Multivariable Advanced Calculus

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58 BASIC LINEAR ALGEBRA3.8.5 Orthonormal BasesNot all bases for an inner product space H are created equal.The best bases areorthonormal.Definition 3.8.11 Suppose {v 1 , · · · , v k } is a set of vectors in an inner productspace H. It is an orthonormal set ifv i · v j = δ ij ={ 1 if i = j0 if i ≠ jEvery orthonormal set of vectors is automatically linearly independent.Proposition 3.8.12 Suppose {v 1 , · · · , v k } is an orthonormal set of vectors. Thenit is linearly independent.Proof: Suppose ∑ ki=1 c iv i = 0. Then taking dot products with v j ,0 = 0 · v j = ∑ ic i v i · v j = ∑ ic i δ ij = c j .Since j is arbitrary, this shows the set is linearly independent as claimed.It turns out that if X is any subspace of H, then there exists an orthonormal basisfor X.Lemma 3.8.13 Let X be a subspace of dimension n whose basis is {x 1 , · · · , x n } .Then there exists an orthonormal basis for X, {u 1 , · · · , u n } which has the property thatfor each k ≤ n, span(x 1 , · · · , x k ) = span (u 1 , · · · , u k ) .Proof: Let {x 1 , · · · , x n } be a basis for X. Let u 1 ≡ x 1 / |x 1 | . Thus for k = 1,span (u 1 ) = span (x 1 ) and {u 1 } is an orthonormal set. Now suppose for some k
3.8. INNER PRODUCT AND NORMED LINEAR SPACES 59The vectors, {u j } n j=1, generated in this way are therefore an orthonormal basis becauseeach vector has unit length.The process by which these vectors were generated is called the Gram Schmidtprocess.3.8.6 The Adjoint Of A Linear TransformationThere is a very important collection of ideas which relates a linear transformation to theinner product in an inner product space. In order to discuss these ideas, it is necessaryto prove a simple and very interesting lemma about linear transformations which mapan inner product space H to the field of scalars, F. This is sometimes called the Rieszrepresentation theorem.Theorem 3.8.14 Let H be a finite dimensional inner product space and letL ∈ L (H, F) . Then there exists a unique z ∈ H such that for all x ∈ H,Lx = (x · z) .Proof: By the Gram Schmidt process, there exists an orthonormal basis for H,{e 1 , · · · , e n } .First note that if x is arbitrary, there exist unique scalars, x i such thatx =n∑x i e iTaking the dot product of both sides with e k yields( n)∑n∑(x · e k ) = x i e i · e k = x i (e i · e k ) =which shows thati=1i=1x =i=1i=1n∑(x · e i ) e iand so by the properties of the dot product,()n∑n∑Lx = (x · e i ) Le i = x· e i Le ii=1i=1n∑x i δ ik = x kso let z = ∑ ni=1 e iLe i . It only remains to verify z is unique. However, this is obviousbecause if (x · z 1 ) = (x · z 2 ) = Lx for all x, then(x · z 1 − z 2 ) = 0for all x and in particular for x = z 1 − z 2 which requires z 1 = z 2 . This proves thetheorem. Now with this theorem, it becomes easy to define something called the adjoint ofa linear operator. Let L ∈ L (H 1 , H 2 ) where H 1 and H 2 are finite dimensional innerproduct spaces. Then letting ( · ) idenote the inner product in H i ,i=1x → (Lx · y) 2
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4 CONTENTS5 Continuous Functions 85
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6 CONTENTS11.5 Integration Of Diffe
Page 8 and 9: 8 CONTENTS
Page 10 and 11: 10 INTRODUCTION
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Page 86 and 87: 86 CONTINUOUS FUNCTIONS≤ |a| |f (
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Page 90 and 91: 90 CONTINUOUS FUNCTIONSProof: First
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Page 98 and 99: 98 CONTINUOUS FUNCTIONS5.6 Polynomi
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Page 102 and 103: 102 CONTINUOUS FUNCTIONSTherefore,
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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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274 BROUWER DEGREEwhich is clearly
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276 BROUWER DEGREEThe product formu
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278 BROUWER DEGREEProof: Let B (y,3
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280 BROUWER DEGREEand y /∈ C. Sin
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282 BROUWER DEGREEwhich is the same
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284 BROUWER DEGREEFor y ∈ S, let
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286 BROUWER DEGREELetdist ( )y, WjC
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288 BROUWER DEGREE∞∑∫∞∑
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290 BROUWER DEGREE11. Establish the
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292 BROUWER DEGREE19. Using Problem
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294 INTEGRATION OF DIFFERENTIAL FOR
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326 THE LAPLACE AND POISSON EQUATIO
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350 THE JORDAN CURVE THEOREMNow the
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352 THE JORDAN CURVE THEOREMA solid
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354 THE JORDAN CURVE THEOREMLemma 1
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356 THE JORDAN CURVE THEOREMwhere R
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358 THE JORDAN CURVE THEOREMa QHKbN
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360 THE JORDAN CURVE THEOREM
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362 LINE INTEGRALSLet x t ≡ tx 1
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364 LINE INTEGRALSActually, people
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366 LINE INTEGRALSand if t is in th
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368 LINE INTEGRALSProof: The functi
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370 LINE INTEGRALSProof of the clai
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372 LINE INTEGRALS(γ s − 2h + t
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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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