sttions tht send signls t the sme time.
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 information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationLinearity, linear operators, and self adjoint eigenvalue problems
Linerity, liner opertors, nd self djoint eigenvlue problems 1 Elements of liner lgebr The study of liner prtil differentil equtions utilizes, unsurprisingly, mny concepts from liner lgebr nd liner ordinry
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
More informationModule 6: LINEAR TRANSFORMATIONS
Module 6: LINEAR TRANSFORMATIONS. Trnsformtions nd mtrices Trnsformtions re generliztions of functions. A vector x in some set S n is mpped into m nother vector y T( x). A trnsformtion is liner if, for
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly welldefined, is too restrictive for mny purposes; there re functions which
More information