Multivariate Calculus - Bruce E. Shapiro

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ii CONTENTS 20 Triple Integrals 161 21 Vector Fields 169 22 Line Integrals 175 23 Green’s Theorem 191 24 Flux Integrals & Gauss’ Divergence Theorem 195 25 Stokes’ Theorem 201 Revised December 6, 2006. Math 250, Fall 2006
Preface: A note to the Student These lecture notes are meant to accompany the textbook, not to replace it. It has been my experience that students who attempt to use lecture notes (even mine, which I have to admit, are practically perfect) in lieu of the textbook receive abysmally poor final term grades. However, properly using the lecture notes can help you. Here is how I would suggest you study in this class. • Read the pertinent sections in the textbook prior to attending the class lecture. • Come to class. • Pay attention. Although I use the notes during class, I will not follow the notes verbatum, and I will most certainly not stand in front of the class and read the notes out loud to you. I will frequently also include additional material so you need to keep sharp. • Take notes in class. Some students find it helpful to have a copy of these lecture notes in front of them, but others do not. It is a personal style, and you will have to decide what is right for you. • Review your notes after class, before you try to do the homework. Compare your notes with the lecture notes; you can use the lecture notes to check to see if you copied the formulas correctly, or if you don’t understand something in your notes. • Reread or at least skim back over the chapter after reviewing your notes. • Do the exercises at the end of the chapter. Note that this step comes AFTER you have read the chapter, not before. • If you don’t understand something, go back over the chapter, your class notes, and the lecture notes, in that order. If that doesn’t help, ask me during office hours, during class, or via email for help. Sometimes you will run into something in either the notes or the text that just plain does not make sense. If you can manage it, skip over the problem, or go on with the next step or sentence in what you are reading. Come back to it later, iii
Page 1 and 2: Multivariate Calculus in 25 Easy Le
Page 3: Contents 1 Cartesian Coordinates 1
Page 7 and 8: CONTENTS v The order in which the m
Page 9 and 10: Examples of Typical Symbols Used Sy
Page 11 and 12: CONTENTS ix Table 1: Symbols Used i
Page 13 and 14: Lecture 1 Cartesian Coordinates We
Page 15 and 16: LECTURE 1. CARTESIAN COORDINATES 3
Page 21 and 22: Lecture 2 Vectors in 3D Properties
Page 23 and 24: LECTURE 2. VECTORS IN 3D 11 Figure
Page 25 and 26: LECTURE 2. VECTORS IN 3D 13 Definit
Page 27 and 28: LECTURE 2. VECTORS IN 3D 15 Hence u
Page 29 and 30: LECTURE 2. VECTORS IN 3D 17 and the
Page 31 and 32: LECTURE 2. VECTORS IN 3D 19 The Equ
Page 33 and 34: Lecture 3 The Cross Product Definit
Page 35 and 36: LECTURE 3. THE CROSS PRODUCT 23 Pro
Page 37 and 38: LECTURE 3. THE CROSS PRODUCT 25 Exa
Page 39 and 40: LECTURE 3. THE CROSS PRODUCT 27 5.
Page 41 and 42: Lecture 4 Lines and Curves in 3D We
Page 43 and 44: LECTURE 4. LINES AND CURVES IN 3D 3
Page 49 and 50: Lecture 5 Velocity, Acceleration, a
Page 51 and 52: LECTURE 5. VELOCITY, ACCELERATION,
Page 53 and 54: LECTURE 5. VELOCITY, ACCELERATION,
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LECTURE 5. VELOCITY, ACCELERATION,
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Lecture 6 Surfaces in 3D The text f
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Lecture 7 Cylindrical and Spherical
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LECTURE 7. CYLINDRICAL AND SPHERICA
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Lecture 8 Functions of Two Variable
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LECTURE 8. FUNCTIONS OF TWO VARIABL
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Lecture 9 The Partial Derivative De
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LECTURE 9. THE PARTIAL DERIVATIVE 6
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Lecture 10 Limits and Continuity In
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LECTURE 10. LIMITS AND CONTINUITY 6
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LECTURE 10. LIMITS AND CONTINUITY 7
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Lecture 11 Gradients and the Direct
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LECTURE 11. GRADIENTS AND THE DIREC
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Lecture 12 The Chain Rule Recall th
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LECTURE 12. THE CHAIN RULE 83 The p
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LECTURE 12. THE CHAIN RULE 85 Examp
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LECTURE 12. THE CHAIN RULE 87 becau
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LECTURE 12. THE CHAIN RULE 89 Solut
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LECTURE 12. THE CHAIN RULE 91 Examp
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Lecture 13 Tangent Planes Since the
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LECTURE 13. TANGENT PLANES 95 Solut
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LECTURE 13. TANGENT PLANES 97 Multi
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LECTURE 13. TANGENT PLANES 99 Accor
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Lecture 14 Unconstrained Optimizati
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LECTURE 14. UNCONSTRAINED OPTIMIZAT
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Lecture 15 Constrained Optimization
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LECTURE 15. CONSTRAINED OPTIMIZATIO
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Lecture 16 Double Integrals over Re
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LECTURE 16. DOUBLE INTEGRALS OVER R
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Lecture 17 Double Integrals over Ge
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LECTURE 17. DOUBLE INTEGRALS OVER G
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Lecture 18 Double Integrals in Pola
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LECTURE 18. DOUBLE INTEGRALS IN POL
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Lecture 19 Surface Area with Double
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LECTURE 19. SURFACE AREA WITH DOUBL
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LECTURE 19. SURFACE AREA WITH DOUBL
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Lecture 20 Triple Integrals Triple
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LECTURE 20. TRIPLE INTEGRALS 163 Fi
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LECTURE 20. TRIPLE INTEGRALS 165 so
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LECTURE 20. TRIPLE INTEGRALS 167 Tr
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Lecture 21 Vector Fields Definition
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LECTURE 21. VECTOR FIELDS 171 Defin
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LECTURE 21. VECTOR FIELDS 173 Examp
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Lecture 22 Line Integrals Suppose t
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LECTURE 22. LINE INTEGRALS 177 Exam
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LECTURE 22. LINE INTEGRALS 179 ener
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LECTURE 22. LINE INTEGRALS 181 The
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LECTURE 22. LINE INTEGRALS 183 so t
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LECTURE 22. LINE INTEGRALS 185 wher
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Lecture 23 Green’s Theorem Theore
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LECTURE 23. GREEN’S THEOREM 193 T
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Lecture 24 Flux Integrals & Gauss
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LECTURE 24. FLUX INTEGRALS & GAUSS
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LECTURE 24. FLUX INTEGRALS & GAUSS
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Lecture 25 Stokes’ Theorem We hav
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LECTURE 25. STOKES’ THEOREM 203 S
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Multivariate Calculus - Bruce E. Shapiro

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