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Advanced Calculus Folland Solutions Manual PDF
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Complete Answers Solutions Manual PDF for Advanced Calculus by Gerald B Folland. Includes the answers for all of the exercises of the book.
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Instructor’s Solution Manual forADVANCED CALCULUS
Gerald B. Folland
,
,Contents
1 Setting the Stage 1
1.1 Euclidean Spaces and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Subsets of Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.7 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.8 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Differential Calculus 8
2.1 Differentiability in One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Differentiability in Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 The Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Functional Relations and Implicit Functions: A First Look . . . . . . . . . . . . . . . . . . 10
2.6 Higher-Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.8 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.9 Extreme Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.10 Vector-Valued Functions and Their Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 17
3 The Implicit Function Theorem and its Applications 19
3.1 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Curves in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Surfaces and Curves in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Transformations and Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Functional Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Integral Calculus 25
4.1 Integration on the Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 Integration in Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.3 Multiple Integrals and Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Change of Variables for Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.5 Functions Defined by Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.6 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.7 Improper Multiple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
iii
, iv Contents
5 Line and Surface Integrals; Vector Analysis 34
5.1 Arc Length and Line Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.2 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.3 Surface Area and Surface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.4 Vector Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.5 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.6 Some Applications to Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.7 Stokes’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.8 Integrating Vector Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 Infinite Series 43
6.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 Series with Nonnegative Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Absolute and Conditional Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.4 More Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.5 Double Series; Products of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7 Functions Defined by Series and Integrals 49
7.1 Sequences and Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2 Integrals and Derivatives of Sequences and Series . . . . . . . . . . . . . . . . . . . . . . . 50
7.3 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.4 The Complex Exponential and Trig Functions . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.5 Functions Defined by Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.6 The Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.7 Stirling’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8 Fourier Series 59
8.1 Periodic Functions and Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.2 Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.3 Derivatives, Integrals, and Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . 61
8.4 Fourier Series on Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
8.5 Applications to Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
8.6 The Infinite-Dimensional Geometry of Fourier Series . . . . . . . . . . . . . . . . . . . . . 65
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