The City College CUNY Class: MATH 32400 Advanced Calc. II

Professor: Thea Pignataro

Read Ross secs 13, 21 & 22.

Review linear algebra: Subspace, spanning set, linear independence, dimension, basis, kernel, range, std. basis, affine subspace.

Prof's notes onĀ http://1drv.ms/1KlFihR, mirror: link1

Sec. 1.6

Def: a normed real vect. space. (Theorem 1.6.6)

Generalization of vector length

Inner product space (theorem 1.6.6 I)

Remark: Every inner product induces a norm, but not every norm can be represented as an inner product. Nor does every metric arises from a norm.

Sec. 1.7 Matrices, etc.

Def. 1. 7.1 A metric space.

Check example 1.7.2 (discrete methods), many set, d(x, y) = 0 if x=y and d(x,y) = 1 if x != y. Check that this is a metric.

This entry was posted in Mathematics, Projects. Bookmark the permalink.

9 Responses to The City College CUNY Class: MATH 32400 Advanced Calc. II

  1. timlyg says:

    02/03/2016
    Ch. 2 Topology of Metric Spaces
    Open sets
    Def. 2.1.1
    Prop. 2.1.3 (i easy, ii proven, iii trivial)

    Interior 2.2
    Def. 2.2.1

    Closed 2.3
    Def.2.3.1

  2. timlyg says:

    02/08/2016
    HW #01
    1.6 #1,3,4;
    1.7 #2,3;
    Ch 1 #10,12, read 14&20
    2.1 #1,3;
    2.2 #1;
    2.3 #1,2,5
    2.4 #4
    2.5 #1,3,4
    2.6 #2,3
    2.7 #1
    2.8 #2,3
    2.9 #3
    Ch 2 #44(read) 19, 22

    2.4 Accumulation Points (pg. 127)
    Theorem (2.4.2)

    2.5 Closure
    Definition 2.5.1
    Remark: So cl(A) is the smallest closed set containing A.

    2.6 Boundary
    Def. 2.6.1
    ie pts of A & pts of M\A = = pts. not in A arbit. close to any boundary pt.
    remark: bd(A) is closed
    Examples given

    2.7 Sequences
    Definition: 2.7.1

    Prop. 2.7.2 (pg. 134)
    Remark: 1. , then any open set containing x contains all but finitely many .
    2. a lot of seq. then carry over to metric or normed spaces (e.g. prop. 2.7.3) but there is no longer a notion of order. Therefore, no notion of monotonic (unless we are talking about R) or sup/inf of subsets of M (only of sets of distances . Need to deal with bdd.
    3. Limits in metric spaces are unique (pf. just like for R since triangle ineq. holds for d).

    Prop. 2.7.4
    Proof. n finite is crucial
    outline of pf. in book.


    So take N =

  3. timlyg says:

    02/10/2016
    Prop. 2.7.6, with proofs.

    Prop. 2.8 Completeness (check all), all of 2.9

    Chapter 3. Compact & Connected Sets

  4. timlyg says:

    02/17/2016
    HW for Mon: 3.1 #4,5

  5. timlyg says:

    02/24/2016
    3.3 Nested Sets
    Def. a seq. of sets is nested iff .

    Theorem (3.3.1) If F_k is a nested seq. of compact, non-empty, sets in a metric sp(M,d), then

    Pf: Method 1, p.157, method 2, p.169

    Theorem: Let C be a collection of closed sets in with the finite prop. s.t. an element of C which is bdd, .
    Pf: let C = {}, closed s.t. bdd.
    &
    so
    s.t.. Suppose not, ie. .
    So no pt. of is in each
    So = open cover of
    ie.

    Since s.t.
    is open and contains y.
    closed & bdd => compact => finite sub-cover
    but then .
    # C has finite prop.

    3.5 Connected Sets (some lectures found on youtube)
    When is a set in "one piece"?
    Def: A set S in a space X is connected IFF whenever , then or .
    Two open sets U, V in X are said to separate S IFF
    i).
    ii).
    iii).
    iv).

    If such U & V, then we call S disconnected.
    So, 1st definition is negation of disconnected.

    Pf: . Then
    So, , open
    , open
    And


    and
    and




    => exercise where you consider & , both closed.

  6. timlyg says:

    03/09/2016
    Quiz #1 on chapter 1, 2, 3 on Wed 3/16 at start of class.
    Quiz #1. hi: 49.5, low: 30, mean: 42.8, median: 45.5

  7. timlyg says:

    05/02/2016
    Catch up:
    Jacobian
    LaGrange
    Lagrange Multiplier (not covered, but read for Math major's sake) also in multivar calc. textbook.

  8. timlyg says:

    05/09/2016
    last day Wed 5/18, Quiz 2:
    No proofs for 7.1, 7.2.

    Final: NAC 4/115, 3:30pm - 5:45pm Monday 5/23

  9. timlyg says:

    05/18/2016
    Final:
    Expect basic proofs.
    Office Hours Th 5/19, 2-4pm, NAC 6/202B
    Review Class NAC 1/511E
    (Artino Classroom) 6pm ->? Pizza

    Final Exam Mon 5/23, 3:30PM _ 5:45PM in NAC 4/115

    Theorem list for knowing proofs from "midterm".

    Definition-type proofs (proofs by def)

Leave a Reply

Your email address will not be published. Required fields are marked *

*

This site uses Akismet to reduce spam. Learn how your comment data is processed.