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math234 third semester calculus fall 2009 1 2 math 234 3rd semester calculus lecture notes version 0 9 fall 2009 this is a self contained set of lecture notes for ...

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                                                                                          MATH234
                                                                             THIRD SEMESTER
                                                                                         CALCULUS
                                                                                                 Fall 2009
                                                                   1
                            2
                                                 Math 234 – 3rd Semester Calculus
                                                 Lecture notes version 0.9(Fall 2009)
                            This is a self contained set of lecture notes for Math 234. The notes were written by Sigurd
                            Angenent, many problems and parts of the text were taken from Guichard’s open calculus
                            text which is available at http://www.whitman.edu/mathematics/multivariable/src/
                                     A
                                The LT X files, as well as the Python and Inkscape-svg files which were used to
                                       E
                            produce the notes before you can be obtained from the following web site
                                        http://www.math.wisc.edu/ angenent/Free-Lecture-Notes
                                                                 ~
                            They are meant to be freely available for non-commercial use, in the sense that “free
                            software” is free. More precisely:
                                Copyright (c) 2009 Sigurd B. Angenent. Permission is granted to copy, distribute and/or
                                modify this document under the terms of the GNU Free Documentation License, Version 1.2
                                or any later version published by the Free Software Foundation; with no Invariant Sections,
                                no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the
                                section entitled ”GNU Free Documentation License”.
                                                                                     Contents
                                       Chapter 1.   Functions of two and more variables.                                                            5
                                           1.  n-dimensional space                                                                                  5
                                           2.  Functions of two or more variables                                                                   5
                                           2.1.  The graph of a function                                                                            5
                                           2.2.  Vector notation                                                                                    5
                                           2.3.  Example                                                                                            5
                                           2.4.  Example                                                                                            6
                                           2.5.  Freezing a variable                                                                                6
                                           2.6.  Example – draw the graph of f(x,y) = xy                                                            6
                                           2.7.  The domain of a function                                                                           7
                                           2.8.  Example                                                                                            7
                                           3.  Open and closed sets in Rn                                                                           8
                                           3.1.  Example                                                                                            8
                                           4.  More examples of visualization of Functions                                                          8
                                           4.1.  Example                                                                                            9
                                           4.2.  Level sets of the saddle surface                                                                   9
                                           4.3.  An example from the “real” world                                                                  10
                                           4.4.  Moving graphs                                                                                     10
                                           Problems                                                                                                11
                                           Problems about movies                                                                                   12
                                           About open and closed sets                                                                              13
                                           5.  Continuity and Limits                                                                               13
                                           5.1.  The limit of a function of two variables                                                          13
                                           5.2.  Definition                                                                                         13
                                           5.3.  Definition of Continuity                                                                           13
                                           5.4.  Iterated limits                                                                                   14
                                           5.5.  Theorem on Switching Limits                                                                       14
                                           5.6.  Limit examples                                                                                    15
                                           6.  Problems                                                                                            16
                                       Chapter 2.   Derivatives                                                                                    17
                                           1.  Partial Derivatives                                                                                 17
                                           1.1.  Definition of Partial Derivatives                                                                  17
                                           1.2.  Examples                                                                                          17
                                           2.  Problems                                                                                            18
                                           3.  The Chain Rule and friends                                                                          18
                                           3.1.  Linear approximation of a graph                                                                   18
                                           3.2.  The tangent plane to a graph                                                                      20
                                           3.3.  Example: tangent plane to the sphere                                                              21
                                           3.4.  Example: tangent planes to the saddle surface                                                     21
                                           3.5.  Example: another tangent plane to the saddle surface                                              22
                                           3.6.  Follow-up problem – intersection of tangent plane and graph                                       22
                                           3.7.  The Chain Rule                                                                                    22
                                           3.8.  The difference between d and ∂                                                                     23
                                           4.  Problems                                                                                            23
                                           5.  Gradients                                                                                           24
                                           5.1.  The gradient vector of a function                                                                 24
                                           5.2.  The gradient as the “direction of greatest increase” for a function f                             24
                                           5.3.  The gradient is perpendicular to the level curve                                                  25
                                           5.4.  The chain rule and the gradient of a function of three variables                                  25
                                           5.5.  Tangent plane to a level set                                                                      27
                                           5.6.  Example                                                                                           27
                                           6.  Implicit Functions                                                                                  28
                                           6.1.  The Implicit Function Theorem                                                                     29
                                           6.2.  The Implicit Function Theorem with more variables                                                 29
                                           6.3.  Example – The saddle surface again                                                                30
                                           7.  The Chain Rule with more Independent Variables;
                                                  Coordinate Transformations                                                                       30
                                                                                              3
                                             4                                                      CONTENTS
                                                 7.1.  An example without context                                                                                      30
                                                 7.2.  Example: a rotated coordinate system                                                                            31
                                                 7.3.  Another example – Polar coordinates                                                                             32
                                                 Problems about the Gradient and Level Curves                                                                          32
                                                 About the chain rule and coordinate transformations                                                                   34
                                                 8.  Higher Partials and Clairaut’s Theorem                                                                            37
                                                 8.1.  Higher partial derivatives                                                                                      37
                                                 8.2.  Example                                                                                                         37
                                                 8.3.  Clairaut’s Theorem – mixed partials are equal                                                                   37
                                                 8.4.  Proof of Clairaut’s theorem                                                                                     37
                                                 8.5.  Finding a function from its derivatives                                                                         38
                                                 8.6.  Example                                                                                                         38
                                                 8.7.  Example                                                                                                         38
                                                 8.8.  Theorem                                                                                                         39
                                                 9.  Problems                                                                                                          39
                                             Chapter 3.    Maxima and Minima                                                                                           41
                                                 1.  Local and Global extrema                                                                                          41
                                                 1.1.  Definition of global extrema                                                                                     41
                                                 1.2.  Definition of local extrema                                                                                      41
                                                 1.3.  Interior extrema                                                                                                41
                                                 2.  Continuous functions on closed and bounded sets                                                                   42
                                                 2.1.  Theorem about Maxima and Minima of Continuous Functions                                                         42
                                                 2.2.  Example – The function f(x,y) = x2 + y2                                                                         42
                                                 2.3.  A fishy example                                                                                                  42
                                                 3.  Problems                                                                                                          43
                                                 4.  Critical points                                                                                                   43
                                                 4.1.  Theorem. Local extrema are critical points                                                                      44
                                                 4.2.  Three typical critical points                                                                                   44
                                                 4.3.  Critical points in the fishy example                                                                             45
                                                 4.4.  Another example: Find the critical points of f(x,y) = x − x3 − xy2                                              46
                                                 5.  When you have more than two variables                                                                             46
                                                 6.  Problems                                                                                                          47
                                                 7.  A Minimization Problem: Linear Regression                                                                         48
                                                 8.  Problems                                                                                                          49
                                                 9.  The Second Derivative Test                                                                                        50
                                                 9.1.  The one-variable second derivative test                                                                         50
                                                 9.2.  Taylor’s formula for a function of several variables                                                            50
                                                 9.3.  Example: Compute the Taylor expansion of f(x,y) = sin2xcosy at the point (1π, 1π)                               51
                                                                                                                   3     3                      6    6
                                                 9.4.  Another example: the Taylor expansion of f(x,y) = x + y − 3xy at the point (1,1)                                52
                                                 9.5.  Example of a saddle point                                                                                       53
                                                 9.6.  The two-variable second derivative test                                                                         53
                                                 Theorem (second derivative test)                                                                                      53
                                                 9.7.  Example: Apply the second derivative test to the fishy example                                                   54
                                                 10.   Problems                                                                                                        54
                                                 11.   Second derivative test for more than two variables                                                              55
                                                 11.1.   The second order Taylor expansion                                                                             55
                                                 12.   Optimization with constraints                                                                                   56
                                                 12.1.   Solution by elimination or parametrization                                                                    56
                                                 12.2.   Example                                                                                                       56
                                                 12.3.   Example                                                                                                       57
                                                 12.4.   Solution by Lagrange multipliers                                                                              57
                                                 12.5.   Theorem (Lagrange multipliers)                                                                                57
                                                 12.6.   Example                                                                                                       58
                                                 12.7.   A three variable example                                                                                      58
                                                 13.   Problems                                                                                                        59
                                             Chapter 4.    Integrals                                                                                                   61
                                                 1.  Overview                                                                                                          61
                                                 1.1.  The one variable integral                                                                                       61
                                                 1.2.  Generalizing the one variable integral                                                                          62
                                                 2.  Double Integrals                                                                                                  62
                                                 2.1.  Definition                                                                                                       63
                                                 2.2.  The integral is the volume under the graph, when f ≥ 0                                                          64
                                                 2.3.  How to compute a double integral                                                                                65
                                                 2.4.  Theorem                                                                                                         67
                                                                                                                                2      2
                                                 2.5.  Example: the volume under the graph of the paraboloid z = x + y                   above the square
                                                                      Q={(x,y):0≤x≤1,0≤y≤1}                                                                            67
                                                 2.6.  Double integrals when the domain is not a rectangle                                                             68
                                                 2.7.  An example–the parabolic office building                                                                          69
                                                 2.8.  Double integrals in Polar Coordinates                                                                           70
                                                 2.9.  Example: the volume under a quarter turn of a helicoid                                                          72
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...Math third semester calculus fall rd lecture notes version this is a self contained set of for the were written by sigurd angenent many problems and parts text taken from guichard s open which available at http www whitman edu mathematics multivariable src lt x les as well python inkscape svg used to e produce before you can be obtained following web site wisc free they are meant freely non commercial use in sense that software more precisely copyright c b permission granted copy distribute or modify document under terms gnu documentation license any later published foundation with no invariant sections front cover texts back included section entitled contents chapter functions two variables n dimensional space graph function vector notation example freezing variable draw f y xy domain closed sets rn examples visualization level saddle surface an real world moving graphs about movies continuity limits limit denition iterated theorem on switching derivatives partial chain rule friends l...

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