MATH 405: Midterm 2
Friday, November 17, 2006
NAME:
D LMT/om g
This exam consists of 5 problems, with points as labelled. Please write each of your answers
clearly and legibly. Dont forget to explain yourself, and show all of your work!
There are no c
MATH 405: Problem Set 7 Additional Problems due Monday 11/6/2006
1. In Kreyszig Section 5.8 the expansion of a function f in terms of a Legendre series f (x) = an Pn (x) is discussed, where it is given that: an = 2n + 1 2
1
f (x)Pn (x)dx
-1
Deriv
MATH 405: Sample Questions (Material Since 2nd Midterm)
1. Evaluate the line integral C B dr along the curve C defined by y = 2x3 between x = 0 and 4, for the vector field B = (yx, x). 2. a) Given the functions F (x, y) and G(x, y) which are bounde
MATH 405: Quiz 10 Name: December 11, 2006
1. Given the vector field E = (2y, 9x, 18z), evaluate the following surface integral (pls show all of your work): ^ E n dA
B
where B is the surface of the sphere defined by x2 + y 2 + z 2 = 1.
2. Solve th
MATH 405: Problem Set 5 Additional Problems due Monday 10/23/2006
1. Consider Laplace's equation:
2
=0
in 2D Cartesian coordinates, i.e. = (x, y). Assume a separation of variables form (x, y) = f (x)g(y) and find the ODEs for f and g. Solve these
MATH 405: Quiz 6 Name: October 30, 2006
1. Write out the general solution (in terms of functions discussed in the text or in class) to the following equation for f (x): x2 f + xf + (x2 - 3)f = 0 You do not need to derive this solution, but discuss o
MATH 405: Quiz 8 Name: November 13, 2006
1. Identify each of the following functions as even, odd, or neither: a) xex b) sin2 (x) c) cos(x3 ) d) x(x2 - 1)
2. Find the general solution to the following ODE for y(t) y + 4y = r(t) given that the drivi
MATH 405: Quiz 2 Name: September 22, 2006
1. Given the matrix:
1 0 A= 1
7 21 3
8 7 49
decompose it as A = S + T , where S is a symmetric matrix and T is skew-symmetric.
2. Find a general solution y1 (t) and y2 (t) for the following differe
MATH 405: Matrices The Six Theorems of Sect.7.4 September 25, 2006 AB
1. Row reducing a matrix does not change its rank (does not really change the matrix)
2. To decide if p given vectors (each with n components) are linearly independent, use them
MATH 405: Sample Midterm Questions
1. Show that the following two functions are linearly independent (hint: use the Wronskian) x1.3 x1.3 log x 2. Find the homogeneous and particular solutions for the following equation for y(t) y + 2y + 31y = e-t si
MATH 405: Quiz 5 Name: October 23, 2006
1. Determine the radius of convergence of the following series, showing the details of your work: 3(n + 3)2 n x (n - 3)4 n=4
2. Solve the following ODE for y(x) xy - y = 0 by going only as far as finding the
MATH 405: Quiz 7 Name: November 6, 2006
1 1. Given that the first 4 Legendre polynomials are P0 = 1, P1 = x, P2 = 2 (3x2 - 1), and 1 3 2 P3 = 2 (5x - 3x), represent the function f (x) = 2x as a Legendre expansion.
2. Consider a Sturm-Liouville equa
MATH 405: Quiz 9 Name: December 4, 2006
1. Show that each of the following integrals is or is not path independent: a)
C
y 2 cos(xy 2 ) dx + 2x cos(xy 2 ) dy
b)
C
F dr
with F = (6x2 y, 2x3 )
2. Given the vector field F = (62y, -31x), evaluate
MATH 405: Quiz 3 Name: September 29, 2006
1. Given the matrix:
1 0 A= -3
-2 0 6
find the rank of A, define its column space, and give a basis for the column space.
2. Consider B to be a real 3 3 matrix. Show that the eigenvalues of B are e
MATH 405: Sample 2nd Midterm Questions
1. Calculate the radius of convergence for the following functions
n=1
n3 xn (n + 1)! (6)/(5)
(13m)! m x (m!)4 m=1
2. Evaluate the following where is the Gamma function. 3. Find a power series solution f
MATH 405: Quiz 4 Name: October 9, 2006
1. Find the spectrum for the matrix: A= 0 b -12 0
where b is a constant. What value of b (if any) would make A skew-symmetric? SkewHermitian? And what happens in general to the spectrum for that value of b?
2
MATH 405: Problem Set 11 due Friday 12/15/2006
1. Using Faraday's Law of Induction E = B/t between the electric field E and the magnetic field B, find the line integral of the electric field E dr
C
around a closed current loop C with area A due t
MATH 405: Quiz 1 Name: September 15, 2006
1. The differential equation for the current I(t) in a circuit involving an inductor L, a resistance R, and a capacitance C is: LI + RI + 1 I=0 C
Derive the condition on L, R, and C such that the current wi