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# 237_course_notes.pdf

Department
Mathematics
Course Code
MATH237
Professor
Diana Parry

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Calculus 3
Course Notes for MATH 237
Edition 4.21
J. Wainwright and D. Wolczuk
Department of Applied Mathematics
Copyright: J. Wainwright, August 1991
2nd Edition, July 1995
D. Wolczuk, 3rd Edition, April 2008
D. Wolczuk, 4th Edition, September 2009 (updated October 2010)

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Contents
Preface .................................... ii
To the Student Reader ........................... iii
Acknowledgements ............................. v
1Graphs of Scalar Functions 2
1.1 Scalar Functions .............................. 2
1.2 Geometric Interpretation of z=f(x, y).................. 4
2Limits 9
2.1 Deﬁnition of a Limit ............................ 9
2.2 Limit Theorems ............................... 10
2.3 Proving a Limit Does Not Exist ...................... 11
2.4 Proving a Limit Exists ........................... 14
3Continuous Functions 20
3.1 Deﬁnition of a Continuous Function .................... 20
3.2 The Continuity Theorems ......................... 22
3.3 Limits revisited ............................... 28
4The Linear Approximation 29
4.1 Partial Derivatives ............................. 29
4.2 Second Partial Derivatives ......................... 32
4.3 The Tangent Plane ............................. 35
4.4 Linear Approximation for z=f(x, y)................... 36
4.5 Linear Approximation in Higher Dimensions ............... 39
5Diﬀerentiable Functions 42
5.1 Deﬁnition of Diﬀerentiability ........................ 42
5.2 Diﬀerentiability and Continuity ...................... 47

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CONTENTS CONTENTS
5.3 Continuous Partial Derivatives and Diﬀerentiability ........... 49
5.4 The Linear Approximation Revisited ................... 52
6The Chain Rule 55
6.1 Basic Chain Rule in Two Dimensions ................... 55
6.2 Extensions of the Basic Chain Rule .................... 62
6.3 The Chain Rule for Second Partial Derivatives .............. 67
7Directional Derivatives and the Gradient Vector 72
7.1 Directional Derivatives ........................... 72
7.2 The Gradient Vector in Two Dimensions ................. 76
7.3 The Gradient Vector in Three Dimensions ................ 80
8Taylor Polynomials and Taylor’s Theorem 82
8.1 The Taylor Polynomial of Degree 2 .................... 82
8.2 Taylor’s Formula with Second Degree Remainder ............ 85
8.3 Generalizations ............................... 89
9Critical Points 91
9.1 Local Extrema and Critical Points ..................... 91
9.2 The Second Derivative Test ........................ 95
9.3 Proof of the Second Partial Derivative Test ................ 106
10 Optimization Problems 109
10.1 Extreme Value Theorem .......................... 109
10.2 Algorithm for Extreme Values ....................... 112
10.3 Optimization with Constraints ....................... 115
11 Coordinate Systems 124
11.1 Polar Coordinates .............................. 124
11.2 Cylindrical Coordinates .......................... 131
11.3 Spherical Coordinates ............................ 133
12 Mappings of R2into R2137
12.1 The Geometry of Mappings ........................ 138
12.2 The Linear Approximation of a Mapping ................. 142
12.3 Composite Mappings and the Chain Rule ................. 145
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