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15.8, 15.9, 16.1 Notes

by: Pawin Jiaravanon

15.8, 15.9, 16.1 Notes MA 26100 - 300

Pawin Jiaravanon
Multivariate Calculus
David W Catlin

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About this Document

Here you go. Added some pictures as well this time.
Multivariate Calculus
David W Catlin
Class Notes
Multivariate Calculus, Calculus, calc 3, vectors, Catlin
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This 11 page Class Notes was uploaded by Pawin Jiaravanon on Friday October 30, 2015. The Class Notes belongs to MA 26100 - 300 at Purdue University taught by David W Catlin in Fall 2015. Since its upload, it has received 42 views. For similar materials see Multivariate Calculus in Mathematics (M) at Purdue University.

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Date Created: 10/30/15
158 Triple Integrals in Cylindrical Coordinates If r6z are cylindrical coordinates then the rectangular coordinates are x r cos6 y r sin6 and z 2 If xyz are rectangular coordinates then the cylindrical coordinates are r2 x2 yz tan0 yx and z 2 Polar x r cos6 y r sin6 Cartesian r2 x2 y2 tan6 yx In 3 dimensions we can use a related system of polar coordinates Eg If a point P has a cylindrical coordinates of 2n63 then the rectangular coordinates are X 2 COST6 2 x32 3 Y2sinn6212 1 23 Eg What are the cylindrical coordinates of the surface x2 y2 6 in the rst octant with Oszs4 r26 OsesnZ r6z x6 0 s 6 s n3 0 s z s 4 fffE fxyz dV IID Izu1xy 2x39y fr cos6 r sin6 2 dz dA You can replace dA with r dr d6 EgFindffszdehereEis bounded bylszsx2y2zandx2y2 1 IIIEdVf02n folfermzdzrdrde 2nf01f1m2 2 dz rdr 2nf01 222 1r 22 r dr nf01r222 1rdr nf01r54r3 3rdr n r66 r4 3r22 01 n 16 1 32 5113 E9 1 22 Iol tx mmwmz x2 y212 dV Ioquot I02 I r dz r dr d6 n L I r2 dz dr nf02r2z r2dr n L 2r2 r3 dr n 2r33 r44 o2 4113 Eg Find the integral HIE z x dV where E is the region in the 1st octant bounded by z x2 y2 and the sphere x2 y2 z2 2 The solid lies above a disc in the xyplane Its radius is x2 r2 D 2 r2 U r4 2 r2 r22ri2 r2zxz r22r2zorr 1 IE X 2 CW No frozm39m X 2 d2 dA Ila X 222 Immml r dr d6 Ion4 I o1 r cos62 2r r33 r5 dr d6 for cos62 2r2 r43 r66 o1 d6 ltCan39t nd pagegt Eg Find the volume of the solid that lies within both the cylinder x2 y2 1 and the sphere x2 y2 z2 4 This region satis es 0s rs 1 056s 2n x4 x2 y2 SZSx4 X2y2 x2y2224D224 x2 y2Dzx4 x2 y2x4 r2 Volume fffE 1 dV f02quotf01f 4rA2 439r 2 1 dz r dr d6 fezquot fol 2 x4 rquot2 r dr d6 4n fol x4 rquot2 r dr 2nf01 x4 rquot2 3r dr 2nf01 4 r232 23 dr 4n3 r r232 01 4n3 332 8 4113 8 332 ex compute fffE z dV where E is in the rst octant and lies between x2 y2 1 and x2 and y2 4 and lies between 2 y IE 2 CW Ion2 I12 forswe 2 dz r dr de Ion2 I12 zquot22 or5i e r dr d6 f0 I12 r3 sin262 dr d6 fem r48 sin26 ll2 d6 158 Ion2 sin26 d6 158 Ion2 1cos262 one 151 16Ioquot 2 1Cos26 d6 15Tl32 15sin2632 loll2 15n32 Ex Let E be the region in the rst octant bounded by x2 z2 4 and bounded between y 4 and y x Use cylindrical coordinates to express fffE xy dV We use cylindrical coordinates with x2 z2 r2 X r cos6 Z r sin6 Y is like 2 The variable y satisfies x s y s 4 or r cos6 s y s 4 HIE x y dV fort2 L f rcose4 r2 cos6e sin6 dy r dr d6 dV dy r dr d6 Ion2 I02 I rcose4 r3 cos6e sin6 dy dr d6 159 Triple Integrals in Spherical Coordinates Spherical coordinates p 6 p E D angle between p and positive zaxis p is the distance from the origin to the surface You can think of it like the radius Useful conversions 29 cos cp D 2 p cosltp rp sinltlgt D r p sinltlgt x r cos6 y r sin6 x psinltp cos6 y psinltp sin6 p xx2 y2 22 p COS391ZX2 y2 22 If y gt 0 D 6 COS391XX2 y2 If y lt 0 D 6 2n COS391XX2 y2 Rules If p is constant it is a sphere If 6 is constant you get a half plane you don39t go the other way or else the angle is 6 Ti If p is constant and is less than n2 you get a cone If p is constant and in between n2 and n you get a cone opening downward In polar coordinates Area of An A6 r An A6 In spherical coordinates Eval Volume of Ap A6 A p 92 sinlcp Therefore when converting from Cartesian to spherical you must include p2 sinltp Eg Let E region of ball about 000 of radius 4 in the rst octant Compute IIIE z dV This is a box in spherical coordinates 0ltplt4 0ltltpltn2 Olt6ltn2 m5 2 W Io forquot2 forquot2 p coslcp 92 sinlcp dlcp dlp dle sinlcp coslcp 944 dlcp dle 64 Ion2 Ion2 sinlcp coslcp dlcp dle Ion2 l2 sinltp o 2 d6 32M 2 d6 1611 Eg Find the center of mass of E xyz 1 s x2 y2 z2 s 4 where z 2 0 Assume density 1 lsp2s4 lsps2 Z 2 0 D 0 s p s n2 M355 I02 Ion2 I12 1 DZ Sinltlgt 19 dltlgt 19 HIE 2 forquot2 I12 92 Sinlltlgtl 2n fem sinltp p33 12 2n3 81 Io l 2 sinltp dlt 14n3 fem sinltJ dltp 14n3 cosltp Ion2 14n3 01 14n3 Mxy HIE z dV I02quot for 2 If p coslcp 92 sinltp mm mm mm 2H for 2 944 sinlltlgtl I12 dltIgt n2 191 foil2 sinltp dltp 15n2 cosltpo 2 15Tl2 O1 15112 E Mxy14rl3 15n214r3 4528 y 0 and 7 0 by symmetry Therefore 292 004528 Eg Find the volume of the region below the sphere x2 y2 z2 1 and above the cone 2 xx2 y2 Set r xx2 y2 ZZ 1 r2andzr22r2 Therefore 1 r2 r2 l 2r2 1 l r 1x2 Therefore tanltp 1x212 I I n4 V I02quot for fol 1 92 sinltp dp mm mm 2n 93310 sinlcp dlcp quotIn spherical coordinates things are much more horrible All you have to do is look at a picture in the book and you ll go 39oh my god that39s horrible quot Catlin 0ltxlt10 0ltylt10 0 lt z lt 105qrt2 Pquot4sinphiquot3sinthetacostheta 0ltplt20 0 lt phi lt pi4 0 lt theta lt pi4 161 Vector Fields E 5 3 5 R 3 2 3 5 L2 3 EH all a h E F T if Tquot TE T I L m D 15L LJEIITampITE a u 1 55 3 5 j u E q E h a F E E 55d J E E IL R W 391 W F l R as 1EEEFEt 2 I I I I I I I 3 E 39 El 1 E 3 I Imagine a map showing the direction of the wind At each point there is an arrow Also if the wind is strong then the arrow is bigger This is a vector eld De nition Let D be a region in R A vector eld F is a function that assigns to each point xy in D a vector 2dimensional Fxy More precisely we can write FXy FxyT QXyT De nition Let E be a region in R3 A vector eld Fon E is a function that assigns each point xyz a 3dimensional vector Fxyz Wr can write Fxyz as Fxy2 PXyZT QXyzli RXy2Tlt You can use vector elds for many things We will say Fis continuous on E if P Q and R are continuous functions of xyz Eg Sketch F ltxygt Fxx ltxxgt lfy O Fx0 ltx0gt lfx O F0y lt0ygt Conclusion The vector eld Fpoints radially outward from the origin Rotational vector eld like a hurricane Fxy ltyxgt Vector eld gets stronger and stronger as you move away from the origin When y 0 F lt0xgt When x 0 F lty0gt When y x F ltxxgt or ltyygt Vector lines point in a circular motion centered around the origin Doldrums Fxy lty0gt Gravity Fxy ltxygt ltxygt xx2 y2 But gravity gets stronger as xy D 00 F ltX outward always has a magnitude 1 Where i ltxygt F ltxygtltxygt ltXX2 y2 yxx2 y2gt We want the magnutide to be inversely proportional to the square of distance from 00 F ltXX2 y23239 yXZ y232gt But we want F to point inward F ltXX2 y23239 yX2 y232gt Finally we have to multiply by the right constant C depending on units F ltX CX2 y23239 y gtllt CX2 y232gt The US Economy or the world economy etc Think of many possible quantities 1 Price of steel 2 Price of oil 3 Interest rate 4 Price of wheat 5 Etc One can model the economy based on many measures say 100 as a vector eld in 100 measures Given P1 P2 P100 the vector eld measures the expected value of how P1 P100 should change e P1 a1 P1 aln Pn 11 P239a2P1a2nPnq1 PnlanP1annPnq1 A LaGrangean Point is a point where things are relatively stable The net effect from the vectors in the vector eld is 0 There are ve hypothesized LaGrangean Points Placing an object in space at a speci c location will make it not move toward a speci c celestial body since the gravity is balanced Think of a building The location of the joints gives quantities P1 Pn Also the velocity at the joints gives quantities Ql Qn By Newton39s 2nCI or 3 law P1 a139 P1 an39 Pn 01 Q1 bn39 Qn Pnn ann P1 ann Pn bln Q bnn Qn Say an earthquake happens This is an external force with a vibration Does the frequency of the earthquake match up with the quotnatural frequencyquot of the building Resonance Fun Fact From Catin Galloping Gertie A building that started oscillating back and forth until it became too violent Kepler39s Laws A planet travels about the sun so the Sun is at the focus of an ellipse Newton invented calculus to show this The grandfather of all these laws is Isaac Newton Given a function fxy the gradient of F is delfxy fxxyT fyxyT The gradient def is always perpendicular to the level sets level surfaces The gradient is a vector As you move about it changes Eg Sketch the vector eld F xT yT When y 0 ltx0gt When x 0 lt0ygt The center of this is a saddle point If you go any bit to the side you will go right back down the side You can only cross the center if you were dead set on that path from the very beginning Whatever eld you may be in you will notice that vector elds are ubiquitous


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