First assignment. Due Friday, August 29.
1.2.1, 1.2.4, 1.2.5, 1.2.10,
1.3.2, 1.3.8, 1.3.15, 1.3.18.
Second assignment. Due Monday, September 8.
1. Use Note 2 (see below) for this problem
Find a polynom of the degree n
P_n(x)=a0 + a1 * x + a2 * x^2 + ... an * x^n
that either passes through the four points minimizing the distances to the points
(0, 0), (1, 8), (2, 8), (3, 20)
or passes as close to these points as possible.
Consider the cases n=2, n=3, n=4 and perform the least square approximation.Achieve the additional goad: minimize the distances to the points (as in the first part) plus the sum
N2=eps * ( a_1^2 + 2 * a2^2 + 3* a3^2 +.. )
where eps is a small positive number. Take eps=1, 0.1, 001. Use Maple (or Mathematica, or MatLab) to draw the pictures of the polynoms.2. Consider a harmonic oscillator: the mass M attached to a linear spring of the length L which compliance is proportional to its length: C* L. Perform two experiments:
(1) Elongate the spring on the distance D, release, and observe harmonic oscillations of the mass.
(2) Elongate a half of the spring by the same distance D, release, and observe harmonic oscillations of the mass.
Show that the second elongation requires twice more energy.
Where is this energy?3. Strang:
1.4.2, 1.4.5, 1.4.10. 1.4.15, 1.5.3.
Third assignment. Due Monday, Sept. 15. Read Section 1.6
1.5.3, 1.5.4, 1.5.6, 1.5.9, 1.5.20.
2.1.3, 2.1.6
Fourth assignment. Due Monday, Sept. 22.
Fifth assignment. Due Wednesday, Oct. 01. Strang: 2.3.3, 2.3.12, 2.4.4, 2.4.7, 2.4.8
An extra-credit challenge! Due Wednesday, Oct. 01.
Crossing a plain, tourists have lost their way in a mist. Suddenly, they find a message at a fallen pole that reads: "A straight road is within a mile from that pole." The direction to the road is lost because the pole is fallen. The tourists need to find the road exploring the plain around. They are shortsighted in the mist: They can see the road only when they step on it.
What is the shortest way to the road even if the road is most unfortunately located?
Sixth assignment. Due Monday, Oct. 13.
Strang: 3.1.4, 3.1.17, 3.3.1, 3.2.6, 3.3.4.
Strang: Read sections 3.1-3.3
Seventh assignment. Due Monday, Oct. 28.
1. The speed $v$ varies with the hight as v= exp( a y). Find the best trajectory of fastest descent from the point (1, 0) to the point (0, 1).2a. Derive Lagrangian and the equation for a two-link pendulum with the sequential links of the length $l_1$ and $l_2$ able to turn around two mutually perpendicular axes one is OX-axis, the other becomes OY at the equilibrium. A mass $m$ is attacted to the end of the second link. The gravitational force is applied along the axis OZ.
Hint. Show that the mass moves around the surface of a torus. You may use toroidal coordinates
x = a*sinh(v)*cos(w)/d
y = a*sinh(v)*sin(w)/d
z = a*sin(u)/d
where
d = cosh(v) - cos(u)
to express kinetic and potential energy. 2b. Assuming that the magnitude of vibration is small, investigate the linearized equations. Find the frequencies of oscillations.3. Find the best approximation to the function f(x)= |x|, -1 < x < 1, with penalization for the integral of the square of derivative of the approximation. Use Green's function.