Multivariable Advanced Calculus

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324 INTEGRATION OF DIFFERENTIAL FORMS
The Laplace And PoissonEquationsThis material is mostly in the book by Evans [14] which is where I got it. It is reallypartial differential equations but it is such a nice illustration of the divergence theoremand other advanced calculus theorems, that I am including it here even if it is somewhatout of place and would normally be encountered in a partial differential equations course.12.1 BallsRecall, B (x, r) denotes the set of all y ∈ R n such that |y − x| < r. By the change ofvariables formula for multiple integrals or simple geometric reasoning, all balls of radiusr have the same volume. Furthermore, simple reasoning or change of variables formulawill show that the volume of the ball of radius r equals α n r n where α n will denote thevolume of the unit ball in R n . With the divergence theorem, it is now easy to give asimple relationship between the surface area of the ball of radius r and the volume. Bythe divergence theorem,∫B(0,r)∫div x dx =∂B(0,r)x· x|x| dσ n−1because the unit outward normal on ∂B (0, r) is x|x| . Therefore,nα n r n = rσ n−1 (∂B (0, r))and soσ n−1 (∂B (0, r)) = nα n r n−1 .You recall the surface area of S 2 ≡ { x ∈ R 3 : |x| = r } is given by 4πr 2 while the volumeof the ball, B (0, r) is 4 3 πr3 . This follows the above pattern. You just take the derivativewith respect to the radius of the volume of the ball of radius r to get the area of thesurface of this ball. Let ω n denote the area of the sphere S n−1 = {x ∈ R n : |x| = 1} . Ijust showed thatω n = nα n . (12.1)I want to find α n now and also to get a relationship between ω n and ω n−1 . Considerthe following picture of the ball of radius ρ seen on the side.325
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4 CONTENTS5 Continuous Functions 85
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6 CONTENTS11.5 Integration Of Diffe
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8 CONTENTS
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10 INTRODUCTION
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12 SOME FUNDAMENTAL CONCEPTSthere i
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14 SOME FUNDAMENTAL CONCEPTSthat is
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16 SOME FUNDAMENTAL CONCEPTSFollow
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18 SOME FUNDAMENTAL CONCEPTSDefinit
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20 SOME FUNDAMENTAL CONCEPTSTheorem
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22 SOME FUNDAMENTAL CONCEPTSNext ob
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24 BASIC LINEAR ALGEBRA6 · (2, 6)(
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26 BASIC LINEAR ALGEBRA3.2 Subspace
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28 BASIC LINEAR ALGEBRATheorem 3.2.
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30 BASIC LINEAR ALGEBRAProof: This
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32 BASIC LINEAR ALGEBRAwhich showsL
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34 BASIC LINEAR ALGEBRAsuch thatl i
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36 BASIC LINEAR ALGEBRA3.4 Block Mu
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38 BASIC LINEAR ALGEBRAnumbers in (
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40 BASIC LINEAR ALGEBRAProof: Let (
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42 BASIC LINEAR ALGEBRABy Corollary
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44 BASIC LINEAR ALGEBRAProposition
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46 BASIC LINEAR ALGEBRADefinition 3
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48 BASIC LINEAR ALGEBRAcontrary to
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50 BASIC LINEAR ALGEBRALemma 3.6.2
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52 BASIC LINEAR ALGEBRA3. Let L ∈
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54 BASIC LINEAR ALGEBRATheorem 3.8.
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56 BASIC LINEAR ALGEBRALemma 3.8.9
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58 BASIC LINEAR ALGEBRA3.8.5 Orthon
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60 BASIC LINEAR ALGEBRAis in L (H 1
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62 BASIC LINEAR ALGEBRA3.8.7 Schur
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64 BASIC LINEAR ALGEBRAProof: Let {
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66 BASIC LINEAR ALGEBRANow for x, y
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68 BASIC LINEAR ALGEBRAby 3.47.Sinc
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70 BASIC LINEAR ALGEBRA8. A normal
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72 SEQUENCESTheorem 4.1.5 Suppose {
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74 SEQUENCESTheorem 4.1.8 Let {x n
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76 SEQUENCESTheorem 4.3.2 The inter
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78 SEQUENCESHowever, this sequence
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80 SEQUENCESGiven R is complete, th
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82 SEQUENCES10. Suppose A ⊆ R n a
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84 SEQUENCES
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86 CONTINUOUS FUNCTIONS≤ |a| |f (
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88 CONTINUOUS FUNCTIONSof f −1 (U
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90 CONTINUOUS FUNCTIONSProof: First
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92 CONTINUOUS FUNCTIONSlet C ∩ (
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94 CONTINUOUS FUNCTIONSProof: Suppo
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96 CONTINUOUS FUNCTIONSThen the fol
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98 CONTINUOUS FUNCTIONS5.6 Polynomi
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100 CONTINUOUS FUNCTIONSThus the ex
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102 CONTINUOUS FUNCTIONSTherefore,
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104 CONTINUOUS FUNCTIONSProof: Let
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106 CONTINUOUS FUNCTIONSand so ( )3
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108 CONTINUOUS FUNCTIONSLet( )a1b k
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110 CONTINUOUS FUNCTIONSTheorem 5.8
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112 CONTINUOUS FUNCTIONS5.9 Ascoli
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114 CONTINUOUS FUNCTIONSBy equicont
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116 CONTINUOUS FUNCTIONS11. By Theo
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118 CONTINUOUS FUNCTIONS28. Let X b
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120 CONTINUOUS FUNCTIONS
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122 THE DERIVATIVEor equivalently,|
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124 THE DERIVATIVEIt follows≡limt
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126 THE DERIVATIVEand soNow dividin
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128 THE DERIVATIVEprovided ||v|| is
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130 THE DERIVATIVEare both linear.T
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132 THE DERIVATIVE6.7.1 Some Standa
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134 THE DERIVATIVEDefinition 6.8.4
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136 THE DERIVATIVEAs implied above,
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138 THE DERIVATIVEwhich requires ea
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140 THE DERIVATIVETheorem 6.10.3 (i
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142 THE DERIVATIVEThen there exist
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144 THE DERIVATIVENow the idea is t
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146 THE DERIVATIVETheorem 6.11.5 If
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148 THE DERIVATIVEare linearly inde
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150 THE DERIVATIVEShow {x 1 , · ·
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152 THE DERIVATIVE25. Let (x, y) be
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154 MEASURES AND MEASURABLE FUNCTIO
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182 THE ABSTRACT LEBESGUE INTEGRALL
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184 THE ABSTRACT LEBESGUE INTEGRALa
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186 THE ABSTRACT LEBESGUE INTEGRALD
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192 THE ABSTRACT LEBESGUE INTEGRALP
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194 THE ABSTRACT LEBESGUE INTEGRALP
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196 THE ABSTRACT LEBESGUE INTEGRALF
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198 THE ABSTRACT LEBESGUE INTEGRALT
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202 THE ABSTRACT LEBESGUE INTEGRALC
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204 THE ABSTRACT LEBESGUE INTEGRALN
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206 THE ABSTRACT LEBESGUE INTEGRALc
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208 THE LEBESGUE INTEGRAL FOR FUNCT
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258 BROUWER DEGREEDefinition 10.1.1
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260 BROUWER DEGREEIt is obvious tha
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262 BROUWER DEGREETherefore, for an
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264 BROUWER DEGREE= −A.Therefore,
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266 BROUWER DEGREEProof: From Lemma
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268 BROUWER DEGREEfor all t ∈ [0,
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270 BROUWER DEGREELemma 10.3.2 Let
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272 BROUWER DEGREELemma 10.4.2 Let
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Page 284 and 285: 284 BROUWER DEGREEFor y ∈ S, let
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Page 290 and 291: 290 BROUWER DEGREE11. Establish the
Page 292 and 293: 292 BROUWER DEGREE19. Using Problem
Page 294 and 295: 294 INTEGRATION OF DIFFERENTIAL FOR
Page 326 and 327: 326 THE LAPLACE AND POISSON EQUATIO
Page 350 and 351: 350 THE JORDAN CURVE THEOREMNow the
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Page 354 and 355: 354 THE JORDAN CURVE THEOREMLemma 1
Page 356 and 357: 356 THE JORDAN CURVE THEOREMwhere R
Page 358 and 359: 358 THE JORDAN CURVE THEOREMa QHKbN
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Page 362 and 363: 362 LINE INTEGRALSLet x t ≡ tx 1
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374 LINE INTEGRALS≤≤n∑∫ tj|
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376 LINE INTEGRALSand so|F (γ (t))
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378 LINE INTEGRALSLet m be the last
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380 LINE INTEGRALSLemma 14.3.5 In t
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382 LINE INTEGRALSLemma 14.3.6 Let
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384 LINE INTEGRALSwheref (x, y) ≡
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386 LINE INTEGRALSwhere B δ is the
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388 LINE INTEGRALSto Green’s theo
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390 LINE INTEGRALSDefinition 14.4.2
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392 LINE INTEGRALS14.5 Interpretati
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394 LINE INTEGRALS✻a × b✣θc
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396 LINE INTEGRALSSuppose then that
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398 LINE INTEGRALSProposition 14.6.
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400 LINE INTEGRALSCombining these l
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402 LINE INTEGRALSIn this case the
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404 LINE INTEGRALSThen as in the pr
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406 LINE INTEGRALSConsider only h
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408 LINE INTEGRALSwhere |g (w)| <
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410 LINE INTEGRALS6. In the situati
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412 LINE INTEGRALS18. Using Problem
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414 LINE INTEGRALS24. Let f, g be a
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416 LINE INTEGRALSShow this equals
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418 LINE INTEGRALS
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420 HAUSDORFF MEASURES AND AREA FOR
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460 BIBLIOGRAPHY[21] Gurtin M. An i
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462 INDEXderivative, 307exact, 322i
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464 INDEXvectors, 25Vitaliconvergen
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Multivariable Advanced Calculus

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