INTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović


 Madison Burns
 3 years ago
 Views:
Transcription
1 THE TEACHING OF MATHEMATICS 2005, Vol. VIII, 1, pp INTERCHANGING TWO LIMITS Zorn Kdelburg nd Milosv M. Mrjnović This pper is dedicted to the memory of our illustrious professor of nlysis Slobodn Aljnčić ( ). Abstrct. By the use of convenient metrics, the ordered set of nturl numbers plus n idel element nd the prtilly ordered set of ll prtitions of n intervl plus n idel element re converted into metric spces. Thus, the three different types of it, rising in clssicl nlysis, re reduced to the sme model of the it of function t point. Then, the theorem on interchnge of iterted its, vlid under the condition tht one of the iterted its exists nd the other one exists uniformly, is used to derive long sequence of sttements of tht type tht re commonly present in the courses of clssicl nlysis. All pprently vried conditions ccompnying such sttements re, then, unmsked nd reduced to one nd the sme: one iterted it exists nd the other one exists uniformly. ZDM Subject Clssifiction: I35; AMS Subject Clssifiction: 00A35 Key words nd phrses: Interchnge of two its, uniform convergence, definite integrl s it. 1. Introduction The simple concept of metric spce proves s convenient setting for unifiction of different types of convergence nd continuity tht rise in clssicl nlysis. For this reson, this concept finds its plce in the contents of the first contemporry courses of mthemticl nlysis. Prticulrly, by replcing the bsolute vlue with distnce function, immedite generliztions of conditions determining clssicl concepts of convergence nd continuity re obtined. Due to different terms, the following three types of it re present in the clssicl nlysis n, f(x), σ(f,p), being respectively: the it of sequence, the it of function t point nd the it of integrl sums. To hve them s prticulr cses of more generl concept of it, generlized sequences (nets) or filters re used (nd they re not common topics on the list of themes for the first courses in nlysis). Our ide of integrting these different types of it is bsed on the procedure of introducing metric on the set N of nturl numbers together with n idel point nd on the set of ll prtitions of n intervl together with n idel point. Then, the model of the it t point of function mpping metric spce into nother one embrces ll three types of it present in the clssicl nlysis. Then, our min
2 16 Z. Kdelburg, M. M. Mrjnović objective in this pper is to use (nd prove) theorem on the interchnge of two its under the condition tht one of them exists nd the other one exists uniformly. Relying on this theorem, we derive proofs of (long) sequence of sttements on interchnge of two its tht re commonly present in the nlysis courses. The effects of such n pproch re lso seen in the fct tht n pprently vried set of conditions ssocited with these prticulr sttements trnsltes uniquely into the requirement: one of the two its exists nd the other one exists uniformly. Now we turn our ttention to the history of the 19th century mthemtics to recll how the concept of uniform convergence, though used explicitly, did not become common property for longer time. In his fmous Course d nlyse, (l821), Cuchy (AL. Cuchy, ) mde misstep with respect to rigor, stting tht the sum of convergent series of continuous functions is continuous function. In 1826, Abel (N. H. Abel, ) restricted this sttement to the cse of uniform convergence, giving correct proof of it (Jour. für Mth. 1, ), but not isolting the property of uniform convergence. Both men, Stokes (G. G. Stokes, ) (Trns. Cmb. Phil. Soc., 85, 1848) nd Seidel (Ph. L. von Seidel, ) (Abh. der Byer. Akd. der Wiss., 1847/49) recognized the distinction between uniform nd pointwise convergence nd the significnce of the former in the theory of infinite series. Further on, from letter of Weierstrss (K. Weierstrss, ) (unpublished till 1894 (Werke)) it is seen tht he must hve drwn this distinction s erly s Nevertheless, Cuchy ws the first to recognize ultimtely the uniform convergence s property ssuring the continuity of the sum of series of continuous functions (Comp. Rend., 36, 1853). It ws Weierstrss who used first the concept of uniform convergence s condition for term by term integrtion of series s well s for differentition under the integrl sign. Through the circle of his students the importnce of this concept ws mde widely known. At the end, let us note tht, when prepring this pper, we used n unpublished mnuscript of the second uthor. 2. Iterted its Let A be nonempty subset of metric spce (M 1,d 1 )ndb nonempty subset of metric spce (M 2,d 2 ). Let f : A B M be mpping into complete metric spce (M,d). Denote by X the set of ccumultion points of subset X of metric spce. Note tht A B (A B), which follows from (x 0,y 0 ) A B ( >0) {[K(x 0,) \{x 0 }] A nd [K(y 0,) \{y 0 }] B } = ( >0) [K(x 0,) K(y 0,) \{(x 0,y 0 )}] A B (x 0,y 0 ) (A B).
3 Interchnging two its 17 Let x 0 A nd y 0 B. A it f(x, y) =ϕ(x) existsforechx A if the following condition holds: ( x A)( >0)( δ >0)( y B)(0 <d 2 (y, y 0 ) <δ = d(f(x, y),ϕ(x)) <); nd one cn formulte similrly the condition for the existence of f(x, y) = ψ(y). Let us strengthen this condition demnding tht the number δ does not depend on x. Prcticlly, this mens tht the quntifier ( x A) hs to be plced fter the quntifier ( δ > 0). Thus we obtin tht f(x, y) =ϕ(x) y y exists uniformly in x A if 0 ( >0)( δ >0)( x A)( y B)(0 <d 2 (y, y 0 ) <δ = d(f(x, y),ϕ(x)) <); nd the condition for the existence of f(x, y) =ψ(y), uniformly in y B is x x formulted similrly. 0 Cuchy condition for the existence of the f(x, y) =ϕ(x), uniformly in y y x A is 0 ( >0)( δ >0)( x A)( y B)( y B) 0 <d 2 (y,y 0 ) <δnd 0 <δ 2 (y,y 0 ) <δ = d(f(x, y ),f(x, y )) <. Since the spce (M,d) is ssumed to be complete, it is cler tht this condition is necessry nd sufficient for the existence of f(x, y) =ϕ(x), uniformly in y y x A. 0 Writing formlly, ϕ(x) = ( f(x, y)), ψ(x) = ( f(x, y)), we cll these expressions iterted its (which, of course need not exist). In the sequel, we shll omit dditionl prentheses. In this context, the expression f(x, y) (x 0,y 0) is clled double it (nd it my eqully be nonexistent). A generl connection between the mentioned its is given by the following Proposition 1. Let f : A B M be function from subset A B M 1 M 2 into M nd (x 0,y 0 ) A B,whereM 1, M 2 nd M re metric spces. If (i) f(x, y) =α exists, (x 0,y 0) (ii) for ech y B, f(x, y) =ψ(y) exists, then ψ(y) exists nd is equl to α.
4 18 Z. Kdelburg, M. M. Mrjnović Proof. Condition (i) implies tht ( >0)( δ >0)( (x, y) A B) (x, y) K(x 0,δ) K(y 0,δ) \{(x 0,y 0 )} = d(f(x, y),α) < 2. According to (ii), using the continuity of function d, there exists Thus, d(f(x, y),α)=d(ψ(y),α) 2. ( >0)( δ >0)( y B) y K(y 0,δ) \{y 0 } = d(ψ(y),α) <, which mens tht ψ(y) =α. Note tht the conditions of the previous proposition re pretty strong, especilly this is the cse with condition (i). Therefore, this proposition is of smll prcticl vlue. However, it cn be used to prove the nonexistence of some double its. Exmple 1. Consider the following three functions: f(x, y) = x y + x2 + y 2 x + y, g(x, y) = x sin 1 x + y, h(x, y) =x sin 1 x + y y, ll three defined on the set A B =(0, +) (0, +). For the function f, f(x, y) =1+x, forechx A nd f(x, y) = 1+y, y 0 x 0 for ech y B, ndso f(x, y) =1 1 = f(x, y). x 0 y 0 y 0 x 0 The double it does not exist. For the function g, g(x, y) =sin 1,forechx A nd g(x, y) =1,for y 0 x x 0 ech y B. Hence, g(x, y) = 1, nd both g(x, y) nd the double y 0 x 0 x 0 y 0 it do not exist. For the function h, using tht 0 x sin 1 y x, we obtin tht h(x, y) = (0,0) 0. On the other hnd, h(x, y) does not exist for ny x A, nd so the respective iterted it does not exist either. The other iterted it exists nd y 0 h(x, y) =0. y 0 x 0 Now we shll modify the conditions of Proposition 1, demnding tht one of the its f(x, y) nd f(x, y) exists, nd tht the other one exists uniformly. We shll conclude tht the three its, both iterted nd the double one, ll exist nd re equl. Such sttement synthesizes then series of clssicl
5 Interchnging two its 19 results on interchnge of it opertions. In terms of generlized sequences it is clled Moore Theorem (E. H. Moore, ) 1. Theorem 1. (The theorem on interchnge of two its) Let f : A B M be mpping into complete metric spce, where A nd B re subsets of metric spces M 1 nd M 2, respectively, nd let x 0 A \ A, y 0 B \ B. If (i) f(x, y) =ψ(y) exists for ech y B; (ii) f(x, y) =ϕ(x) exists uniformly in x A, then the three its f(x, y), f(x, y), nd (x 0,y 0) f(x, y) ll exist nd re equl. Proof. Let >0 be rbitrry. According to condition (ii), we hve (1) ( δ >0)( y B) 0<d 2 (y, y 0 ) <δ = ( x A) d(f(x, y),ϕ(x)) < 6. Let y K(y 0,δ) \{y 0 }. Using condition (i) we obtin (2) ( δ >0)( x A) 0<d 1 (x, x 0 ) < δ = d(f(x, y),ψ(y)) < 6. Let us tke neighborhood of the point (x 0,y 0 )oftheform U = K(x 0, δ) K(y 0,δ) nd let points (x,y ), (x,y )belongtothesetu \{(x 0,y 0 )}. Using the tringle inequlity, we get d(f(x,y ),f(x,y )) d(f(x,y ),ϕ(x )) + d(ϕ(x ),f(x, y)) + d(f(x, y),ψ(y)) + d(ψ(y),f(x, y)) + d(f(x, y),ϕ(x )) + d(ϕ(x ),f(x,y )). By (2), the third nd the fourth summnd on the righthnd side re both less thn /6, nd by (1), ech of the other four summnds is less thn /6. Thus, ( (x,y ) A B)( (x,y ) A B) (x,y ), (x,y ) U \{(x 0,y 0 )} = d(f(x,y ),f(x,y )) <, nd so the function f stisfies Cuchy condition t the point (x 0,y 0 ). Therefore, since the spce M is complete, there exists f(x, y) =α. (x 0,y 0) 1 See N. Dunford nd J. T. Schwrtz, Liner Opertors, Prt I, Interscience Publishers, New York, 1958.
6 20 Z. Kdelburg, M. M. Mrjnović Using Proposition 1 nd condition (i), we hve nd, by (ii), ψ(y) =α = ϕ(x) =α = f(x, y), f(x, y). We shll pply the obtined result to sequence (f n ) of functions with domin A which is subset of metric spce M 0 nd codomin which is metric spce M. Then, we cn consider f n (x) s function of two vribles, tking ϕ(n, x) =f n (x). Since N cn be understood s subset of the metric spce N = N {} with the metric d(m, n) = 1 m 1 ( ) 1 n, =0 nd since N, for the function ϕ: N A M, the fct tht for ech x A, ϕ(n, x) =f(x) exists, simply mens tht the sequence (f n) converges to the function f on the set A. The fct tht ϕ(n, x) =f(x) exists uniformly in x A, is equivlent to the fct tht the sequence (f n ) converges uniformly on A. The respective condition ( >0)( δ >0)( n N)( x A) 0<d(n, ) <δ = d(ϕ(n, x),f(x)) <, cn be written s ( >0)( m N)( n N)( x A) n>m = d(f n (x),f(x)) <, tking tht d(n, ) = 1 [ ] 1 n nd = m. δ As n exmple of using such tretment of functionl sequences s functions of two vribles, we shll derive now the wellknown theorem bout continuity of the it function. Corollry 1. Let (f n ) be uniformly convergent sequence of continuous functions from metric spce M 0 into complete metric spce M. Then the it function f = f n is lso continuous. Proof. If x 0 M 0 is n isolted point, the ssertion is trivil. Let x 0 M 0. For the function ϕ(n, x) =f n (x), (M 0 M 0, N N ) we hve lso (i) ϕ(n, x) exists uniformly in x M 0, nd by the continuity of ll the functions f n t the point x 0,
7 Interchnging two its 21 (ii) for ech n N, ϕ(n, x) =f n (x 0 ) exists, nd by Theorem 1, it follows tht f(x) = = f n(x 0 )=f(x 0 ). ϕ(n, x) = ϕ(n, x) In similr wy, the following ssertion on termwise differentition cn be proved. Corollry 2. Let (f n ) be sequence of relvlued functions, converging on [, b] tofunctionf, ndlet (i) f n be differentible function for ech n N, (ii) the sequence (f n) converges uniformly on [, b]. Then the function f is differentible nd the equlity f (x) = f n(x) holds. The respective properties of functionl series cn be derived now s esy consequences. 3. Double sequences A function : N N R is clled double sequence. As usul, we shll denote the vlue of the function t (i, j) s ij, nd this sequence will be denoted by ( ij ). The following theorem gives sufficient conditions for the interchnge of the order of summtion. Theorem 2. Let ( ij ) be double sequence such tht: (i) ( i) ij = b i < +, (ii) Then i=1 j=1 b i is convergent series. ij = ij. i=1 j=1 j=1 i=1 Proof. Let f i : N = N {} R be the function defined by f i (k) = k ij. j=1 For ech i, the function f i is continuous on N, becuse f i(k) = ij = f i (). k j=1
8 22 Z. Kdelburg, M. M. Mrjnović The series f i (k) converges uniformly ccording to the Weierstrss Test, becuse Therefore, i=1 (i) ( i) f i (k) b i nd (ii) k i=1 The desired equlity follows from k i=1 i=1 f i (k) = i=1 b i converges. i=1 f i (k) = f i(k). k k ij = k i=1 j=1 f i(k) = ij. k i=1 j=1 k ij = ij, k j=1 i=1 j=1 i=1 4. The Toeplitz Limit Theorem The following theorem ws proved by Toeplitz (O. Toeplitz, ) in Prce mt.fiz. 22 (1911), Theorem 3. The coefficients of the mtrix n n k0 k1... kn re ssumed to stisfy the following two conditions: () for ech fixed n, kn 0 s k, (b) there exists constnt K such tht for ech fixed k nd ny n, k0 + k1 + + kn <K. Then, for every null sequence (z 0,z 1,...,z n,...), the numbers lso form null sequence. Proof. Denote The ssumption () implies tht ( ) ϕ(n, k) = 0, for ech n. k z k = k0 z 0 + k1 z kn z n + ϕ(n, k) = k0 z 0 + k1 z kn z n.
9 Interchnging two its 23 On the other hnd, (b ) ϕ(n, k) = kj z j = ψ(k) exists, uniformly in k. n j=0 To prove it, observe first tht, by (b), the series kj converges bsolutely. Fix >0. Let j j=0 0 be such tht Then, for j>j 0,wehve ψ(k) ϕ(n, k) = j=n+1 j>j 0 = z j < K. kj z j j=n+1 kj mx z j <K j K =, nd so ϕ(n, k) exists uniformly in k. n Now, by Theorem 1, we obtin tht z k = ϕ(n, k) = ϕ(n, k) =0. k k n n k Corollry 3. If, in ddition to ssumption () of Theorem 3, ( k) k0 + k1 + + kn 1 s n, nd z n z 0, then z k z 0 s k. 5. Integrl s it Denote by Π the set of ll prtitions P of segment [, b]. For the prtition P given by = x 0 <x 1 < <x n = b, the number P = mx{ x i x i 1 i =1, 2,...,n} is clled the norm of prtition P. Let be n element which does not belong to Π nd let =0. ThesetΠ =Π {}, together with the metric { P1 + P 2, P 1 P 2, d Π (P 1,P 2 )= 0, P 1 = P 2, is metric spce which will be clled the prtition spce nd will be denoted by (Π,d Π ). If, for two prtitions P 1 nd P 2, P 1 P 2, we sy tht P 1 is smller thn P 2. Note tht finer prtition is lso smller, but the converse is not true. For prtition P,letP be the prtition which hs s prtitioning points ll the points of prtition P nd, moreover, ll the midpoints of segments of P. Then P = 1 2 P. Let P 0 be the prtition with = x 0 <x 1 = b. Let us define sequence (P n ) of prtitions, tking P n+1 = P n. Then P n = b 2 n,ndso d b Π(,P n ) = 2 n =0, wherefrom it follows tht is n ccumultion point of the set Π in the spce Π.
10 24 Z. Kdelburg, M. M. Mrjnović Let f :[, b] R be bounded function, s(f,p) nds(f,p) be its lower nd its upper sum, with respect to the prtition P. These sums cn be considered s functions s: Π R, S:Π R from the spce of prtitions into R. The it s(f,p) =α exists if the condition ( >0)( δ >0)( P Π) 0 <d Π (P, ) <δ = α s(f,p) < holds, or, equivlently, if the condition ( >0)( δ >0)( P Π) P <δ = α s(f,p) < holds. The following proposition ssocites this it with the lower (upper) integrl of the function f. Proposition 2. Let f :[, b] R be bounded function. Then s(f,p) (resp. S(f,P)) existsndisequltosup P s(f,p) (resp. inf P S(f,P)). Proof. Let sup P s(f,p) =I. Since the function f is bounded, there exists constnt M>0, such tht for ech x [, b], f(x) M. Let>0. There exists prtition P 1 with prtitioning points = x 0 <x 1 < <x k = b, such tht I s(f,p 1 ) < 2. Let δ =. Let P Π be such tht P <δ. Denote by P the superposition 8Mk of prtitions P nd P 1. We hve now s(f,p ) s(f,p) = m v( ) m v( ), P where m =inf{ f(x) x } nd v( ) is the length of segment. Since prtition P is finer thn prtition P, for ech P there exists unique P such tht.letm = m,ndso m v( ) = m v( ). Now, P P P s(f,p ) s(f,p) = (m m )v( ). P We hve m m = 0 when hs no endpoints belonging to prtition P 1, except points nd b, ndm m 0 when n endpoint of segment belongs to the set {x 1,...,x k 1 }, nd this endpoint does not belong to P. The number of such segments is t most 2(k 1). Since m m 2M, wehve s(f,p ) s(f,p) 2(k 1) 2M P < 2(k 1) 2M 8Mk = 2.
11 Interchnging two its 25 Hence, when P <δ,wehve I s(f,p) (I s(f,p )+(s(f,p ) s(f,p)) < =, becuse P is lso finer tht P 1 nd so I s(f,p ) I s(f,p 1 ) < 2. This completes the proof tht s(f,p) =I. The equlity S(f,P) =I =inf P S(f,P) cn be proved in similr wy. If the function f is integrble, then s(f,p) = S(f,P) = f(x) dx. For prtition P,letξ P be choice function, i.e., let ξ P ( ). The sum σ(f,p) = { f(ξ P ( ))v( ) P } is clled the integrl sum of the function f, corresponding to the prtition P. Since s(f,p) σ(f,p) S(f,P), for ech ξ P,wehve σ(f,p) = f(x) dx, for ech integrble function f. We shll consider now the question of interchnging it nd n integrl. First, we need n uxiliry sttement. Lemm 1. Let f :[, b] R nd g :[, b] R be two functions, such tht ( x [, b]) f(x) g(x) <k. Then, for ech nonempty set S [, b], the inequlities hold true. inf f(x) inf g(x) < 2k, x S x S sup x S f(x) sup g(x) < 2k. x S Proof. Let x S be point, such tht f(x) <m S (f) +k, where m S (f) = inf f(x). Then, x S m S (f) >f(x) k>g(x) 2k m S (g) 2k, where m S (g) = infg(x). Similrly, x S m S (g) >m S (f) 2k, which proves tht m S (f) m S (g) < 2k. The proof of the other prt of the proposition is similr.
12 26 Z. Kdelburg, M. M. Mrjnović Theorem 4. Let (f n ) be sequence of functions, integrble on the segment [, b], ndlet(f n ) converges uniformly to function f. Then, the function f is integrble nd the equlity is vlid. f n (x) dx = f(x) dx Proof. Let >0 be given. Since the sequence (f n ) converges uniformly to the function f, there exists n m N such tht ( n N)( x [, b]) n m = f n (x) f(x) < 2(b ). The lower sum of the function f n for the prtition P, s(f n,p)= { m n ( )v( ) P }, is function from N Π N Π into R. Forn m nd ech P,wehve s(f.p) s(f n,p) m n ( ) m( ) v( ). P Applying Lemm 1, we obtin s(f,p) s(f n,p) < 2 (b ) =, 2(b ) nd so s(f n,p)=s(f,p) exists uniformly in P.Forechn N, s(f n,p)= f n (x) dx P exists, becuse f n is n integrble function. Applying Theorem 1, it follows tht both iterted its exist nd re equl. Hence, we hve s(f n,p) = s(f,p) = f(x) dx, P n P nd s(f n,p) = f n (x) dx, n P n b f(x) dx = n f n (x) dx, Repeting the previous rguments nd using upper sums, the equlity b f(x) dx = f n (x) dx n cn lso be proved.
13 Interchnging two its 27 Note tht the previous theorem remins vlid when the domin of the given functions is n rbitrry set A R, which is Jordn mesurble (C. M. E. Jordn, ). Nmely, such set A is bounded nd there exists segment [, b] such tht A [, b]. If the functions f n : A R re integrble, so re the functions { fn (x), x A, f n :[, b] R given by f n (x) = 0, x [, b] \ A. If the sequence (f n ) converges uniformly to the function f : A R, then the sequence (f n ) converges uniformly to the function { f(x), x A, f(x) = 0, x [, b] \ A, which is lso integrble. According to the previous theorem, we hve f n (x) dx = f n (x) dx = f(x) dx = f(x) dx. A Assuming tht the it function of convergent sequence (f n ) is integrble, the ssumption of uniform convergence cn be wekened. First, we formulte nd prove n uxiliry sttement, which is of interest on its own. Let (f n ) be sequence of rel functions, converging to function f on set A R. LetA be the closure of the set A in R. We shll sy tht (f n ) converges uniformly round point x 0 A if there exists n open neighborhood U of the point x 0, such tht (f n )converges uniformly on the set U A. Proposition 3. Let f n, n N, be rel functions with domin A R. The sequence (f n ) converges uniformly in x A if nd only if it converges uniformly round ech point of the set A R. Proof. If the sequence (f n ) converges uniformly in x A, then it obviously converges uniformly round ech point of the set A. Conversely, suppose tht (f n ) converges uniformly round ech point of A. Then (f n ) is certinly pointwise convergent to function f. If A contins one of the elements, + (or both of them), choose respective neighborhood U or U + (or both of them), such tht (f n ) converges uniformly on A U, resp. A U +. For ny other point x A, letu x be neighborhood such tht (f n ) converges uniformly on U x A. The set A \ (U U + ) (if or + does not belong to A, then U, resp. U + is n empty set) will be compct. Hence, there exist finitely mny neighborhoods U x1, U x2,..., U xk covering the set A \ (U U + ). Adding U nd/or U + to them, we obtin finitely mny subsets of A, such tht (f n ) converges uniformly on them. Since ech point of the set A belongs to one of the sets U ξ A, ξ {x 1,x 2,...,x k,, +}, the sequence (f n ) converges uniformly on A. A
14 28 Z. Kdelburg, M. M. Mrjnović Roughly speking, more points re there round which the sequence (f n )isnot uniformly convergent, less regulr is the convergence of (f n ). On the other hnd, Proposition 3 cn be formulted s follows: The sequence (f n ) does not converge uniformly in x A if nd only if it does not converge uniformly round point x A. Hence, nonuniform convergence cn lwys be spotted loclly. The following is the Lebesgue theorem (H. L. Lebesgue, ) on bounded convergence for the Riemnn integrl (G. F. B. Riemnn, ). Theorem 5. Let (f n ) be sequence of functions, integrble on [, b], converging to function f. If (i) the function f is integrble, (ii) ( M R)( n N)( x [, b]) f n (x) M, (iii) the set A = { x [, b] (f n ) does not converge uniformly round x } isclosedsetoflebesguemesure0, then f n (x) dx = f(x) dx. Proof. Let >0 be given. According to (iii), there exists sequence ( i ) i N of open intervls covering the set A, such tht { v( i ) i N } < 4M. The set A is closed nd bounded, nd so, by BorelLebesgue Theorem (F. E. J. E. Borel, ), there exist finitely mny intervls i1,..., ik which cover A, s well. Let D = i1 ik. The set D is Jordn mesurble nd m(d) v( i1 )+ + v( ik ) < 4M. The sequence (f n ) converges uniformly round ech point of the set [, b] \ D, nd so, by Proposition 3, it converges uniformly on [, b] \ D. Hence, there exists n n 0 N such tht ( n N)( x [, b] \ D) n n 0 = f n (x) f(x) < Now, for n n 0,wehve f(x) dx f n (x) dx which proves the theorem. D < 2M 2(b ). f(x) f n (x) dx + f(x) f n (x) dx [,b]\d 4M + (b ) =, 2(b )
15 Interchnging two its 29 Exmple 2. The sequence of functions nx f n (x) =, x [0, 1], 1+(nx) 2 converges to the function 0: [0, 1] R, nd it does not converge uniformly only round the point 0. The set A = {0} is closed nd of mesure 0, while ( n N)( x [, b]) nx 1+(nx) Hence, we hve 1 nx dx = (nx) 2 At the end, let us remrk tht we hve not included severl other cses of interchnge of two its: cses of differentition nd integrtion of integrls depending on prmeter, equlity f xy = f yx, etc., which cn be treted in the sme wy. Zorn Kdelburg, Fculty of Mthemtics, Studentski trg 16/IV, Beogrd, Serbi & Montenegro Emil: Milosv M. Mrjnović, Mthemticl Institute, Knez Mihil 35/I, Beogrd, Serbi & Montenegro Emil:
Uniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationOstrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias
Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationLecture 3 Basic Probability and Statistics
Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: x n+ x n n + + C, dx = ln x + C, if n if n = In prticulr, this mens tht dx = ln x + C x nd x 0 dx = dx = dx = x + C Integrl of Constnt:
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationSection 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More information6: SEQUENCES AND SERIES OF FUNCTIONS, CONVERGENCE
6: SEQUENCES AND SERIES OF FUNCTIONS, CONVERGENCE STEVEN HEILMAN Contents 1. Review 1 2. Sequences of Functions 2 3. Uniform Convergence nd Continuity 3 4. Series of Functions nd the Weierstrss Mtest
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810193 Aveiro, Portugl
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationAe2 Mathematics : Fourier Series
Ae Mthemtics : Fourier Series J. D. Gibbon (Professor J. D Gibbon, Dept of Mthemtics j.d.gibbon@ic.c.uk http://www.imperil.c.uk/ jdg These notes re not identicl wordforword with my lectures which will
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More informationMatrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA
CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationWritten Homework 6 Solutions
Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions
More informationArc Length. P i 1 P i (1) L = lim. i=1
Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationReal Analysis and Multivariable Calculus: Graduate Level Problems and Solutions. Igor Yanovsky
Rel Anlysis nd Multivrible Clculus: Grdute Level Problems nd Solutions Igor Ynovsky 1 Rel Anlysis nd Multivrible Clculus Igor Ynovsky, 2005 2 Disclimer: This hndbook is intended to ssist grdute students
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationm(r n ) = Bor(R n ),
6. The Lebesgue mesure In this section we pply vrious results from the previous sections to very bsic exmple: the Lebesgue mesure on R n. Nottions. We fix n integer n 1. In Section 21 we introduced the
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationThe Riemann Integral. Chapter 1
Chpter The Riemnn Integrl now of some universities in Englnd where the Lebesgue integrl is tught in the first yer of mthemtics degree insted of the Riemnn integrl, but now of no universities in Englnd
More informationKarlstad University. Division for Engineering Science, Physics and Mathematics. Yury V. Shestopalov and Yury G. Smirnov. Integral Equations
Krlstd University Division for Engineering Science, Physics nd Mthemtics Yury V. Shestoplov nd Yury G. Smirnov Integrl Equtions A compendium Krlstd Contents 1 Prefce 4 Notion nd exmples of integrl equtions
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationDETERMINANTS. ] of order n, we can associate a number (real or complex) called determinant of the matrix A, written as det A, where a ij. = ad bc.
Chpter 4 DETERMINANTS 4 Overview To every squre mtrix A = [ ij ] of order n, we cn ssocite number (rel or complex) clled determinnt of the mtrix A, written s det A, where ij is the (i, j)th element of
More informationThe Velocity Factor of an Insulated TwoWire Transmission Line
The Velocity Fctor of n Insulted TwoWire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
More informationMATLAB: Mfiles; Numerical Integration Last revised : March, 2003
MATLAB: Mfiles; Numericl Integrtion Lst revised : Mrch, 00 Introduction to Mfiles In this tutoril we lern the bsics of working with Mfiles in MATLAB, so clled becuse they must use.m for their filenme
More informationBabylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity
Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University
More informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More informationITS HISTORY AND APPLICATIONS
NEČAS CENTER FOR MATHEMATICAL MODELING, Volume 1 HISTORY OF MATHEMATICS, Volume 29 PRODUCT INTEGRATION, ITS HISTORY AND APPLICATIONS Antonín Slvík (I+ A(x)dx)=I+ b A(x)dx+ b x2 A(x 2 )A(x 1 )dx 1 dx 2
More informationThe Fundamental Theorem of Calculus for Lebesgue Integral
Divulgciones Mtemátics Vol. 8 No. 1 (2000), pp. 75 85 The Fundmentl Theorem of Clculus for Lebesgue Integrl El Teorem Fundmentl del Cálculo pr l Integrl de Lebesgue Diómedes Bárcens (brcens@ciens.ul.ve)
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More informationArea Between Curves: We know that a definite integral
Are Between Curves: We know tht definite integrl fx) dx cn be used to find the signed re of the region bounded by the function f nd the x xis between nd b. Often we wnt to find the bsolute re of region
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationNull Similar Curves with Variable Transformations in Minkowski 3space
Null Similr Curves with Vrile Trnsformtions in Minkowski spce Mehmet Önder Cell Byr University, Fculty of Science nd Arts, Deprtment of Mthemtics, Murdiye Cmpus, 45047 Murdiye, Mnis, Turkey. mil: mehmet.onder@yr.edu.tr
More information1. The leves re either lbeled with sentences in ;, or with sentences of the form All X re X. 2. The interior leves hve two children drwn bove them) if
Q520 Notes on Nturl Logic Lrry Moss We hve seen exmples of wht re trditionlly clled syllogisms lredy: All men re mortl. Socrtes is mn. Socrtes is mortl. The ide gin is tht the sentences bove the line should
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationSequences and Series
Centre for Eduction in Mthemtics nd Computing Euclid eworkshop # 5 Sequences nd Series c 014 UNIVERSITY OF WATERLOO While the vst mjority of Euclid questions in this topic re use formule for rithmetic
More information1. Inverse of a tridiagonal matrix
PréPublicções do Deprtmento de Mtemátic Universidde de Coimbr Preprint Number 05 16 ON THE EIGENVALUES OF SOME TRIDIAGONAL MATRICES CM DA FONSECA Abstrct: A solution is given for problem on eigenvlues
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationReal Analysis HW 10 Solutions
Rel Anlysis HW 10 Solutions Problem 47: Show tht funtion f is bsolutely ontinuous on [, b if nd only if for eh ɛ > 0, there is δ > 0 suh tht for every finite disjoint olletion {( k, b k )} n of open intervls
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationTriangles, Altitudes, and Area Instructor: Natalya St. Clair
Tringle, nd ltitudes erkeley Mth ircles 015 Lecture Notes Tringles, ltitudes, nd re Instructor: Ntly St. lir *Note: This M session is inspired from vriety of sources, including wesomemth, reteem Mth Zoom,
More informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
More information4 Approximations. 4.1 Background. D. Levy
D. Levy 4 Approximtions 4.1 Bckground In this chpter we re interested in pproximtion problems. Generlly speking, strting from function f(x) we would like to find different function g(x) tht belongs to
More informationQuasilog concavity conjecture and its applications in statistics
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 R E S E A R C H Open Access Qusilog concvity conjecture nd its pplictions
More informationOnline Multicommodity Routing with Time Windows
KonrdZuseZentrum für Informtionstechnik Berlin Tkustrße 7 D14195 BerlinDhlem Germny TOBIAS HARKS 1 STEFAN HEINZ MARC E. PFETSCH TJARK VREDEVELD 2 Online Multicommodity Routing with Time Windows 1 Institute
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationEuler Euler Everywhere Using the EulerLagrange Equation to Solve Calculus of Variation Problems
Euler Euler Everywhere Using the EulerLgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More information19. The FermatEuler Prime Number Theorem
19. The FermtEuler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout
More informationThe Calculus of Variations: An Introduction. By Kolo Sunday Goshi
The Clculus of Vritions: An Introduction By Kolo Sundy Goshi Some Greek Mythology Queen Dido of Tyre Fled Tyre fter the deth of her husbnd Arrived t wht is present dy Liby Irbs (King of Liby) offer Tell
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationm, where m = m 1 + m m n.
Lecture 7 : Moments nd Centers of Mss If we hve msses m, m 2,..., m n t points x, x 2,..., x n long the xxis, the moment of the system round the origin is M 0 = m x + m 2 x 2 + + m n x n. The center of
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationIntroduction to Integration Part 2: The Definite Integral
Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More information14.2. The Mean Value and the RootMeanSquare Value. Introduction. Prerequisites. Learning Outcomes
he Men Vlue nd the RootMenSqure Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men
More informationAlgorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationComplementary Coffee Cups
Complementry Coffee Cups Thoms Bnchoff Tom Bnchoff (Thoms Bnchoff@brown.edu) received his B.A. from Notre Dme nd his Ph.D. from the University of Cliforni, Berkeley. He hs been teching t Brown University
More information5.1 SecondOrder linear PDE
5.1 SecondOrder liner PDE Consider secondorder liner PDE L[u] = u xx + 2bu xy + cu yy + du x + eu y + fu = g, (x,y) U (5.1) for n unknown function u of two vribles x nd y. The functions,b nd c re ssumed
More informationAn OffCenter Coaxial Cable
1 Problem An OffCenter Coxil Cble Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Nov. 21, 1999 A coxil trnsmission line hs inner conductor of rdius nd outer conductor
More informationNovel Methods of Generating SelfInvertible Matrix for Hill Cipher Algorithm
Bibhudendr chry, Girij Snkr Rth, Srt Kumr Ptr, nd Sroj Kumr Pnigrhy Novel Methods of Generting SelfInvertible Mtrix for Hill Cipher lgorithm Bibhudendr chry Deprtment of Electronics & Communiction Engineering
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationRegulated evolution quasivariational inequalities
Regulted evolution qusivritionl inequlities Pvel Krejčí Mtemtický ústv Akdemie věd ČR Žitná 25, CZ 11567 Prh 1 Czech Republic Lectures held t the University of Pvi My, 23 Supported by the Grnt Agency of
More informationTwo special Righttriangles 1. The
Mth Right Tringle Trigonometry Hndout B (length of )  c  (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Righttringles. The
More informationFor a solid S for which the cross sections vary, we can approximate the volume using a Riemann sum. A(x i ) x. i=1.
Volumes by Disks nd Wshers Volume of cylinder A cylinder is solid where ll cross sections re the sme. The volume of cylinder is A h where A is the re of cross section nd h is the height of the cylinder.
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More information