Math 227 Winter 2011
Homework 3
1. In the previous homeworks you showed that a harmonic function satises the mean value
property. This means that if u = 0 for all x D and if Br (x0 ) D,
u(x0 ) =
1
|Br (x0 )|
u(x) dS(x)
(0.1)
Br (x0 )
where Br (x0 ) = cfw_
Math 227 Queens University, Department of Mathematics
Vector Analysis, Homework 1-Solutions
September 2013
1. Question number 8, section 15.1 Stewart.
The midpoint rule we will use consists of taking the average of the minimum=m
and the maximum= M value o
Math 227 Winter 2011
Homework 4
1. A digital option is one with payo
g(x) =
0, x < K,
b xK
(0.1)
at the expiration time T . By transforming the Black-Scholes equation to the heat equation
and solving the resulting initial value problem explicitly, nd the
Homework 1
Notation: given 1 p < + we say that f Lp (Rn ) if the integral
denote f
set g
p
=
Rn
|f (x)|p dx
1/p
Rn
|f (x)|p dx < + and
. For p = + we do as follows: for a bounded function g(x) we
= supxRn |g(x)|. For a function u(x, t) we denote u(t)
p
=
Homework 2
1. Use the fact that sj+1 Bsj+1 sj Bsj = sj (Bsj+1 Bsj ) + (sj+1 sj )Bj+1 to show directly that
t
t
s dBs () = tBt
Bs ds
0
(0.1)
0
Would the same formula hold if Bt were actually dierentiable and the rst integral were
given by a Riemann-Stielj
Math 227 Winter 2011
Final Homework
1. Suppose that the function u(x, t; K, T ) solves the Black-Scholes equation (in (x, t) with terminal data u(x, T ; K, T ) = (x K)2 . Derive a PDE (and either the terminal or initial
+
condition) for the function v(K,
Math 227 Queens University, Department of Mathematics
Vector Analysis, Homework 2
September 2013
1. Evaluate the double integral
y = 0, x = 1 and the curve x =
D
x cos(y)dA where D is bounded by the lines
y. Evaluate this integral in two dierent ways by
r
Math 227 Queens University, Department of Mathematics
Vector Analysis, Homeworkz 3-solutions
September 2013
1. Evaluate the triple integral
W
dV
x2 +y 2 +z 2
where W is the solid region between
the upper hemispheres of two concentric spheres of radii a <
Math 227 Queens University, Department of Mathematics
Vector Analysis, Homeworkz 4-Solutions
October 2013
1. Consider the spiral curve which in polar coordinates is given by the equation r = .
a) Find a vector parameterization for this curve in rectanglua
Math 227 Queens University, Department of Mathematics
Vector Analysis, Homework 2-solutions
September 2013
1. Evaluate the double integral
y = 0, x = 1 and the curve x =
D
x cos(y)dA where D is bounded by the lines
y. Evaluate this integral in two dierent