Prt I ome reminders 1.1. Eucliden spces 1.2. ubsets of Eucliden spce 1.3. Limits nd continuity Definition (Eucliden n-spce) The set of ll ordered n-tuples of rel numbers, x = (x1,..., xn), is the Eucliden n-spce, R n. Let x, y R n, α R. Addition: x + y = (x1 + y1,..., xn + yn). Multipliction by sclr: αx = (αx1,..., αxn). Dot product: x y = x1y1 + xnyn. x y x y x 2 = x x x y = x y ; cos θ, where θ is the ngle between the vectors x nd y. (Cuchy s inequlity) Let x, y R n, then x y x y p. 1 1.1. Eucliden spces p. 2 Definition (Open bll) The open bll of rdius r > 0 bout the point R n is Br () = {x R n : x < r} Definition (Cross product) For two vectors x, yr 3, i j k x y = x1 x2 x3 = (x2y3 x3y2, x3y1 x1y3, x1y2 x2y1) y1 y2 y3 Also clled r-neighborhood (commonly used in tht cse with δ insted of r, nd clled δ-neighborhood). Definition (Bounded set) A set R n is bounded if there exists r > 0 such tht Br (0) Definition (Complement of set) The complement (in R n ) of R n is c = {x R n : x } 1.1. Eucliden spces p. 3 1.2. ubsets of Eucliden spce p. 4
Interior nd boundry sets Definition x R n is n interior point of if there is bll centered t x tht is entirely in. The interior of is int = {x : r > 0, Br (x) } Open sets, closed sets, closure, neighborhood Definition (Open set) is open if it contins none of its boundry points, i.e., for ll x, x is n interior point of. Definition (Closed set) is closed if it contins ll of its boundry points. Definition x R n is boundry point of if every open bll centered t x hs points in nd points in c. The boundry of is Definition (Closure) The closure of is = = {x R n : r > 0, Br (x) nd Br (x) c } Definition (Neighborhood) A neighborhood of x is n open set contining x, i.e., n open set of which x is n interior point. 1.2. ubsets of Eucliden spce p. 5 1.2. ubsets of Eucliden spce p. 6 closed c open R n nd re both open nd closed. The union of countbly mny open sets is open. The intersection of infinitely mny open sets is open. The union of infinitely mny closed sets is closed. The intersection of countbly mny closed sets is closed. Definition (Limit) Let f : R n R, nd R n. Then iff L = lim x f (x) ε > 0, δ > 0, 0 < x < δ f (x) L < ε Definition (Continuous function) The function f : R n R is continuous t R n if lim f (x) = f () x 1.2. ubsets of Eucliden spce p. 7 1.3. Limits nd continuity p. 8
Definition (Limit) Let f : R m R n, nd R m. Then iff L = (L1,..., Ln) = lim x f (x) lim fk(x) = Lk, k x = 1,..., n f continuous on U iff ε > 0, U, δ > 0, 0 < x < δ f (x) f () < ε. uppose f : R n R k continuous on U R n, nd g : R m R k continuous on f (U) R m. Then g f : R n R k continuous on U. Corollry The sum, product or difference of two continuous functions is continuous. The quotient of two continuous functions is continuous on the set where the denomintor in nonzero. Let f : R n R k be continuous, U R k nd Then open if U open closed if U closed = {x R n : f (x) U} 1.3. Limits nd continuity p. 9 Prt II 1.3. Limits nd continuity p. 10 Little-o nd Big-O nottions Differentil clculus 2.1. Differentibility in one vrible 2.2. Differentibility in severl vribles 2.3. The chin rule 2.4. The men vlue theorem 2.5. Implicit functions I Definition (Little o nottion) Let f, g : I R R. f is o(g(x)) s x iff f (x) lim x g(x) = 0 Definition (Big o nottion) Let f, g : I R R. f is O(g(x)) s x iff δ > 0, M > 0 such tht f (x) M g(x) for x < δ. 2.6. Higher-order prtil derivtives p. 11 2.1. Differentibility in one vrible p. 12
Definition f : R R is differentible t if m such tht with lim h 0 o(h) = 0. f ( + h) = f () + mh + o(h) Proposition Let f : I R R nd I. If f hs locl extremum t nd f is differentible t, then f () = 0. (Rolle) uppose f : R R continuous on [, b] nd differentible on (, b). If f () = f (b), then c (, b) such tht f (c) = 0. (Men vlue) Let f : R R continuous on [, b] nd differentible on (, b). Then c (, b) such tht f f (b) f () (c) = b Let f, g be continuous on [, b], differentible on (, b), nd g 0 on (, b). Then c (, b), f (c) f (b) f () g = (c) g(b) g() 2.1. Differentibility in one vrible p. 13 2.1. Differentibility in one vrible p. 14 L Hospitl s rule uppose f : R R differentible on I R. If f (x) c for ll x I, then f (b) f () M b for ll, b I. If x I, f (x) = 0 then f constnt on I. If x I, f (x) 0 (resp. f (x) > 0, f (x) 0 nd f (x) < 0), then f is incresing (resp. strictly incresing, decresing, strictly decresing) on I. Assume f, g differentible on (, b) nd lim x + f (x) = lim x + g(x) = 0. uppose tht x (, b), g (x) 0, nd tht f (x) lim x + g (x) = L Then nd x (, b), g(x) 0 f (x) lim x + g(x) = L Works with left-hnd limits, limits (two-sided) nd limits t infinity. Also works if the limit is ±. 2.1. Differentibility in one vrible p. 15 2.1. Differentibility in one vrible p. 16
Differentibility for vector functions Definition A function f : R R n is differentible if ech of its component functions fk is differentible. Let f, g : R R n, α : R R. Then (αf ) = α f + αf (f g) = f g + f g (f g) = f g + f g (if n = 3). Definition (Prtil derivtive) Let f : R n R. The prtil derivtive of f with respect to xj is f xj = lim h 0 f (x1,..., xj + j,..., xn) f (x1,..., xn) h when this limit exists. Other nottions: fxj, fj, xj f, jf. If f (t) M for ll t [, b], then f (b) f () M b 2.1. Differentibility in one vrible p. 17 2.2. Differentibility in severl vribles p. 18 Definition Let f : R n R, open. f is differentible t if c R n such tht f ( + h) f () f () h lim = 0 h 0 h where f () is the grdient of f t. z = f () + f () (x ) is the tngent plne to z = f (x) t x =. If f differentible t, then the prtils fj() ll exist, nd f () = (f1(),..., fn()) If f is differentible t, then f is continuous t Be creful! The converses of these two theorems re flse 2.2. Differentibility in severl vribles p. 19 2.2. Differentibility in severl vribles p. 20
Differentils Let f : R n R with open,. uppose tht the prtils fxj ll exist in some neighborhood of, nd re continuous t. Then f differentible t. The differentil of f, differentible function, t, is the increment f ( + h) f (), i.e., df (; h) = df(h) = f () h 2.2. Differentibility in severl vribles p. 21 2.2. Differentibility in severl vribles p. 22 Directionl derivtive Definition Let, u R n, with u = 1. The directionl derivtive of f t in the direction of u is uf () = Duf () = lim t 0 f ( + tu) f () t (Chin rule) uppose g(t) differentible t t =, f (x) differentible t x =, nd b = g(). Then the function ϕ(t) = f (g(t)) is differentible t t =, with derivtive ϕ (t) = f (b) g () If f is differentible t, then the directionl derivtives ll exist t, nd Duf () = f () u or df dt = f dx1 x1 dt + + f dxn xn dt 2.2. Differentibility in severl vribles p. 23 2.3. The chin rule p. 24
(Chin rule) uppose g1,..., gn re functions of t = (t1,..., tm) differentible t t = nd f differentible t b = g(). Let ϕ = f g. Then ϕ is differentible t, with prtils given by ϕ = f x1 + + f xn tk x1 tk xn tk with prtils f / xj evluted t b nd derivtives ϕ/ tk nd xj/ tk evluted t. The result cn lso be stted with g of clss C 1 ner nd f of clss C 1 ner b, in which cse the conclusion is tht ϕ is of clss C 1 ner. uppose tht F is differentible function on some open set U R 3, nd ssume tht the set = {(x, y, z) U; F (x, y, z) = 0} is smooth surfce. If nd F () 0, then F () is orthogonl to t. The eqution of the tngent plne to t is F () (x ) = 0. 2.3. The chin rule p. 25 2.3. The chin rule p. 26 Another formultion of these results Definition (Differentibility, generl cse) Let U R n be n open set, nd f : U R m. We sy f is differentible t x0 U if the prtil derivtives of f exist t x0 nd if lim h x0 f (x) f (x0) Df (x0)(x x0) = 0, x x0 where Df (x0) is the m n mtrix with elements fi/ xj evluted t x0. In the cse m = 1, i.e., f : U R n R, then ( f Df (x) =,..., f ) = f x1 xn (Chin rule) Let U R n nd V R m be open sets. Let g : U V nd f : V R p, so tht f g : U R p. uppose g differentible t x0 nd f differentible t y0 = g(x0). Then f g differentible t x0 nd D(f g)(x0) = Df (y0)dg(x0), where the right-hnd side is the mtrix product of Df (y0) with Dg(x0). 2.3. The chin rule p. 27 2.3. The chin rule p. 28
(Men vlue theorem) Let R n contining nd b nd the line segment L joining them. Let f be defined on, continuous on L nd differentible on L except perhps t nd b. Then there exists c L such tht f (b) f () = f (c) (b ) Definition (Convex set) A set R n is convex if for ll, b, the line segment from to b lies in. Corollry Let f be differentible on n open convex set, nd suppose f (x) M for ll x. Then, for ll, b, f (b) f () M b Corollry uppose f differentible on n open convex set nd f (x) = 0 for ll x. Then f constnt on. 2.4. The men vlue theorem p. 29 2.4. The men vlue theorem p. 30 Implicit functions Problem sttement Consider the functionl reltionship F (x1,..., xn, y) = 0 uppose f differentible on n open connected set nd f (x) = 0 for ll x. Then f constnt on. Is it possible to obtin y s function of x = (x1,..., xn)? For now, ssume tht there exists g(x1,..., xn) differentible, defined on R n, such tht for ll (x1,..., xn) F (x1,..., xn, g(x1,..., xn)) = 0 Wht cn we sy bout g? 2.4. The men vlue theorem p. 31 2.5. Implicit functions I p. 32
Exmples Let F = F (x, y) be continuously differentible, nd y be differentible of x such tht F (x, y) = 0. Then t the points (x, y) where F / y 0, dy / x = F dx F / y Let F = F (x, y, z) be continuously differentible, nd z = z(x, y) be differentible such tht F (x, y, z) = 0. Then t the points (x, y, z) where F / z 0, z / x = F x F / z nd z / y = F y F / z (Clirut) uppose f defined on the open set R n. uppose ll first-order nd second-order prtil derivtives exist, nd tht second derivtives re continuous t. Then for ll i, j = 1,..., n. 2 f () = 2 f () xi xj xj xi Corollry If f C 2 (), with open, then for ll i, j = 1,..., n, on. 2 f xi xj = 2 f xj xi 2.5. Implicit functions I p. 33 Clirut s theorem for higher-order prtils 2.6. Higher-order prtil derivtives p. 34 Prt III The implicit function theorem nd pplictions If f C k (), with open, then k f k f xi1 = xik xj1 xjk on, where {j1,..., jk} is some permuttion of {i1,..., ik}. 3.1. The implicit function theorem 3.2. Curves in the plne 3.3. Curves in spce 3.4. Trnsformtions nd coordinte systems 2.6. Higher-order prtil derivtives p. 35 p. 36
ttement of the problem The Implicit Function, v. 1.0. Let F (x, y) C 1 (R n R) nd (, b) R n R such tht F (, b) = 0. We wnt to know if there exists 1. function f (x), defined on some set in R n, 2. n open set U R n+1 contining (, b), such tht for (x, y) U, F (x, y) = 0 y = f (x). (IFT, single eqution) Let F (x, y) C 1 on some neighborhood of (, b) R n R. uppose tht F (, b) = 0 nd y F (, b) 0. Then r0, r1 > 0 such tht: 1. x Br0 (),!f (x) such tht f (x) b < r1 nd F (x, f (x)) = 0. In prticulr, f () = b. 2. The function f thus defined for x Br0 () is C 1, nd its prtil derivtives re given, for j = 1,..., n, by F f (x) (x, f (x)) xj jf (x, f (x)) = = F xj y (x, f (x)) y F (x, f (x)) 3.1. The implicit function theorem p. 37 3.1. The implicit function theorem p. 38 More generl version of the problem Corollry Let F C 1 (R n ) nd = {x : F (x) = 0}. Then such tht F () 0, N, neighborhood of, such tht N is the grph of C 1 function. Let F (x, y) C 1 (R n R k R k ) nd (, b) R n R k such tht F (, b) = 0. We wnt to know if there exists 1. function f (x), defined on some set in R n, with vlues in R k, 2. n open set U R n+k contining (, b), such tht for (x, y) U, F (x, y) = 0 y = f (x). 3.1. The implicit function theorem p. 39 3.1. The implicit function theorem p. 40
IFT, v. 2.0. (IFT, system of equtions) Let F : R n R k R k be C 1 on some neighborhood of (, b) R n R k, nd Bij = Fi (, b), yj for i, j = 1,..., k. uppose tht F (, b) = 0 nd detb 0. Then r0, r1 > 0 such tht 1. x Br0 (),!f (x) such tht f (x) b < r1 nd F (x, f (x)) = 0. In prticulr, f () = b. 2. The function f thus defined for x Br0 () is C 1, nd its prtil derivtives xj f cn be computed by differentiting the equtions F (x, f (x)) = 0 with respect to xj nd solving the resulting liner system of equtions for xj f1,..., xj fk. Let F : R 2 R C 1 on n open set in R 2, nd = {(x, y) : F (x, y) = 0}. If nd F () 0, then there exists neighborhood N of in R 2 such tht N is the grph of C 1 function f : R R. (Either y = f (x) or x = f (y)). Let f : (, b) R 2 be C 1. If f (t0) 0, there exists n intervl I contining t0 such tht the set {f (t) : t I } is the grph of C 1 function f : R R (either y = f (x) or x = f (y)). Definition (mooth curve) A set R 2 is smooth curve if is connected nd, there is neighborhood N of such tht N is the grph of C 1 function f : R R. 3.1. The implicit function theorem p. 41 3.2. Curves in the plne p. 42 k-mnifold, nonprmetric form Let F : R 3 R defined on n open set in R 3, nd = {(x, y, z) R 3 : F (x, y, z) = 0}. If nd F () 0, then there exists neighborhood N R 3 of such tht N is the grph of C 1 function f : R 2 R. Let f : R 2 R 3 defined on n open set in R 2. If [ uf v f ](u, v) 0, then there exists neighborhood N R 2 of (u0, v0) such tht {f (u, v) : (u, v) N} is the grph of C 1 function. Let F1,..., Fn k : U R n R be C 1 functions on the open U, F = (F1,..., Fn k) : R n R n k. Then the set = {x R n : F (x) = 0} is k-dimensionl mnifold in R n. The mnifold is smooth if i.e., F1(x),..., Fn k(x) linerly independent x, DF (x) hs rnk n k for ll x 3.3. Curves in spce p. 43 3.3. Curves in spce p. 44
k-mnifold, prmetric form Let f : V R k R n be C 1 on the open V. Consider is k-mnifold in R n. is smooth if = {f (u) : u V }. u1f (u),..., uk f (u) linerly independent u V, The function f : R n R n of clss C 1 cn be considered s trnsformtion of R n. A trnsformtion of R n moves points in R n. Cn lso be interpreted s coordinte system. A coordinte system should be one-to-one. i.e., Df (u) hs rnk k for ll u V 3.3. Curves in spce p. 45 Inverse mpping theorem Let U, V R n open sets, U nd b = f (). uppose tht f : U V is C 1, with invertible Fréchet derivtive Df () invertible, i.e., Df () 0. Then there exists neighborhood M U nd Nb V of nd b, respectively, so tht f is one-to-one mp from M onto Nb, nd the inverse f 1 : Nb M is lso C 1. Moreover, if y = f (x) N, then Df 1 (y) = (Df (x)) 1 3.4. Trnsformtions nd coordinte systems p. 46 Prt IV Integrl Clculus 4.1. Integrtion on the line 4.2. Integrtion in higher dimensions 4.3. Multiple integrls nd iterted integrls 4.4. Chnge of vribles for multiple integrls 4.5. Functions defined by integrls 4.6. Improper integrls 3.4. Trnsformtions nd coordinte systems p. 47 p. 48
Prtition of the line Riemnn sums Let f be bounded function, f : [, b] R, nd P = {x0,..., xj} prtition of [, b]. Let Let [, b] be n intervl. A prtition P of [, b] is subdivision of [, b] into subintervls, P = {x0,..., xj}, with = x0 < x1 < < xj = b. A prtition P is refinement of the prtition P if P P, i.e., if P is obtined from P by dding more points. mj = inf {f (x)}, Mj = sup {f (x)} xj 1 x xj xj 1 x xj The lower Riemnn sum is J spf = mj(xj xj 1) j=1 nd the upper Riemnn sum is J Pf = Mj(xj xj 1) j=1 4.1. Integrtion on the line p. 49 Properties of Riemnn sums on prtitions 4.1. Integrtion on the line p. 50 Riemnn integrble function The lower nd upper integrls of f on [, b] re Lemm If P is refinement of P, then Lemm If P, Q prtitions of [, b], then sp f spf nd P f Pf spf Qf I b (f ) = sup spf P nd Ī b (f ) = inf P respectively. We lwys hve nd if I b (f ) Ī b (f ), I b (f ) = Ī b (f ) then f is Riemnn integrble, nd is the Riemnn integrl. I b (f ) = Ī b (f ) = f (x)dx 4.1. Integrtion on the line p. 51 4.1. Integrtion on the line p. 52
Lemm Let f be bounded on [, b], then the following conditions re equivlent: 1. f integrble on [, b]. 2. ε > 0, there exists prtition P of [, b] such tht Pf spf < ε If f bounded nd monotone on [, b], then f integrble on [, b]. If f continuous on [, b], then f integrble on [, b]. We dopt the convention tht f (x)dx = f (x)dx b 4.1. Integrtion on the line p. 53 More on integrbility If f bounded on [, b], continuous t ll except finitely mny points in [, b], then f integrble on [, b]. In fct, do not need finitely mny points of discontinuity. Definition Z R hs zero content if ε > 0, I1,..., IL, finite collection of intervls, such tht Z L l=1 Il nd L l=1 L(Il) < ε, where L(Il) is the length of the intervl Il. If f bounded on [, b] nd the set of discontinuous points of f in [, b] hs zero content, then f integrble on [, b]. 4.1. Integrtion on the line p. 55 Importnt properties Let f, g integrble on [, b], α, β R. Then: 1. If f integrble on [b, c], then f integrble on [, c] nd c c f (x)dx = f (x)dx + f (x)dx b 2. αf + βg integrble on [, b], nd αf (x) + βg(x) dx = α f (x)dx + β g(x)dx 3. If [c, d] [, b], f integrble on [c, d]. 4. If f (x) g(x) for x [, b], then f (x)dx g(x)dx 5. f integrble on [, b], nd f (x)dx f (x) dx 4.1. Integrtion on the line p. 54 Proposition uppose f, g integrble on [, b], nd f (x) = g(x) for ll x [, b] except t finitely mny points x [, b]. Then f (x)dx = g(x)dx (Fundmentl theorem of clculus) 1. Let f integrble on [, b]. For x [, b], define F (x) = x f (t)dt. Then F continuous on [, b], F (x) exists nd equls f (x) t every x t which f is continuous. 2. Let F continuous on [, b], differentible except t perhps finitely mny points in [, b], nd f defined on [, b] such tht f (x) = F (x) where F is defined. If f integrble on [, b], then f (x)dx = F (b) F () 4.1. Integrtion on the line p. 56
Prtitions for R 2 Proposition Let f integrble on [, b]. Given ε > 0, δ > 0 such tht if P = {x0,..., xj} is ny prtition of [, b] stisfying mx (xj xj 1) < δ, 1 j J then the sums spf nd Pf stisfy spf f (x)dx < ε nd Pf f (x)dx < ε. In R 2, prtition of the rectngle R = [, b] [c, d] tkes the form { = x0 < < xj = b P = {x0,..., xj, y0,..., yk }, c = y0 < < yk = d nd ech sub-rectngle Rjk = [xj 1, xj] [yk 1 yk] hs re Ajk = (xj xj 1)(yk yk 1). For given prtition P, define mjk = inf {f (x, y)}, Mjk = sup {f (x, y)} (x,y) Rjk (x,y) Rjk nd the lower nd upper Riemnn sums of f corresponding to P, J K J K spf = mjk Ajk, Pf = Mjk Ajk j=1 k=1 j=1 k=1 4.1. Integrtion on the line p. 57 Riemnn integrl for function of two vribles Given Riemnn sums J K J K spf = mjk Ajk, Pf = Mjk Ajk j=1 k=1 j=1 define the lower nd upper integrls of f on R, respectively. k=1 I R (f ) = sup spf P nd ĪR(f ) = inf P As for functions f : R R, we hve I R (f ) ĪR(f ), nd if the two integrls coincide, f is Riemnn integrble nd the common limit is the Riemnn integrl of f on R, denoted fda or f (x, y)dxdy R R 4.2. Integrtion in higher dimensions p. 58 Double integrls over generl regions To define double integrls over more generl regions, we use the indictor function of set, { 1 if x χ(x) = 0 otherwise Then, let R 2, bounded, nd f bounded on R 2. Let R be rectngle. f integrble on is f χ integrble on R, nd then fda = f χda R 4.2. Integrtion in higher dimensions p. 59 4.2. Integrtion in higher dimensions p. 60
Importnt properties of double integrls 1. Let f, g integrble on bounded set, α, β R, then αf + βg da = α fda + β gda 2. If f bounded function, integrble on 1 nd 2, 1 2 =, then f integrble on 1 2 nd fda = fda + fda 1 2 1 2 3. If f, g integrble on, nd f (x) g(x) for x, then fda gda 4. If f integrble on, then f integrble on, nd fda f da 4.2. Integrtion in higher dimensions p. 61 Definition Z R 2 hs zero content if ε > 0, R1,..., RM, finite collection of rectngles, such tht Z M m=1 Rm nd M m=1 A(Rm) < ε, where A(Rm) is the re of Rm. uppose f bounded on the the rectngle R. If the set of points in R t which f is discontinuous hs zero content, then f integrble on R. Proposition 1. If Z R 2 hs zero content nd U Z, then U hs zero content. 2. If Z1,..., Zk hve zero content, so does k j=1 Zj. 3. If f (0, b0) R 2 is C 1, then f ([, b]) hs zero content whenever 0 < < b < b0. 4.2. Integrtion in higher dimensions p. 62 Lemm χ discontinuous t x iff x, the boundry of. Definition (Jordn mesurble) R 2 is Jordn mesurble if it is bounded nd its boundry hs zero content. Let R 2, Jordn mesurble. Let f : R 2 R bounded, nd suppose tht the set of points in t which f discontinuous hs zero content. Then f integrble on. uppose Z R 2 hs zero content. If f : R 2 R is bounded, then f integrble on Z, nd Z fda = 0. Corollry 1. uppose f integrble on R 2. If g(x) = f (x) except for x in set of zero content, then g integrble on, nd gda = fda 2. uppose f integrble on nd on T, nd T hs zero content. Then f integrble on T, nd fda = fda + fda T T 4.2. Integrtion in higher dimensions p. 63 4.2. Integrtion in higher dimensions p. 64
Higher dimensions The sme notions extend to functions of n vribles. Elements of re da become elements of n-dimensionl volume, so integrtion is dv n, d n x or dx1 dxn. We write fdv n = f (x)d n x = f (x1,..., xn)dx1 dxn Men vlue theorem for integrls Let compct connected mesurble subset of R n, f, g continuous on with g 0. Then such tht f (x)g(x)d n x = f () g(x)d n x In prticulr, if g 1, Corollry Let compct connected mesurble subset of R n, nd f continuous on. Then such tht f (x)d n x = f ()Vn() where Vn() is the n-dimensionl volume of. 4.2. Integrtion in higher dimensions p. 65 Fubini s theorem 4.2. Integrtion in higher dimensions p. 66 Regions Let R = {(x, y) : x b, c y d}, nd f integrble on R. uppose tht y [c, d], f (x, y) integrble on [, b], nd f (x, y)dx integrble on [c, d]. Then d ( ) fda = f (x, y)dx dy R c If f (x, y) integrble on [c, d] for ll x [, b], nd d c f (x, y)dy integrble on [, b], then ( d ) fda = f (x, y)dy dx R c Remember tht it is then possible to consider integrls over regions of the type = {(x, y) : x b, ψ1(x) y ψ2(x)}, = {(x, y) : c y d, φ1(y) x φ2(y)} nd higher dimensionl versions, etc. = {(x, y, z) : c(x, y) U, ζ1(x, y) z ζ2(x, y)} 4.3. Multiple integrls nd iterted integrls p. 67 4.3. Multiple integrls nd iterted integrls p. 68
Let A Mn be invertible, nd G(u) = Au be the corresponding liner trnsformtion of R n. uppose R n mesurble, nd f integrble on. Then G 1 () = {A 1 x : x } mesurble, f G integrble on G 1 (), nd f (x)d n x = det A f (Au)d n u G 1 () Let U, V R n, open sets, G : U V one-to-one C 1 trnsformtion with derivtive DG(u) invertible u U. uppose T U nd V mesurble such tht G(T ) =. If f integrble on, then f G on T nd f (x)d n x = f (G(u)) det DG(u) d n u T 4.4. Chnge of vribles for multiple integrls p. 69 4.4. Chnge of vribles for multiple integrls p. 70 Let f : (x, y), for x R m nd y R n. If f is integrble over R n, s function of y for ech fixed x, we cn define F (x) = f (x, y)d n y uppose R n nd T R m compct, nd mesurble. If f (x, y) continuous on T, then F is continuous on T. uppose R n compct nd mesurble, nd T R m open. If f continuous on T, C 1 s function of x T for ll y, then F is C 1 on T, nd F f (x) = (x, y)d n y (x T ) xj xj (Bounded convergence theorem) Let R n mesurble, nd {fj} sequence of integrble functions on. uppose fj(y) f (y) for ll y, with f integrble on, nd tht there exists C R such tht fj(y) C for ll j nd y. Then lim j fj(y)d n y = f (y)d n y 4.5. Functions defined by integrls p. 71 4.5. Functions defined by integrls p. 72
Corollry Let R n mesurble nd T R m. uppose f : T R integrble s function of y for ech x T, nd F (x) = f (x, y)d n y 1. If f (x, y) continuous s function of x T for ech y nd there exists C R such tht f (x, y) C for ll x T nd y, then F continuous on T. 2. uppose T open. If f (x, y) is C 1 s function of x T for ll y, nd there exists C R such tht xf (x, y) C for ll x T nd y, then F is C 1 on T nd F f (x) = (x, y)d n y (x T ) xj xj Improper integrls of the first kind f defined on [, ) nd integrble on [, b] for ll b >. The improper integrl is f (x)dx = lim 0 b f (x)dx The integrl 0 f (x)dx converges if the limit exists, otherwise 0 f (x)dx diverges. 4.5. Functions defined by integrls p. 73 4.6. Improper integrls p. 74 uppose 0 f (x) g(x) for ll sufficiently lrge x. If g(x)dx converges, so does f (x)dx. If f (x)dx diverges, so does g(x)dx. Corollry uppose f, g > 0, nd f (x)/g(x) l s x. 1. If 0 < l <, then f (x)dx nd g(x)dx re both convergent or both divergent. 2. If l = 0, then g(x)dx convergent f (x)dx convergent. 3. If l =, then g(x)dx divergent f (x)dx divergent Corollry If 0 f (x) Cx p for ll sufficiently lrge x, with p > 1, then f (x)dx converges. If f (x) cx 1 (c > 0) for ll sufficiently lrge x, then f (x)dx diverges. If f (x) dx converges, then f (x)dx converges (we sy f (x)dx bsolutely convergent, or converges bsolutely). 4.6. Improper integrls p. 75 4.6. Improper integrls p. 76
Improper integrls of the second kind f defined on (, b] nd integrble on [c, b] for ll c >. Then f (x)dx = lim c c f (x)dx The integrl f (x)dx converges if the limit exists, otherwise f (x)dx diverges. uppose 0 f (x) g(x) for ll x sufficiently close to. If g(x)dx converges, so does f (x)dx. If f (x)dx diverges, so does g(x)dx. Corollry If 0 f (x) C(x ) p for ll x sufficiently close to, with p < 1, then f (x)dx converges. If f (x) > c(x ) 1 (c > 0) for ll x sufficiently close to, then f (x)dx diverges. 4.6. Improper integrls p. 77 Cuchy principl vlue Let < c < b, nd f integrble on [, c ε] nd on [c + ε, b] for ll ε > 0. The (Cuchy) principl vlue of f (x)dx is ( c ε ) P.V. f (x)dx = lim ε 0 f (x)dx + f (x)dx c+ε when this limit exists. If converges, then P.V. f (x)dx = f (x)dx Proposition uppose < c < b. If ϕ continuous on [, b] nd differentible t 0, then P.V. ϕ(x)dx exists. 4.6. Improper integrls p. 78 Prt V Line nd surfce integrls: vector nlysis 5.1. Arc length nd line integrls 5.2. Green s theorem 5.3. urfce re nd surfce integrls 5.4. Grd, curl nd div 5.5. The divergence theorem 5.7. tokes theorem 5.6. Applictions 4.6. Improper integrls p. 79 p. 80
Types of functions Arc length We hve seen vector vlued functions f : R R n functions of severl vribles f : R n R n vector vlued functions of severl vribles f : R m R n Here, we will consider vector fields Let C be smooth curve in R n, nd suppose C given by x = g(t), t b, with g : R R n of clss C 1, nd g (t) 0. Then the rc length of C is given by ds = g (t) dt C where ds is tken long the curve C nd represents the distnce between two very close points long C. f : R n R n p. 81 5.1. Arc length nd line integrls p. 82 Arc length is independent of prmetriztion Piecewise smooth curves Let t = φ(u), with φ one-to-one smooth from [c, d] to [, b]. Then x = g(t) is lso described by x = (g φ)(u), for c u d. Then rc length is d d (g φ) (u) du = g (φ(u)) φ (u) du c c by Chin rule, nd thus, since φ 1 ([, b]) = [c, d], g (φ(u)) φ (u) du = g (t) dt φ 1 ([,b]) [,b] A piecewise smooth curve is curve tht is the union of finitely mny smooth curves. More formlly, C piecewise smooth if C continuous nd C 1, except t finitely mny points tj where the exist. lim g (t) t t ± j by the chnge of vribles theorem. 5.1. Arc length nd line integrls p. 83 5.1. Arc length nd line integrls p. 84
Line integrl of sclr vlued function Let f : R n R be continuous, defined on the curve C, C D R n. uppose C prmetrized by x = g(t), g : R R n, for t b, then fds = f (g(t)) g (t) dt C is the integrl of f over C. Proposition Let F : R R m defined on [, b]. Then F (t)dt F (t) dt We cn then consider line integrls for vector fields: let F = (F1,..., Fm), with ech Fj : R n R, then ( ) Fds = F1ds,..., Fmds C C C 5.1. Arc length nd line integrls p. 85 clr-vlued line integrl for vector fields Let C be piecewise smooth in R n, nd F : R n R n be continuous vector field defined on some neighborhood of C. The line integrl of the vector field F over C is F dx = (F1dx1 + + Fndxn) C C nd, if C described prmetriclly by x = g(t), t b, then F dx = F (g(t)) g (t)dt C 5.1. Arc length nd line integrls p. 86 Rectifible curves uppose C prmetrized by g : [, b] R n, g continuous nd one-to-one. Let P = {t0,..., tj} be prtition of [, b], then J LP(C) = g(tj) g(tj 1) is the length of the segments joining the points g(tj). If the set 1 L = {LP(C) : P is prtition of [, b]} is bounded, then C is rectifible, nd the rc length L(C) is L(C) = sup{lp(c) : P is prtition of [, b]} 5.1. Arc length nd line integrls p. 87 5.1. Arc length nd line integrls p. 88
A few definitions If P refinement of P, then LP (C) LP(C). If C1 nd C2 re two curves prmetrized by g(t), for t [, c] nd t [c, b], respectively, then L(C) = L(C1) + L(C2) uppose g C 1 prmetrizes C, for t [, b], then C rectifible nd L(C) = g (t) dt A simple closed curve hs initil nd end points mtching, nd does not intersect itself otherwise. A regulr region is compct R n tht is the closure of its interior. A regulr region R 2 hs piecewise smooth boundry if consists of finite union of disjoint, piecewise smooth simple closed curves. The positive orienttion of hs on the left hnd side of, when is followed in the positive orienttion. 5.1. Arc length nd line integrls p. 89 5.2. Green s theorem p. 90 (Green s theorem) uppose R 2 regulr region with piecewise smooth boundry. uppose tht F C 1 ( ). Then ( ) F2 F d x = F1 da, x1 x2 or, writing F = (P, Q) nd x = (x, y), P dx + Q dy = ( Q x P y ) da, Corollry uppose R 2 is regulr region with piecewise smooth boundry, nd n(x) be the unit outwrd norml vector to t x. uppose F C 1 ( ), then ( ) F1 F n ds = + F2 da. x1 x2 5.2. Green s theorem p. 91 5.2. Green s theorem p. 92
5.6. Applictions p. 93