Lecture Summaries for Multivariable Integral Calculus M52B

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1 These leture summries my lso be viewed online by liking the L ion t the top right of ny leture sreen. Leture Summries for Multivrible Integrl Clulus M52B Chpter nd setion numbers refer to the 6th edition. The orresponding hpter nd setion in the 7th edition n be found by subtrting one from the hpter number of the 6th edition. Leture 01: Introdution. An introdution to the ourse, listing the topis to be overed; remrks onerning the types of proofs given throughout the ourse. Leture 02: eview of Ares under Curves nd the Definite Integrl. A review of the reltionship between res under urves nd definite integrtion of funtion of one vrible; disussion of iemnn sum pproximtions to the re; definition of re nd net signed re s limit of iemnn sums; definition of the definite integrl of funtion; sttement of theorems sying tht ontinuity is suffiient ondition for integrbility nd tht definite integrl n be omputed by pproprite evlution of n ntiderivtives. Leture 03 ( 16.1) : Volumes Under Surfes nd Double Integrls. Disussion of proedure for finding the volume of solid lying between region in the xy-plne nd the grph of non-negtive funtion f(x, y) defined on ; definition of the double integrl of f over, f(x, y) da, in terms of limit of iemnn sums, nd its geometri interprettion s net signed volume; onditions on nd f whih gurntee the existene of the double integrl; theorem nd proofs of the bsi properties of double integrls. Leture 04 ( 16.1) : Computing Double Integrls. Definition nd exmples of prtil definite integrtion; definition of iterted integrl; theorem nd (geometri) proof tht if f(x, y) is ontinuous on the retngulr region = {(x, y) x b, y d}, then f(x, y) da = b d f(x, y) dy dx = d b f(x, y) dx dy; omments on the impliit ssumptions bout volume used in this geometri proof. Leture 05 ( 16.2) : Double Integrls over Nonretngulr egions: Theory. Disussion nd exmples of iterted integrls hving the forms b g2(x) g 1(x) f(x, y) dy dx nd d h2(y) h 1(y) f(x, y) dx dy; definitions nd exmples of type I nd type II regions; theorem nd (geometri) proof tht double integrls of ontinuous funtions over these types of regions n be omputed using these more generl forms of iterted integrls. Leture 06 ( 16.2) : Double Integrls over Nonretngulr egions: Exmples. Exmples omputing double integrls over non-retngulr regions of types I nd II. Leture 07 ( 16.3) : Polr Double Integrls: Theory. Definition, disussion, nd exmples of simple polr regions; definition of the polr double integrl f(r, θ) da s limit of polr iemnn sums; theorem nd proof tht if is the simple polr region whose boundries re the rys θ = α nd θ = β nd the polr urves r = r 1 (θ) nd r = r 2 (θ), nd if f(r, θ) is ontinuous on, then β r2(θ) f(r, θ) da = f(r, θ)r dr dθ. α 1 r 1(θ)

2 Leture 08 ( 16.3) : Polr Double Integrls: Exmples. Exmples omputing polr double integrls. Leture 09 ( 16.4) : Prmetri epresenttions of Surfes. Introdution to prmetriztion of surfes; definitions of onstnt u-urves nd onstnt v-urves; exmples illustrting how to prmetrize surfes defined by equtions where either x, y, or z is funtion of the other two, or surfes obtined by revolving the grph of y = f(x) round the x-xis; exmples illustrting how to find prmetriztions with given onstnt u- or onstnt v-urves. Leture 10 ( 16.4) : Prmetri Surfes: Vetor Equtions nd Tngent Plnes. Disussion of the vetor form of prmetriztion for surfe; definition of ontinuity for vetor-vlued funtions of two vribles; definition of the prtil derivtives r r u nd v of vetor-vlued funtion r(u, v); geometri interprettion of these prtils s tngent vetors to onstnt v- nd onstnt u-urves respetively; derivtion of the eqution of the tngent plne to prmetri surfe t given point; tehnil remrks regrding existene nd uniqueness of tngent plnes; definition of the prinipl unit norml vetor to surfe. Leture 11 ( 16.4) : Prmetri Surfes: Surfe Are. Definition of smooth prmetri surfe; definition of the surfe re of smooth prmetri surfe; disussion of the motivtion behind this definition. Leture 12 ( 16.4) : Exmples Computing Surfe Are. Exmple showing tht the surfe re of sphere of rdius is 4π 2 ; observtion tht if f(x, y) is differentible funtion defined on domin D, then the surfe re of the grph of f is given by fx 2 + fy da; exmple illustrting the observtion. Leture 13 ( 16.5) : Triple Integrls. Definition nd disussion of the triple integrl of funtion of three vribles over solid region in 3-spe; bsi properties of triple integrls; theorem sserting tht if G is the retngulr box defined by the inequlities x b, y d, k z l, nd f is ontinuous on G, then f(x, y, z) dv = G b d l k f(x, y, z) dz dy dx, nd the iterted integrl on the right n be repled by ny of the other 5 iterted integrls obtined by hnging the order of integrtion; exmple illustrting the theorem. Leture 14 ( 16.5) : Triple Integrls over More Generl egions. Definition nd exmple of simple xy-solid; theorem nd (sketh of) proof tht if is region in the xy-plne nd G is the simple xy-solid defined by G = {(x, y, z) (x, y) in, g 1 (x, y) z g 2 (x, y)} (where g 1 g 2 re ontinuous funtions on ), then ssuming f is ontinuous on G. ( g2(x,y) f(x, y, z) dv = G g 1(x,y) ) f(x, y, z) dz da, Leture 15 ( 16.5) : Triple Integrls: Exmples on Simple Solids. Exmple illustrting the theorem proven in the previous leture; orresponding definitions nd theorems for simple xz-solids nd simple yz-solids; exmple of triple integrtion over these types of solids. Leture 16 ( 16.6) : Mss nd Density of Lmin. Definition nd exmples of lmin; definitions of homogeneous nd inhomogeneous; definition of density for homogeneous lmin; D 2

3 definition of density funtion for n inhomogeneous lmin; theorem nd proof tht if lmin with ontinuous density funtion δ(x, y) oupies region in the xy-plne, then the totl mss M of the lmin is δ(x, y) da; similrity between this result nd the fundmentl theorem of lulus; exmples illustrting the theorem. Leture 17: ( 16.6) Center of Grvity. Disussions nd derivtions of the enter of grvity for system of two point-msses lying on line, nd system of n point msses lying on line. Leture 18: ( 16.6) Center of Grvity of Lmin. Disussions nd derivtions of the enter of grvity for system of n point msses lying in plne, nd lmin with mss distributed ording to the density funtion δ(x, y). Leture 19: ( 16.6) Theorem of Pppus nd Center of Grvity of Solid. Definition nd disussion of the entroid of lmin or region; proof of the theorem of Pppus relting the entroid of region to the volume of prtiulr solid of revolution; definition of the density nd density funtion for homogeneous nd inhomogeneous solids in 3-spe; theorem sserting tht the totl mss of solid in 3-spe is the triple integrl of its density funtion; disussion of enter of grvity for system of disrete prtiles in 3-spe; formuls for the oordintes of the enter of grvity of solid in 3-spe. Leture 20: ( 16.7) Triple Integrls in Cylindril Coordintes. Definition of ylindril wedge; omputtion of the volume of ylindril wedge; definition of the triple integrl in ylindril oordintes of funtion f(r, θ, z) over solid G; sttement of theorem expressing the triple integrl in ylindril oordintes s the pproprite triple iterted integrl; disussion of the reltionship between triple integrls in retngulr oordintes nd triple integrls in ylindril oordintes; exmple illustrting the theorem. Leture 21: ( 16.7) Triple Integrls in Spheril Coordintes. Definition of spheril wedge; sttement of the volume of spheril wedge; definition of the triple integrl in spheril oordintes of funtion f(ρ, θ, φ) over solid G; sttement of theorem expressing the triple integrl in spheril oordintes s the pproprite triple iterted integrl; exmple illustrting the theorem; exmple illustrting the reltionship between triple integrls in retngulr, ylindril, nd spheril oordintes. Leture 22: ( 16.8) Chnging Vribles in Integrls: Prt I. Disussion nd derivtion of the hnge of vrible formul for single integrls; definition of the Jobin; sttement of the hnge of vrible formul for double integrls; definition, disussion, nd exmple of one-to-one trnsformtion from the uv-plne to the xy-plne. Leture 23: ( 16.8) Chnging Vribles in Integrls: Prt II. Definition of the Jobin of trnsformtion; disussion of the reltionship between the Jobin of trnsformtion nd the re of the imge of region in the domin; theorem nd sketh of proof of the hnge of vrible formul for double integrls. Leture 24: ( 16.8) Chnging Vribles in Integrls: Prt III. Exmples illustrting how to pply the hnge of vribles formul for double integrls, nd why previous method for omputing double integrls in polr oordintes is speil se of the hnge of vribles formul; definitions of trnsformtion from uvw-spe to xyz-spe, nd the Jobin of suh trnsformtion; sttement of the hnge of vribles theorem for triple integrls. Leture 25: ( 17.1) Vetor Fields. Definition of vetor field; exmples of vetor fields; disussion of grphil nd oordinte representtions of vetor fields; definition of inverse-squre fields. 3

4 Leture 26: ( 17.1) Del, Div, Curl, nd the Lplin. Definitions, nottions, nd exmples of the grdient (del), divergene, url, nd Lplin opertors; disussion of the oordinte independene of the divergene nd the url. Leture 27: ( 17.2) Line Integrls with espet to Ar Length. Definition nd geometri interprettion of the line integrl of f with respet to r length (s) long smooth urve C in 2-spe; derivtion of formul for evluting line integrl with respet to r length in the se where the urve C is prmetrized by r length; derivtion of the more generl formul for evluting suh line integrl diretly from ny smooth prmetriztion of the urve C; exmple illustrting this more generl formul. Leture 28: ( 17.2) Other Types of Line Integrls. Definition of line integrls with respet to r length in 3-spe; formul for evluting suh line integrls; definition nd disussion of line integrls with respet to x, y, nd z; derivtion of the formul for evluting suh integrls. Leture 29: ( 17.2) Prmetriztion Independene nd Pieewise Smooth Curves. Proof tht line integrls with respet to r length long urve do not depend on the prmetriztion or the orienttion of tht urve; proof tht line integrls with respet to x, y nd z do not depend on the prmetriztion of n oriented urve, but these line integrls hnge sign if the orienttion of the urve is hnged; definition of pieewise smooth urve; definition of line integrl long pieewise smooth urve. Leture 30: ( 17.2) Line Integrls nd Work. Motivtion for nd definition, in terms of line integrls, of work done on prtile s it moves long urve through fore (vetor) field; remrks on why the work hnges sign if the orienttion of the urve is hnged; derivtion of formuls for evluting the line integrls whih ompute work; disussion of vrious nottions for these work integrls. Leture 31: ( 17.3) Conservtive Fields nd Independene of Pth: Prt I. Sttement nd proof of the Fundmentl Theorem of Work Integrls, implying tht the vlue of work integrl of onservtive vetor field long pieewise smooth urve does not depend on the urve itself, but rther only on the endpoints of the urve; nottion of the form (x1,y 1) (x 0,y 0) F dr to denote the work integrl of the onservtive field F long ny pieewise smooth urve strting t (x 0, y 0 ) nd ending t (x 1, y 1 ). Leture 32: ( 17.3) Conservtive Fields nd Independene of Pth: Prt II. Definitions nd exmples of losed urve nd onneted set; theorem nd proof tht on onneted domin, F is onservtive if nd only if work integrls of F round pieewise smooth urves re zero if nd only if work integrls of F re independent of pth. Leture 33: ( 17.3) Conservtive Field Test. Definitions, exmples, nd disussions of simple urves nd simply onneted regions; sttement nd prtil proof of the Conservtive Field Test for 2-spe; exmples illustrting the theorem; exmple illustrting how to find potentil for given onservtive field; sttement of the Conservtive Field Test for 3-spe; re-sttement of this theorem in terms of url. Leture 34: ( 17.4) Green s Theorem for Simply Conneted egions. Sttement nd (prtil) proof of Green s Theorem for simply onneted regions; exmples illustrting the usefulness of Green s Theorem; orollry showing tht under pproprite onditions f y = g x implies f(x, y)i + g(x, y)j is onservtive; orollry showing tht Are() = C x dy = C y dx = 1 2 y dx+x dy, where C is simple, losed, pieewise smooth urve, oriented ounterlokwise, C omprising the boundry of ; nottion for line integrls round simple losed urves. 4

5 Leture 35: ( 17.4) Green s Theorem for Multiply Conneted egions. Definition of positive orienttion; sttement nd proof of Green s Theorem for multiply onneted regions; sttement nd proof of orollry tht underlies the Priniple of Deformtion of Pth in omplex nlysis. Leture 36: ( 17.5) Surfe Integrls. Definition of surfe integrls; derivtion of formul for evluting surfe integrl over smooth prmetri surfe; exmples illustrting this formul nd the reltionship between surfe re nd surfe integrls; derivtion of formul for evluting surfe integrl where the surfe is the grph of some funtion of x nd y; exmple illustrting this formul. Leture 37: ( 17.6) Flows nd Orienttions of Surfes. Informl definition of flux; disussion nd exmples of two-sided surfes nd one-sided surfes (Mobius strip); definition of orientble nd orienttion; disussion of how ontinuous unit norml vetor field on surfe defines n orienttion; exmples of suh fields for smoothly prmetrized orientble surfes; definitions of positive orienttion, positive diretion, negtive orienttion, nd negtive diretion. Leture 38: ( 17.6) Flux. Derivtion of the formul for flux of vetor field ross n oriented surfe; disussion of the evlution of flux integrls in the ses where the surfe is given prmetrilly nd where the surfe is the grph of some funtion of x nd y. Leture 39: ( 17.7) The Divergene Theorem. Definitions of losed surfe nd pieewise smooth surfe; sttement nd prtil proof of the divergene theorem. Leture 40: ( 17.7) Divergene nd Flux Density. Exmple illustrting the Divergene Theorem; disussion of the physil interprettion of the divergene s the outwrd flux density (flux per unit volume) t point; disussion of soures nd sinks in n inompressible fluid; sttement of Guss s Lw for Inverse-Squre Fields. Leture 41: ( 17.8) Stokes Theorem. Definition of positive orienttion of boundry urve reltive to the orienttion of the enlosed surfe; sttement of Stokes Theorem; remrks bout Stokes Theorem, inluding proof tht Green s Theorem is speil se of Stokes Theorem. Leture 42: ( 17.8) Curl nd Cirultion Density. Exmples illustrting Stokes Theorem; disussion of the physil interprettion of the url s the diretion of mximl irultion density in n inompressible, stedy-stte fluid flow. Leture 43: Wht s Next?. Brief disussion of the ourse(s) student ould/should tke fter ompleting Multivrible Integrl Clulus. 5