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Calculus Volker Runde

Text of Calculus Volker Runde

  • Advanced Honors Calculus, I and II

    (Fall 2004 and Winter 2005)

    Volker Runde

    August 17, 2006

  • Contents

    Introduction 3

    1 The real number system and finite-dimensional Euclidean space 4

    1.1 The real line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    1.3 The Euclidean space RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    1.4 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    2 Limits and continuity 40

    2.1 Limits of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

    2.2 Limits of functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

    2.3 Global properties of continuous functions . . . . . . . . . . . . . . . . . . 49

    2.4 Uniform continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

    3 Differentiation in RN 55

    3.1 Differentiation in one variable . . . . . . . . . . . . . . . . . . . . . . . . . 55

    3.2 Partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

    3.3 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

    3.4 Total differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

    3.5 Taylors theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

    3.6 Classification of stationary points . . . . . . . . . . . . . . . . . . . . . . . 76

    4 Integration in RN 82

    4.1 Content in RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

    4.2 The Riemann integral in RN . . . . . . . . . . . . . . . . . . . . . . . . . 85

    4.3 Evaluation of integrals in one variable . . . . . . . . . . . . . . . . . . . . 98

    4.4 Fubinis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

    4.5 Integration in polar, spherical, and cylindrical coordinates . . . . . . . . . 107

    1

  • 5 The implicit function theorem and applications 117

    5.1 Local properties of C1-functions . . . . . . . . . . . . . . . . . . . . . . . . 1175.2 The implicit function theorem . . . . . . . . . . . . . . . . . . . . . . . . . 120

    5.3 Local extrema with constraints . . . . . . . . . . . . . . . . . . . . . . . . 127

    6 Change of variables and the integral theorems by Green, Gau, and

    Stokes 132

    6.1 Change of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

    6.2 Curves in RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

    6.3 Curve integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

    6.4 Greens theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

    6.5 Surfaces in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

    6.6 Surface integrals and Stokes theorem . . . . . . . . . . . . . . . . . . . . 172

    6.7 Gau theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

    7 Infinite series and improper integrals 183

    7.1 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

    7.2 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

    8 Sequences and series of functions 201

    8.1 Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

    8.2 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

    8.3 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

    A Linear algebra 227

    A.1 Linear maps and matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

    A.2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

    A.3 Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

    B Stokes theorem for differential forms 237

    B.1 Alternating multilinear forms . . . . . . . . . . . . . . . . . . . . . . . . . 237

    B.2 Integration of differential forms . . . . . . . . . . . . . . . . . . . . . . . . 238

    B.3 Stokes theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

    C Limit superior and limit inferior 244

    C.1 The limit superior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

    C.2 The limit inferior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

    2

  • Introduction

    The present notes are based on the courses Math 217 and 317 as I taught them in the

    academic year 2004/2005. The main difference to the old version, which remains available

    online, is the addtion of figures

    These notes are not intended replace any of the many textbooks on the subject, but

    rather to supplement them by relieving the students from the necessity of taking notes

    and thus allowing them to devote their full attention to the lecture.

    It should be clear that these notes may only be used for educational, non-profit pur-

    poses.

    Volker Runde, Edmonton August 17, 2006

    3

  • Chapter 1

    The real number system and

    finite-dimensional Euclidean space

    1.1 The real line

    What is R?

    Intuitively, one can think of R as of a line stretching from to . Intuitition,however, can be deceptive in mathematics. In order to lay solid foundations for calculus,

    we introduce R from an entirely formalistic point of view: we demand from a certain set

    that it satisfies the properties that we intuitively expect R to have, and then just define

    R to be this set!

    What are the properties of R we need to do mathematics? First of all, we should be

    able to do arithmetic.

    Definition 1.1.1. A field is a set F together with two binary operations + and satisfyingthe following:

    (F1) For all x, y F, we have x+ y F and x y F as well.

    (F2) For all x, y F, we have x+ y = y + x and x y = y x (commutativity).

    (F3) For all x, y, z F, we have x + (y + z) = (x + y) + z and x (y z) = (x y) z((associativity).

    (F4) For all x, y, z F, we have x (y + z) = x y + x z (distributivity).

    (F5) There are 0, 1 F with 0 6= 1 such that for all x F, we have x+0 = x and x 1 = x(existence of neutral elements).

    (F6) For each x F, there is x F such that x + (x) = 0, and for each x F \ {0},there is x1 F such that x x1 = 1 (existence of inverse elements).

    4

  • Items (F1) to (F6) in Definition 1.1.1 are called the field axioms.

    For the sake of simplicity, we use the following shorthand notation:

    xy := x y;x+ y + z := x+ (y + z);

    xyz := x(yz);

    x y := x+ (y);x

    y:= xy1 (where y 6= 0);

    xn := x x n times

    (where n N);

    x0 := 1.

    Examples. 1. Q, R, and C are fields.

    2. Let F be any field then

    F(X) :=

    {p

    q: p and q are polynomials in X with coeffients in F and q 6= 0

    }

    is a field.

    3. Define + and on {A,B} through the following tables:

    + A B

    A A B

    B B A

    and

    A BA A A

    B A B

    This turns {A,B} into a field.

    4. Define + and on {,,}:

    +

    and

    This turns {,,} into a field.

    5. Let

    F[X] := {p : p is a polynomial in X with coefficients in F}.Then F[X] is not a field.

    5

  • 6. Both Z and N are not fields.

    There are several properties of a field that are not part of the field axioms, but which,

    nevertheless, can easily be deduced from them:

    1. The neutral elements 0 and 1 are unique: Suppose that both 01 and 02 are neutral

    elements for +. Then we have:

    01 = 01 + 02, by (F5),

    = 02 + 01, by (F2),

    = 02, again by (F5).

    A similar argument works for 1.

    2. The inverses x and x1 are uniquely determined by x: Let x 6= 0, and let y, z Fbe such that xy = xz = 1. Then we have:

    y = y(xz), by (F5) and (F6),

    = (yx)z, by (F3),

    = (xy)z, by (F2),

    = z(xy), again by (F2),

    = z, again by (F5) and (F6).

    A similar argument works for x.

    3. x0 = 0 for all x F.

    Proof. We have

    x0 = x(0 + 0), by (F5),

    = x0 + x0, by (F4).

    This implies

    0 = x0 x0, by (F6),= (x0 + x0) x0,= x0 + (x0 x0), by (F3),= x0,

    which proves the claim.

    4. (x)y = xy holds for all x, y R.

    6

  • Proof. We have

    xy + (x)y = (x x)y = 0.Uniqueness of xy then yields that (x)y = xy.

    5. For any x, y F, the identity

    (x)(y) = (x(y)) = (xy) = xy

    holds.

    6. If xy = 0, then x = 0 or y = 0.

    Proof. Suppose that x 6= 0, so that x1 exists. Then we have

    y = y(xx1) = (yx)x1 = 0,

    which proves the claim.

    Of course, Definition 1.1.1 is not enough to fully describe R. Hence, we need to take

    properties of R into account that are not merely arithmetic anymore:

    Definition 1.1.2. An ordered field is a field O together with a subset P with the following

    properties:

    (O1) For x, y P , we have x+ y P as well.

    (O2) For x, y P , we have xy P , as well.

    (O3) For each x O, exactly one of the following holds:

    (i) x P ;(ii) x = 0;

    (iii) x P .

    Again, we introduce shorthand notation:

    x < y : y x P ;x > y : y < x;x y : x < y or x = y;x y : x > y or x = y.

    As for the field axioms, there are several properties of odered fields that are not part of

    the order axioms (Definition 1.1.2(O1) to (O3)), but follow from them without too much

    trouble:

    7

  • 1. x < y and y < z implies x < z.

    Proof. If yx P and zy P , then (O1), im

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