MATH 634: Advanced Calculus II

Journal

Practice Final:
p 366 B 9
p 376, A 2
p 403, A 6 (justify all your steps!)
p 404, B 7
p 426, A 1
p 442, C 2

HW 13 (the last one!!!! Due 5/5, 6pm):

Page 415, A 7 b), B 1, 8, C 2.
Page 426, A1 a), d), 2, B 3, C 1, 2, 3 (you cabn deduce them all from C 3 a) ).
Page 439, A 1 a), f), 2, 5 e), 7 c), B 3, 4.

HW 12 (due 4/28, at 6pm):

Page 393, A 1 d), 3, B 4 (for thm 11.1.g)), C 5 (for thms 11.2.a and 11.2.d).
Page 402, A 1 a), 3 b), B 2, 6, 7, C 1.
A 6 a), d); B 4, 7, C 2.
Page 415, A 2 a), 3, 6, B 2, 6, C 2 .

HW 11 (due 4/19, at 6 pm):

Page 364, B9 and C3.
Page 376, A 6, B 2, 5, C 1, 2.
Page 392, B 1 a), B 2, B 4 (Thm 11.1e only for now).

HW 10 (due 4/12, at 6 pm):

Page 364, A 5, 8, B 4, 6, C 1, 2.
Page 376, A 1 f), 3, 9, B 1, 3, 4, C 3 (this one I find cool :) ).

HW 9 (due 4/5, at 6 pm):

Part a: the midterm. 4 copies are out... Jeff, Ron, ask someone to let you copy theirs...
Part b: Page 364, A 1 (how about a linear approximation?), 2, 3.
A 6, B 2, 10.
B 3, 9, 11, C 3.

HW 8 (due 3/15, at 6pm):

Page 348, A1 d), g), i); 2 (for g), 3, 5, 7, 8, 9.

There may be more - it depends on how much we cover by Monday.

HW 7 (due 3/10, at 6pm):

Page 327, A1 d), e); A5, B5, C4. Page 338, A1 e), B2.
Page 337, A1 h), 2 b), 4, 5, B 1, 2, 3, C 1.

Practice Test:
p 328 B4
p 306 A2 / B2-4
p 288 B7
p 327 A4
p 276 B2
p 337 A1a)

HW 6 (due 3/1, at 6pm):

Page 305, A 2 a), e), d); 4. B 1, 2, 3. C 1.
A 3, B 4, 7.
Page 314, A 3, B 1, 2.

HW 5 (due 2/22, at 6pm):

Page 287, A 2, 3. B 1, 2, 3, 5 (this is a tool to later prove the change of variables formula; I seem to remember there's
a pretty worked out proof in Strichartz' "The Way of Analysis" as part of the proof of the change of variables formula - you
may consider looking it up), 7, C 1 ( we did an outline of this in class: to show that every linear transformation is continuous,
or continuous at zero, which is enough to make it continuous - show this in detail).
Page 298, A 2 a), 4 a), 5.

HW 4 (due 2/15, at 6pm):

Page 259, B 3, 7, 10, C 3, 4, 5, 6.
Page 266, A 1 a) and f) (Ban your calculators. Now draw the intersections of the graph with panes parallel to the coordinates axes, then sketch a
graph of the function), 3 a), g), e), i
Page 275, A 1 b), d), 4, 5.

HW 3 (due 2/8, at 6pm):

Page 245, A 5, B 4, C 1, 2, 4.
(Hint for C 4: Then show that F(ns)=nF(s) (by induction); then that (F(1/n s)=1/n F(s) - using that n/n=1 and factors n now
pull out; then use continuity and the fact that rational factors pull out to show that real factors pull out). You are actually proving
that any additive functional (= a function taking values in the field of scalars) in vector space over the real numbers is linear.

Page 259, A 2, B 1 (for A 1 a) ), B 2 b), B 7, C 1.

HW 2 (due 2/1, at 4pm):

Page 227, A 4, 5, 7 (write the *parametric* equation...), 8, 9.
C 1.
Page 245, A 1 c), 2, 3, 4, B 1, 3, 5.
(Hint: use b) from HW 1 for B 1, page 245).

HW 1 (due 1/25, at 5 pm):

a) Page 205: A 1, B1, 2.

b) Consider the vector space R^3 over the field of real numbers, and for x=(x_1,x_2,x_3) in R^3 , the functions
||x||_e (the Euclidean norm), ||x||_1= |x_1| + |x_2|+|x_3| (the discrete 1-norm), and the discrete infinity norm
given by max {|x_1|, |x_2|,|x_3|}.
i) Draw the unit balls ( {x:||x||<1}) for each of these norms.
ii) Show that the three norms are equivalent.
iii) "Draw a geometric interpretation" of ii).
iv) Show that ||x||_?=|x_1| does not define a norm on R^3.

c) Let the field of scalars be R, and the vector space be the continuous functions on the interval [0,1], C([0,1]).
i) Show that both || f||= sup { |f(x)|, x in [0,1]} and ||f||_1=integral over [0,1] of |f|, define norms on C([0,1]).
ii) Show that there is a positive real number k such that for all f in C([0,1]),
||f||_1 is less or equal that k ||f||.
iii) How does ii) relate to the statement "a sequence of continuous functions converging uniformly over a closed
interval has converging integrals (to the integral of the limit)?
iv) Show that the two norms are not eqivalent (by exhibiting a unbounded sequence of continuous functions with bounded
integrals). (Adjust some examples you know from working with uniform convergence).
v) Interpret ii) and iv) in terms of "for which of the norms balls are bigger" (in the sense of having more elements), and for which
norm there are more sequences x_n with norms ||x_n|| tending to zero.

d) Page 218, A 6, 8, C 1, 2, 3.

e) Page 217, A 3, B 3, C 4.