3. Analysis

数学英语



周维祺

Analysis

The part of mathematics in which functions and their generalizations are studied by the method of limits

Index

  • Topological terms

  • Limits and continuity

  • Differentiation and integration

(Punctured) Neighborhood

Interior/Exterior/Boundary

Open sets/Closed sets

Connected/Disconnected

Accumulation Point

Isolated Point

Convex vs. Concave

Bounded/Unbounded/Compact

  • A set is bounded if it is in some neighborhood of the origin

  • A set is unbounded if it is not bounded

  • A set is compact if every open cover of it admits a finite subcover
    In a set is compact if it is bounded and closed

Game


Two players are placing stones alternatively in a standard checkerboard (an chess board)

Game


Player A starts from the top left corner

In each turn stones must be placed in an empty square that is adjacent to the square occupied in the last turn

Game


The player who can not make a move in his/her turn loses

Play and find a best strategy

Game Variations

  • I: Now we are playing on a board

  • II: In addition, now A starts from a square that is adjacent to the top left one

Distances

  • In , is the absolute value of

  • In , is the modulus of

  • In a normed space, is the norm of a vector

  • or is the distance between

Sequences

  • Arithmetic progression:

  • Geometric sequence:

Series and Partial Sums

Limits

  • The limit of over as goes to infinity is , the sequence to

  • The sequence has no limit, it

Exercise

Find where

Find the value of (and prove its convergence)

Continuity

  • The function is continuous

  • The function is discontinuous / not continuous

  • The function has a pole / singularity at

Differentiability

  • The function is differentiable everywhere
    Its (first order) derivative is

  • The function is not differentiable at

  • A function is twice (resp. times) differentiable if its second order derivative (resp. -th order derivative ) exists

  • A function is smooth if all orders of derivatives exist for it

Notations

  • : prime

  • : bar / overline

  • : hat

  • : check

  • : tilde

  • : naught

Ordinary/Partial Derivatives

  • is the ordinary derivative

  • Suppose now is a differentiable function

  • is the partial derivative with respect to (the variable)
    (partial over partial )

  • is the directional derivative

Rules of Differentiations

  • Leibniz rule (product rule):

  • Chain rule:

  • Implicit functions (functions hidden in equations)

Integration

  • Riemann integral: limit of a Riemann sum

  • Lebesgue integral: integrate against a measure

  • Other types of integrals

Rules of Integrations

  • Change of variable:

  • Integration by parts:

Expansions

  • Taylor expansion (at ):
    is analytic at : it has a positive radius of convergence at

  • Laurent expansion (at ):

  • Fourier expansion (in unit interval):

Typical Subfields of Analysis

  • Harmonic analysis

  • Complex analysis

  • Functional analysis

  • Real analysis

  • Geometric analysis

  • Numerical analysis

Lemma

  • Lemma is a technical statement (for proving a theorem later)
    Plural: lemmata / lemmas

  • Whether a statement should be a lemma or a theorem depends on its significance and technality in your writings

  • Examples: Zorn's Lemma, Fatou's Lemma, Schwartz Lemma

Tasks

Make the exercise in this lecture into a lemma, and write a proof for it

Summary

  • Topological terms and concepts

  • Limits

  • Calculus

  • Lemma