Math 404 Quiz 5.
Let f (x) = x on [-, ]. a) Compute Fourier coefficients of f 1 2
c0 = if n = 0 we have cn = 1 2
xdx = 0,
xe-inx dx =
1 e-inx x 2 -in
(-1)n e-inx dx = -in -in
b) Apply Parseval's theorem to conclude that 2 1 = 2
Math 404 Quiz 8.
Let F : (x, y) (sin(x + y), ex-y - 1). represent F as a composition of two primitive mappings in a neighborhood of (0, 0).
Take G1 : (x, y) (x, ex-y - 1) if u = x, v = ex-y - 1 then x = u, y = u - ln(v + 1). Therefore for G2 :
Math 404 Quiz 1.
If "yes" prove it, if "no" give a counterexample: A continuous function f : R R has the following properties: f (0) = 0 and f (x) < 0 if |x| > 1
(i) Is it true that f has to attain its maximum?
(ii) Is it true that f has to att
Math 404, Midterm 3.
5. Let F : R2 R2 be a C 1 -mapping and F (0, 0) = I (here I denotes the
identity mapping). Prove that there is a neighborhood U of (0, 0), such that
restriction of F on U is a composition of two primitive mappings.
F(x, y ) = (z (
Each problem costs 16 points. In the first two problems you will need to prove a theorem (or its part) from the list (the choice is mine): 1. 8.11, 8.12, 8.14, Fejr's theorem(prob. 15), 8.16. e 2. 9.2, 9.5, 9.7, 9.8, 9.12, 9.15, 9.17, 9.21, 9.2
Math 404 Quiz 2.
Let be a monotonically increasing function on [a, b]. Prove that R() if and only if is continuous
IF: Assume is not continuous at y [a, b] and yi y such that |(yi )-(y)| > 0 for any i. R(), therefore > 0 partition P such that U
Math 404 Quiz 4.
inx . - cn e
Let f be a smooth 2-periodic function. Assume f that
in cn einx .
Sinse f is smooth, f R on [-, ]. Let f n = 1 -inx fe 2
n einx . Then
1 2 1 2
f e-inx dx =
f (-ine-inx )dx = 0 + incn .
Math 404 Quiz 7. Your name:
Consider mapping R2 R2 , f : (x, y) (u, v) defined by u = x2 - y 2 , v = 2xy Find [f (x, y)] in standard basis. 2x -2y Find Jacobian Jf (x, y). 2x -2y 2y = 4(x2 + y 2 ) 2x 2y 2x
Jf (x, y) = det
Conclude that f is one-to-one in
Math 404 Quiz 6.
Let A be a linear transformation from Rn to Rm and cfw_ai j is its matrix in the standard bases. Prove that A |ai j |.
|ai j |xj |
|ai j | |x|
|ai j |.
Math 404, Midterm 1. Your name: 1. Function f is monotonic on [a, b], and is continuous on [a, b]. Prove that f R(). 2. Function f R on [a, b] and there is a differentiable function F on [a, b] such that F = f . Prove that
f (x)dx = F (b) - F (a).