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Department
Mathematics
Course Code
MAT237Y1
Professor
all

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Instructor’sSolution Manual for
ADVANCED CALCULUS
Gerald B. Folland
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Contents
1Setting 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
2Differential 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: AFirst Look . . . . . . . . . . . . . . . . . . 10
2.6 Higher-Order Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 Taylor’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.8 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.9 Extreme Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.10 Vector-Valued Functions and Their Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 17
3The 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
4Integral 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
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