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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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