Flux across a cylinder Let S be the cylinder x 2 + y 2 = a 2 , for –L ≤ z ≤ L. a. Find the outward flux of the field F = 〈 x , y, 0〉 across S . b. Find the outward flux of the field F = 〈 x , y , 0 〉 ( x 2 + y 2 ) p / 2 = r | r | p across S. where | r| is the distance from the z -axis and p is a real number. c. In part (b), for what values of p is the outward flux finite as a → ∞ (with L fixed)? d. In part (b), for what values of p is the outward flux finite as L → ∞ (with a fixed)?
Flux across a cylinder Let S be the cylinder x 2 + y 2 = a 2 , for –L ≤ z ≤ L. a. Find the outward flux of the field F = 〈 x , y, 0〉 across S . b. Find the outward flux of the field F = 〈 x , y , 0 〉 ( x 2 + y 2 ) p / 2 = r | r | p across S. where | r| is the distance from the z -axis and p is a real number. c. In part (b), for what values of p is the outward flux finite as a → ∞ (with L fixed)? d. In part (b), for what values of p is the outward flux finite as L → ∞ (with a fixed)?
Flux across a cylinder Let S be the cylinder x2 + y2 = a2, for –L ≤ z ≤ L.
a. Find the outward flux of the field F = 〈x, y, 0〉 across S.
b. Find the outward flux of the field
F
=
〈
x
,
y
,
0
〉
(
x
2
+
y
2
)
p
/
2
=
r
|
r
|
p
across S. where |r| is the distance from the z-axis and p is a real number.
c. In part (b), for what values of p is the outward flux finite as a → ∞ (with L fixed)?
d. In part (b), for what values of p is the outward flux finite as L → ∞ (with a fixed)?
The OU process studied in the previous problem is a common model for interest rates.
Another common model is the CIR model, which solves the SDE:
dX₁ = (a = X₁) dt + σ √X+dWt,
-
under the condition Xoxo. We cannot solve this SDE explicitly.
=
(a) Use the Brownian trajectory simulated in part (a) of Problem 1, and the Euler
scheme to simulate a trajectory of the CIR process. On a graph, represent both the
trajectory of the OU process and the trajectory of the CIR process for the same
Brownian path.
(b) Repeat the simulation of the CIR process above M times (M large), for a large
value of T, and use the result to estimate the long-term expectation and variance
of the CIR process. How do they compare to the ones of the OU process?
Numerical application: T = 10, N = 500, a = 0.04, x0 = 0.05, σ = 0.01, M = 1000.
1
(c) If you use larger values than above for the parameters, such as the ones in Problem
1, you may encounter errors when implementing the Euler scheme for CIR. Explain
why.
#8 (a) Find the equation of the tangent line to y = √x+3 at x=6
(b) Find the differential dy at y = √x +3 and evaluate it for x=6 and dx = 0.3
Q.2 Q.4 Determine ffx dA where R is upper half of the circle shown below.
x²+y2=1
(1,0)
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